Obsidian/Zotero/001_artiklid/simkuscfmiflowmatchingmissing2025.md

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category: literaturenote citekey: simkuscfmiflowmatchingmissing2025 title: "CFMI: Flow Matching for Missing Data Imputation" authors: "Simkus, Vaidotas; Gutmann, Michael U." year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.09258v1" zotero_key: 93BM6BBP zotero_storage: 9B94749N collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Simkus, Vaidotas"

status: converted

CFMI: Flow Matching for Missing Data Imputation

Vaidotas Simkus vaidotas.simkus@ed.ac.uk Michael U. Gutmann michael.gutmann@ed.ac.uk School of Informatics University of Edinburgh

Abstract

We introduce conditional flow matching for imputation (CFMI), a new general-purpose method to impute missing data. The method combines continuous normalising flows, flowmatching, and shared conditional modelling to deal with intractabilities of traditional multiple imputation. Our comparison with nine classical and state-of-the-art imputation methods on 24 small to moderate-dimensional tabular data sets shows that CFMI matches or outperforms both traditional and modern techniques across a wide range of metrics. Applying the method to zero-shot imputation of time-series data, we find that it matches the accuracy of a related diffusion-based method while outperforming it in terms of computational efficiency. Overall, CFMI performs at least as well as traditional methods on lower-dimensional data while remaining scalable to high-dimensional settings, matching or exceeding the performance of other deep learning-based approaches, making it a go-to imputation method for a wide range of data types and dimensionalities.

1 Introduction

Missing data is a persistent challenge that hinders the full potential of machine learning in critical domains such as healthcare and the sciences. Data in these domains are often incomplete due to issues like participant dropout in longitudinal studies or the malfunction of measurement devices. Addressing this issue is essential to enable robust analyses and downstream applications.

A fundamental statistical procedure to address this issue is multiple imputation (Rubin, 1987; Little & Rubin, 2020), which aims to sample (or impute) the missing data with draws from the corresponding conditional data distribution. While modern, deep learning-based imputation methods (e.g. Yoon et al., 2018; Mattei & Frellsen, 2019; Miao et al., 2021; Tashiro et al., 2021) excel in high-dimensional settings, they frequently underperform compared to traditional imputation techniques (e.g. van Buuren & Oudshoorn, 2000; Stekhoven & Bühlmann, 2012) when applied to small- or moderate-dimensional data. This highlights the need to look for a more generalisable approach that can deliver robust performance across diverse data types and dimensionalities.

Recently, advances in diffusion (Sohl-Dickstein et al., 2015; Ho et al., 2020) and continuous normalising flow (CNF, Chen et al., 2019; Lipman et al., 2023) literature has demonstrated remarkable flexibility, robustness, and generalisability across various domains. Building on these developments, we propose an imputation method based on CNFs trained via flow-matching (Lipman et al., 2023; Tong et al., 2023). To facilitate efficient learning and imputation, the model is specified conditionally, enabling all required conditional distributions for imputation to be captured by a single shared model, drawing on insights from prior work (Ivanov et al., 2019; Li et al., 2020; Tashiro et al., 2021; Strauss & Oliva, 2021). Our approach, CFMI (fig. 1), combines the strengths of CNFs, flow-matching, and shared conditional modelling to achieve imputation performance on par with traditional methods for lower-dimensional data while maintaining scalability to highdimensional settings, matching and outperforming the capabilities of other deep learning-based methods.

We summarise our main contributions as follows:

Figure 1: Imputation using CFMI on synthetic 2D data. The left-most column shows the kernel density estimate (KDE) of the ground truth distribution (top) and the KDE of the imputed data distribution (bottom). The red-tinted rows and columns in top-left illustrate that some data-points may be missing one or both (not visualized) of the dimensions, and thus imputation requires sampling the corresponding (conditional) data distribution. CFMI learns a model of all conditional distributions that are needed for imputation (illustrated in the grey box on the right, corresponding to the red-tinted rows and columns). This model is then used to impute the data by sampling the appropriate conditional distribution for each incomplete data point. A sample of imputations is shown as red dots in the bottom-left, and the KDE of the imputed data closely matches that of the ground truth distribution. Additional synthetic-data examples are in fig. 4 of the appendix.

• In section 3, we propose conditional flow matching for imputation (CFMI), a conditional CNF model trained via flow matching for missing data imputation.
• In section 4.1, we evaluate the ability of the CNF model to capture all necessary conditional distributions for missing data imputation on a synthetic data set, where ground truth conditionals can be computed.
• In section 4.2, we thoroughly evaluate the method's imputation effectiveness on 24 small-to-moderate dimensional tabular UCI data sets, matching or outperforming the performance of traditional and modern techniques across a wide range of metrics.
• In section 4.3, we also evaluate the method's performance for zero-shot imputation of held-out time-series data, matching the accuracy of a related state-of-the-art diffusion-based method while outperforming it in terms of computational efficiency.

These contributions represent a step toward a general-purpose imputation framework that provides robust, accurate, and scalable performance across a wide variety of data types and dimensionalities.

2 Background

2.1 Missing data imputation

Let x ^o and x ^m be the observed and missing parts of the i-th data-point x (i) ∈ X ^D drawn from an unknown distribution p ∗ (x). The missingness pattern for x is indicated by a binary mask m(i) ∈ {0, 1} ^D, which follows a generally unknown missingness mechanism p ∗ (m | x ). Formally, the observed and missing data are thus defined as

$$\boldsymbol{x}{o}^{(i)} \stackrel{\text{def}}{=} \boldsymbol{x}^{(i)}[\boldsymbol{m}^{(i)}] \quad \text{and} \quad \boldsymbol{x}{m}^{(i)} \stackrel{\text{def}}{=} \boldsymbol{x}^{(i)}[1 - \boldsymbol{m}^{(i)}],$$ (1)

where there are up to 2 ^D possible missingness patterns, and hence combinations of x^o and *x*m. To simplify the notation, we omit the sample index i in the remainder of this discussion.

Using the common assumption that the data are missing-at-random (MAR, e.g. Little & Rubin, 2020, Section 1.3), where the probability of missingness depends only on the observed values and not on the missing values, we would like to sample p ∗ (x^m | x ^o ), such that the imputations follow the true distribution of the missing variables (e.g. van Buuren, 2018, Section 2.2.6). However, as the ground truth distribution p ∗ (x) is generally unknown approximations are needed.

Given an incomplete data set {x ^o ,m(i)} N i=1, one way to obtain imputations is to approximate the unknown joint distribution p ∗ (x) with a probabilistic model pθ(x) (Dempster et al., 1977), and then perform probabilistic inference to sample from pθ(x^m | *x*o). However, estimating modern jointly-specified models pθ(x) directly from incomplete data is generally challenging (Simkus et al., 2023). Moreover, performing probabilistic inference to impute the incomplete data-points can be computationally demanding.

2.2 Missing data imputation with conditional models

A practical alternative to learning a joint model pθ(x) is to model the conditional distributions pθ(x^m | *x*o) directly. However, since the number of missingness patterns grows exponentially with the dimensionality D (i.e., up to 2 ^D patterns), learning a separate model for each pattern is computationally infeasible.

To address similar challenges, several works have proposed modelling all conditional distributions using a single model (Douglas et al., 2017; Ivanov et al., 2019; Li et al., 2020; Tashiro et al., 2021; Strauss & Oliva, 2021). Such models can be interpreted as learning a set of conditional distributions, {pθ(x^m | *x*o) | x^o = x[m], x^m = x[1 − m], ∀m ∈ {0, 1} ^D}, with shared parameters θ across all missingness patterns. This conditional modelling approach enables efficient imputation by directly sampling from the appropriate conditional distribution for any given missingness pattern.

When fully-observed training data are available, arbitrarily-conditional models are typically trained by randomly splitting each data-point x into target x^t and conditioning x^c parts according to some heuristic splitting strategy (see e.g. Tashiro et al., 2021, Section 4.3). The training objective is then to approximate the true conditional distribution, such that pθ(x^t | *x*c) ≈ p ∗ (x^t | *x*c) for any x^t and *x*c. When the training data are incomplete, the target and conditioning parts are similarly obtained by partitioning the observed parts x^o of the incomplete data-points.

An important caveat to note is that when the models are trained on incomplete data, these imputation methods rely on the (often implicit) assumption that a model trained to approximate pθ(x^t | *x*c), where (*xt, x*c) = *x*o, can generalise to the conditional distribution of missing variables pθ(x^m | *x*o), where (*xo, x*m) = x. Nevertheless, the conditional modelling approach remains appealing due to its simplicity, scalability, and empirical success.

2.3 Continuous normalising flows via flow matching

A continuous normalising flow (CNF, Chen et al., 2019) is a time-dependent probabilistic model p θ (x), also called a probability path, where t ∈ [0, 1] is a continuous time index. At t = 0, p θ (x) is a simple base distribution, while at t = 1, p θ (x) approximates the target distribution. The probability path is generated by a time-dependent vector field vθ(·;t) : R ^D → R ^D, which yields an invertible flow ϕθ(·;t) : R ^D → R D:

$$\phi{\theta}(\mathbf{x}^{0};t) = \mathbf{x}^{0} + \int{0}^{t} v_{\theta}(\mathbf{x}^{\tau};\tau) d\tau \stackrel{\text{def}}{=} \mathbf{x}^{t},$$ (2)

where ϕθ(x 0 ;t) maps a sample x from the base distribution p 0 θ (x) to a sample x ^t ∼ p t θ (x) along the probability path. The target distribution p θ (x) is determined using the change-of-variables formula (e.g. Murphy, 2021, Section 2.8.3):

$$p{\theta}^{1}(\boldsymbol{x}) = p{\theta}^{0}(\phi{\theta}^{-1}(\boldsymbol{x};1)) \det \left[ \frac{\partial \phi{\theta}^{-1}(\boldsymbol{x};1)}{\partial \boldsymbol{x}} \right], \tag{3}$$

where ϕ -1 θ (·; 1) is the inverse of the flow at t = 1. Calculating this density involves solving for the inverse of the integral in eq. (2) and computing the determinant of the Jacobian, both of which can be computationally demanding. As a result, maximum-likelihood estimation (MLE) for CNFs may be resource-intensive and potentially unstable.

Recently, inspired by the connection between CNFs and diffusion models (Sohl-Dickstein et al., 2015; Song et al., 2022), Lipman et al. (2023) proposed an alternative method for fitting CNFs that avoids computationally expensive simulation. This method, called flow matching, constructs a fixed target probability path q t (x) along with a corresponding (conditional) target vector field u(x | x 1 ;t). The CNF is then fitted by minimising the following conditional flow matching objective:

$$\mathcal{L}{CFM}(\boldsymbol{\theta}) = \mathbb{E}{q(t)q(\boldsymbol{x})q^{t}(\boldsymbol{x}^{t}|\boldsymbol{x})} \left| v{\boldsymbol{\theta}}(\boldsymbol{x}^{t};t) - u(\boldsymbol{x}^{t} \mid \boldsymbol{x};t) \right|{2}^{2}, \tag{4}$$

where q(t) is a distribution over the time indices, q(x) is the empirical data distribution formed by samples from the ground truth p ∗ (x), and q t (x t | x) is the conditional probability path such that q t (x t ) = Eq(x) [q t (x t | x)]. The flow matching objective, unlike MLE, does not need computing the density in eq. (3), making the training process significantly more scalable and stable.

Yet, using the joint model p θ (x) for missing data imputation is challenging since fitting a joint CNF model from incomplete data is difficult (e.g. Simkus et al., 2023) and conditional sampling of the density in eq. (3) is computationally demanding. Instead, as discussed in section 2.2, we will use a conditional modelling approach.

3 Conditional normalising flows for missing data imputation via flow matching

We propose conditional flow matching for imputation (CFMI), a conditional continuous normalising flow (CNF) model trained via flow matching for missing data imputation. Our approach leverages flow matching to enable both fast training and accurate model fitting. Building on the motivation discussed in section 2.2, CFMI learns a shared model that captures all conditional distributions needed for imputation, making it particularly effective for high-dimensional data sets. Through extensive evaluation on both synthetic, real-world tabular, and time-series data (see section 4), we demonstrate that CFMI achieves substantial improvements in both computational efficiency and imputation accuracy compared to existing methods.

3.1 Conditional CNF model for missing data imputation

Continuous normalising flows, like diffusion models, are well-known for their highly-flexible modelling capabilities despite being conceptually relatively simple. In general, a CNF operates by taking as input a D-dimensional sample x^t from the learnt probability path p t θ (x) (or the target path q t (x) during training), along with (an encoding of) the time index t, and predicts a D-dimensional vector field vθ(xt;t). The vectorfield can be parametrised using virtually any function approximator, such as neural networks (Lipman et al., 2023) or random forests (Jolicoeur-Martineau et al., 2024). This allows CNFs to adapt to a wide range of tasks while maintaining simplicity and flexibility in their architectural choices. When using a neural network for parametrisation, the CNF framework further enables the modelling of multiple conditional distributions within a single model.

For conditional modelling, the CNF model takes additional inputs, namely the observed conditioning variables *x*o, alongside the sample from the probability path and the time index. To accommodate arbitrary configurations of missing (target) and observed (conditioning) variables, the model maintains a fixed output dimensionality of D. Similarly, the dimensionalities of the probability-path sample and the conditioning variables are also padded to D dimensions before being passed as inputs to the model. Specifically, let x^o and x^m represent the observed and missing variables, respectively, and m denote the missingness pattern. Then, *^x*f^o and *^x*f^m are zero-padded versions of ^xo and ^xm such that the padded vectors have ^D dimensions and the indices are respected, that is, *^x*f^o is zero where ^m is zero, and *^x*f^m is zero where ^{1 − m} is zero. The conditional density ^pθ(x^{m |} *^x*o) of the CNF can then be expressed via the vector field ^vθ(*xf^τ* ^m; *^xf^o, τ* )[1 ^{− m}]:

$$p{\theta}(\mathbf{x}{\mathrm{m}} \mid \mathbf{x}{\mathrm{o}}) = p{\theta}^{0}(\phi{\theta}^{-1}(\mathbf{x}{\mathrm{m}}; \mathbf{x}{\mathrm{o}}, 1)) \det \left[ \frac{\partial \phi{\theta}^{-1}(\mathbf{x}{\mathrm{m}}; \mathbf{x}{\mathrm{o}}, 1)}{\partial \mathbf{x}_{\mathrm{m}}} \right]$$ (5)

with

$$\phi{\theta}(\boldsymbol{x}{m}^{0};\boldsymbol{x}{o},t) = \boldsymbol{x}{m}^{0} + \int{0}^{t} v{\theta}(\widetilde{\boldsymbol{x}{m}^{\tau}};\widetilde{\boldsymbol{x}{o}},\tau)[1-\boldsymbol{m}] d\tau$$ (6)

where p θ (·) is an unconditional base distribution, [1 − m] denotes indexing the outputs of the model using the mask as in eq. (1) and *^xf^τ* ^m is a zero-padded sample from the (conditional) probability path p τ θ (x τ ^m | *x*o). This specification ensures that the same model can seamlessly handle any missingness pattern m.

While the vector field can be parametrised using any function approximator, the choice of a network architecture can significantly impact the performance, and the optimal choice often depends on the specific data set. For the evaluations in this paper, we used both residual fully-connected network and transformer architectures, grounding the choice on the target task. Additionally, while incorporating the binary missingness mask m as an input is theoretically unnecessary in the MAR case (section 2.1), we include it as an auxiliary input to the model ^vθ(*xf^t* ^m; *^xfo,m, t*)[1 ^{− m}]. This generally improves the model's ability to effectively learn the required conditional distributions by providing additional information to the neural network that specifies which conditional distribution is being estimated.

3.2 Training CNF for imputation from incomplete data via flow matching

To perform missing data imputation, we first train a conditional CNF model, defined by the conditional vector field ^vθ(*xf^t* ^m; *^xfo,m, t*)[1 ^{− m}] as introduced in section 3.1. However, since the true missing data ^xm are typically unavailable, we follow prior works (Ivanov et al., 2019; Li et al., 2020; Tashiro et al., 2021) and train the model by partitioning the observed part x^o of each potentially incomplete data point x.

Partitioning observed data. Given an incomplete data point x^o and its missingness mask m, we randomly partition the available variables into two subsets: conditioning variables x^c and target variables *x*t. This partitioning is determined by a stochastic splitting strategy q(*st, s*^c | m), where s^t and s^c are binary masks indicating the target and conditioning variables, respectively. These splitting masks are constrained such that elements corresponding to missing variables (m^j = 0) remain zero in both *starg,j* and *scond,j* for ∀j ∈ [1, . . . , D], and 1−*s*c[m] = *s*t[m]. Thus, missing dimensions cannot be selected as target or conditioning variables. Using these masks, the target and conditioning variables are defined as follows:

$$\boldsymbol{x}{c} \stackrel{\text{def}}{=} \boldsymbol{x}{o}[\boldsymbol{s}{c}[\boldsymbol{m}]] \quad \text{and} \quad \boldsymbol{x}{t} \stackrel{\text{def}}{=} \boldsymbol{x}{o}[\boldsymbol{s}{t}[\boldsymbol{m}]].$$ (7)

We formalise the joint distribution of *x*t, *x*c, *s*t, and s^c as:

$$q(\boldsymbol{x}{\mathrm{t}}, \boldsymbol{x}{\mathrm{c}}, \boldsymbol{s}{\mathrm{t}}, \boldsymbol{s}{\mathrm{c}}) = \iint q(\boldsymbol{x}{\mathrm{o}}, \boldsymbol{m}) q(\boldsymbol{s}{\mathrm{t}}, \boldsymbol{s}{\mathrm{c}} \mid \boldsymbol{m}) \delta(\boldsymbol{x}{\mathrm{t}} - \boldsymbol{x}{\mathrm{o}}[\boldsymbol{s}{\mathrm{t}}[\boldsymbol{m}]]) \delta(\boldsymbol{x}{\mathrm{c}} - \boldsymbol{x}{\mathrm{o}}[\boldsymbol{s}{\mathrm{c}}[\boldsymbol{m}]]) \, \mathrm{d}\boldsymbol{x}{\mathrm{o}} \, \mathrm{d}\boldsymbol{m}, \tag{8}$$

where q(*xo,m*) is the empirical distribution of incomplete data-points and missingness masks, and δ(·) is the Dirac delta, ensuring that the split sample (*xt, x*c) is derived from x^o using the masks s^t and *s*c.

The choice of the splitting strategy q(*st, s*^c | m) introduces some flexibility. An effective strategy should closely align with the true missingness mechanism to help the model prioritise learning the most important conditional distributions. In this work, we use two simple heuristic strategies based on the data type:

• Random strategy: For synthetic and tabular data sets in sections 4.1 and 4.2, we split the observed dimensions randomly into conditioning and target variables. This ensures a diverse set of patterns for training the model without introducing systematic biases.

• Random-Historical strategy: For time-series data in section 4.3, we adopt the heuristic approach from Tashiro et al. (2021). This strategy uses a mixture of the random strategy above and a heuristic strategy that uses a missingness pattern from another training data point and determines the target and conditioning variables by intersecting the two patterns. The heuristic component of this strategy aims to mirror real-world missingness structures, such as the consecutive sequences of missing values often observed in time-series data, whereas the random component of this strategy attemps to ensure that all conditional distributions are learnt. While we apply this strategy primarily to time-series data, it is versatile and can also be applied to other data types.

For more examples of splitting strategies see Tashiro et al. (2021, Section 4.3).

Training objective. The conditional flow matching training objective is then defined as:

$$\mathcal{L}{\text{CFMI}}(\boldsymbol{\theta}) = \mathbb{E}{q(t)q(\boldsymbol{x}{\text{t}}, \boldsymbol{x}{\text{c}}, \boldsymbol{s}{\text{t}}, \boldsymbol{s}{\text{c}})} q^{t}(\boldsymbol{x}{\text{t}}^{t}|\boldsymbol{x}{\text{t}}, \boldsymbol{x}{\text{c}})} \left[ |\boldsymbol{s}{\text{t}}|{0}^{-1} |\boldsymbol{v}{\boldsymbol{\theta}}(\widetilde{\boldsymbol{x}{\text{t}}^{t}}; \widetilde{\boldsymbol{x}{\text{c}}}, \boldsymbol{s}{\text{c}}, t)[\boldsymbol{s}{\text{t}}] - u(\boldsymbol{x}{\text{t}}^{t} | \boldsymbol{x}{\text{t}}; \boldsymbol{x}{\text{c}}, t) \right]{2}^{2} \right]. \tag{9}$$

Here, q(t) is the distribution over the time index t, and q t (x t t | *xt, x*c) represents the fixed conditional probability path for the target variables ^xt conditioned on *^x*c, and *^xf^t* t and *^x*f^c are the zero-padded versions of x t ^t and ^xc such that *^xf^t* t is zero where ^st is zero and *^x*f^c is zero where ^sc is zero. The additional scaling factor ∥*s*t∥ −1 0 , normalises the objective by the number of target variables, ensuring comparable magnitudes for different target variable sets. This normalisation balances the learning of the shared model for all conditional distributions, regardless of the number of target variables. Importantly, the scaling term does not change the optimum of the flow matching objective. As a result, similar to the original flow-matching objective, this loss function minimises the L2 norm between the predicted vector field ^vθ(·; *^xfc, ^sc, t*) and the true conditional vector field u(·; *xc, t*), ensuring that the learned CNF can faithfully model the conditional distributions needed for imputation.

For the probability path q t (x t t | *xt, x*c) we adopt the independent coupling scheme from Tong et al. (2023), where q t (x t t | *xt, x*c) = N (x t t | tx^t + (1 − t)x t , σ² ) and u(x t t | *x*t; *xc, t*) = x^t − x t , where x^t is a sample from the empirical distribution and x 0 t is a sample from the base distribution which is assumed to be standard normal. We leave the evaluation of the effect of the other probability paths from the flow-matching literature (Lipman et al., 2023; Tong et al., 2023; Gat et al., 2024) to missing data imputation as future work.

3.3 Imputation of missing data using a trained model

Once the conditional CNF model has been trained to approximate the conditional distributions using eq. (9), it can be used to impute missing data. The data to be imputed may be the incomplete training data itself, or a separate test data set. The imputation process involves solving the flow equation in eq. (6) using the trained vector-field model ^vθ(*xf^t* ^m; *^xf^o,m, t*)[1 ^{− m}]. While a variety of numerical ODE solvers can be used for this purpose, we observe that a simple Euler integrator performs effectively across all experiments in our evaluation.

It is important to highlight a potential generalisation gap between the training and imputation phases. During training, the model learns to predict the conditional distribution of the target variables *x*t, given the conditioning variables *x*c. These subsets are derived by splitting incomplete training data points x^o into target and conditioning variables based on a splitting strategy q(*st, s*^c | m).

In contrast, during imputation, the task typically involves predicting the missing values x^m from the observed incomplete data points *x*o. This difference between the two stages means that the average dimensionality of the target and conditioning variables is generally higher during imputation phase than during training, which can potentially affect the performance of the model. Despite this potential gap, our evaluations demonstrate that the trained model generalises well and achieves accurate missing data imputation. This robustness highlights the model's capacity to adapt from the training setup to the imputation task.

4 Evaluation

In this section, we evaluate the proposed CFMI method for missing data imputation. The evaluation begins with synthetic 2D data sets, where we examine the method's capacity to approximate complex conditional distributions and demonstrate its flexibility. Next, CFMI is extensively evaluated on 24 tabular data sets using a wide range of metrics, providing insights into its effectiveness in imputing incomplete training data. Finally, the method is applied to time-series data in a zero-shot imputation setting, highlighting its generalisation capabilities across diverse domains.

Throughout these experiments, CFMI is consistently compared to the score-based diffusion method CSDI (Tashiro et al., 2021) to assess its relative performance, since flow matching and diffusion models are related (Albergo et al., 2023). Details of our implementation of CSDI are provided in appendix A.1. The code to reproduce our method and the following results is available at https://github.com/vsimkus/cfmi.

4.1 Learning conditionals from synthetic 2D data

As discussed in section 2.2, accurately imputing missing data requires estimating up to 2 ^D − 1 conditional distributions pθ(x^m | *x*o), one for each pattern of missingness. In this section, we evaluate the performance of CFMI for missing data imputation on synthetic 2D data sets with known distributions from Wenliang et al. (2019). This setting allows reference (conditional) distributions to be estimated accurately via numerical integration, providing a robust baseline for comparison. For the evaluation, samples from the true distribution were made 50% missing uniformly at random before training the CFMI model. Further details about the experimental setup are provided in appendix A.2.

For the cosine data set, fig. 1 demonstrates CFMI's ability to accurately capture all three (conditional) distributions: p(x¹ | *x*2), p(x² | *x*1), and p(*x1, x*2). This is achieved even when the model is trained directly on incomplete data. Further results on four additional synthetic datasets are provided in fig. 4, where CFMI consistently captures the necessary conditional distributions for imputation. These results highlight the flexibility and effectiveness of the model for various data distributions.

Interestingly, the CFMI distributions exhibit a slight mode-seeking trait, reminiscent of variational inference with the reverse KL objective (e.g. Murphy, 2021, Section 6.2.6). This behaviour likely arises from the flowmatching objective, which does not directly correspond to maximum-likelihood estimation (MLE) (Albergo et al., 2023; Dax et al., 2023) that typically has a mass-covering trait instead. While flow matching does not generally correspond to MLE, Dax et al. (2023) demonstrated that if we assume some strong regularity conditions on the vector field model, then flow matching can upper-bound MLE, which would have masscovering properties. However, our findings contrast with theirs, potentially due to differences in the target probability paths used. Specifically, while we use the independent coupling method from Tong et al. (2023), they used the optimal transport coupling from Lipman et al. (2023), which may explain the discrepancy. Despite this mode-seeking tendency, kernel density estimates of the imputed datasets (figs. 1 and 4) reveal that the imputed distributions with CFMI closely match the ground truth distributions. This suggests that the mode-seeking behaviour is inconsequential, provided the degree of missingness remains moderate.

In figs. 5 and 6, we compare CFMI with the diffusion-based alternative, CSDI (Tashiro et al., 2021). CFMI not only achieves comparable imputation accuracy but also trains faster. These findings highlight the potential advantages of flow-matching for scalable and efficient missing data imputation. In the following sections, we extend our investigation to realistic tabular and time-series data sets, evaluating CFMI's performance in practical scenarios.

4.2 Imputing incomplete tabular training data sets

In this section, we evaluate CFMI on 24 tabular data sets from the UCI repository (Dua & Graff, 2017), detailed in appendix A.3.1. The classical imputation scenario is considered, where the goal is to impute an incomplete data set. As before, for this evaluation, the complete UCI data were first made 25/50/75% missing-completely-at-random (MCAR) and 25% missing-at-random (MAR) (Schouten et al., 2018). Impu-

Table 1: Average rank of each method for all metrics (mean and standard error) under 25% MCAR missingness. Lowest means for each metric are bold and underlined; values within one standard error of the lowest are bolded.

	W2	RMSE	CRPS	Energy MMD	Gaussian MMD	Laplacian MMD	1-AUROC	Pred. CRPS	Reg. PB	Reg. 1-CR	Reg. AW
Method
Mean	9.67(0.51)	8.96(0.58)	10.75(0.11)	10.54(0.12)	9.46(0.26)	10.67(0.12)	9.53(0.61)	10.83(0.08)	7.43(1.46)	6.86(1.81)	3.00(1.41)
OT imputer	5.00(0.43)	4.04(0.46)	7.33(0.37)	6.92(0.30)	6.79(0.43)	6.92(0.30)	4.47(0.58)	7.25(0.34)	4.57(0.90)	5.71(1.15)	3.86(1.35)
SoftImpute	5.17(0.64)	3.33(0.53)	8.00(0.57)	8.12(0.39)	5.88(0.78)	8.21(0.38)	8.63(0.29)	8.12(0.55)	6.29(1.02)	4.57(1.46)	8.43(1.21)
MICE	7.21(0.63)	8.00(0.60)	4.38(0.70)	4.46(0.61)	4.08(0.48)	4.46(0.61)	7.32(0.64)	4.04(0.69)	5.00(1.76)	5.57(1.74)	7.43(1.72)
MissForest		1.08(0.26) 0.75(0.20)	3.62(0.40)	3.42(0.51)	3.04(0.49)	3.42(0.51)	5.37(0.78)	3.25(0.37)	4.00(1.02)	3.14(1.03)	8.14(1.22)
ForestDiffusion	7.04(0.51)	8.00(0.41)	4.58(0.57)	5.38(0.48)	7.38(0.53)	5.25(0.47)	2.74(0.79)	4.33(0.58)	4.71(1.29)	5.86(1.32)	2.86(0.77)
GAIN	9.04(0.25)	8.50(0.32)	8.58(0.39)	9.75(0.23)	10.33(0.19)	9.67(0.21)	6.68(0.89)	8.54(0.40)	8.86(0.86)	8.29(0.92)	4.86(0.94)
HyperImpute	4.25(0.64)	3.29(0.67)	6.04(0.65)	7.00(0.50)	5.42(0.66)	7.04(0.46)	5.95(0.76)	6.00(0.67)	5.57(1.39)	4.71(1.15)	7.00(1.23)
CSDI	3.25(0.42)	3.96(0.40)	3.38(0.57)	3.04(0.52)	3.04(0.40)	3.00(0.52)	2.68(0.34)	3.58(0.60)	4.86(1.32)	4.57(1.17)	5.57(0.90)
CFMI (ours)	2.17(0.43)	3.04(0.41)	3.12(0.61)	2.25(0.43)	1.92(0.45)	2.38(0.42)	2.11(0.46)	3.46(0.58)		3.14(1.26) 3.86(1.42)	6.00(0.93)
CSDI (short)	6.62(0.63)	7.50(0.50)	3.38(0.44)	2.83(0.38)	4.92(0.56)	2.71(0.40)	5.47(0.50)	3.29(0.45)	7.14(1.34)	7.71(1.06)	4.86(1.39)
CFMI (ours, short)	5.50(0.44)	6.62(0.39)	2.83(0.28)	2.29(0.32)	3.75(0.46)	2.29(0.32)	5.05(0.54)	3.29(0.34)	4.43(1.17)	5.14(0.59)	4.00(0.85)

Figure 2: Average rank over all metrics and data sets versus MCAR missingness fraction. Left: long run CFMI and CSDI. Right: short run CFMI and CSDI. CFMI remains competitive even at very small training budgets, whereas CSDI's performance drops significantly.

tation was then performed using various methods based solely on the observed data. The true missing values were used for assessing imputation accuracy, with evaluation metrics outlined in appendix A.3.3.

We compare CFMI against a range of traditional (van Buuren & Oudshoorn, 2000; Stekhoven & Bühlmann, 2012; Hastie et al., 2015) and modern (Yoon et al., 2018; Muzellec et al., 2020; Jarrett et al., 2022; Jolicoeur-Martineau et al., 2024) imputation methods, as well as, the related score-based diffusion method (CSDI, Tashiro et al., 2021), comparing the proposed method to nine other methods in total. For CFMI and CSDI, we use the same neural network architecture, and evaluate with two different training budgets: 5000 and 75000 training steps, where the conservative training budget often imputes faster than the traditional methods (including the training time), whereas the more generous training budget generally takes longer but often achieves better overall imputation results. Further details of the experimental setup, including hyperparameters, are provided in appendix A.3.2.

Table 1 shows the per-metric ranks of each method averaged over the 24 data sets for the 25% MCAR data.1 Firstly, we observe that the traditional missForest method performs as well as, or better than, the other traditional and modern baselines. Despite being a deterministic imputation method, missForest interestingly often outperforms the probabilistic baselines, suggesting that conditional distributions of missing data in these settings may have low entropy. This trend reverses with increasing missingness levels, as seen in appendix B.2.4, where missForest's performance declines and is eventually outperformed by the less-flexible

¹See appendix B.2.4 for more MCAR and MAR results.

Figure 3: CRPS and RMSE versus number of training epochs. Solid lines represent the median imputation performance over 5 runs, with shaded areas denoting 95% confidence intervals.

but probabilistic imputation baseline, MICE, likely due to growing entropy in the missing data distribution. This is further highlighted in fig. 2.

The proposed CFMI method achieves a comparable or better performance than missForest across most metrics, except for Wasserstein-2 and RMSE, where missForest holds a slight advantage but CFMI still outperforms the other methods. Notably, CFMI outperforms missForest at higher missingness levels, as shown in appendix B.2.4, maintaining robust performance as the degree of missingness increases. Figure 2 further summarises the results by showing the average rank over all data sets and all metrics against the missingness fraction, which further highlights that the proposed method outperforms all other approaches.

Furthermore, CFMI consistently outperforms CSDI across missingness levels and training budgets, as illustrated in fig. 2. While CFMI remains competitive even with small training budgets, CSDI's performance degrades significantly under similar conditions (fig. 2, right). This aligns with prior findings that flowmatching methods train faster than diffusion-based ones (Lipman et al., 2023). Additionally, we plot the imputation metrics versus training budget in appendix B.2.6, which further confirms this observation on a more granular level.

Overall, in this section, we have established that CFMI is a robust imputation method for a diverse range of tabular data sets. Importantly, CFMI performs comparably or surpasses the best traditional method, miss-Forest, even in low- to moderate-dimensional scenarios, where deep methods often struggle, and maintains strong performance across varying missingness levels and training budgets, demonstrating its scalability and efficiency.

To aid future meta-analyses, we provide supplementary results in appendix B.2: tables and box plots with raw metrics for each data set (appendices B.2.1 and B.2.2), average relative performance of methods for each metric (appendix B.2.3), and average rank versus missingness fraction for individual metrics (appendix B.2.5).

4.3 Imputing held-out time-series data

In this section, we extend our evaluation of CFMI to time-series data imputation. Unlike the previous section, we consider the zero-shot imputation scenario, where a model trained on one incomplete data set is used to impute a held-out incomplete data set.

Following Tashiro et al. (2021), we evaluate on two time-series data sets: PhysioNet (Silva et al., 2012), a clinical time series with 35 variables sampled hourly for 48 hours from an intensive care unit, and PM2.5 (Yi et al., 2016), an air quality data set from 36 weather stations for 12 months. The PhysioNet data has 80% inherent missingness, and for evaluation, we further mask 10/50/90% of the observed data uniformly at random as true missing values. The PM2.5 data set contains 13% missing values not-at-random (MNAR), with artificial true missing values provided for evaluation. Further experimental details and evaluation metrics are outlined in appendices A.4.1 and A.4.2.

We here focus on a comparison to the score-based diffusion method, CSDI, which has previously achieved state-of-the-art performance on these data sets. In fig. 3 we plot the performance of the methods in terms of CRPS and RMSE against the number of training epochs. As before, we observe that CFMI train significantly faster than CSDI, often achieving comparable or better performance in less time than CSDI. This advantage is particularly evident on the RMSE metric, where CFMI exceeds CSDI by orders of magnitude in terms of training speed. In addition, we provide the final metric values and comparisons with additional methods in appendix B.3.1, as well as results for additional metrics in appendix B.3.2, reinforcing CFMI's robust performance across various metrics.

Overall, these results further demonstrate the effectiveness of CFMI for missing data imputation, extending the benefits beyond tabular data to time-series data. Notably, CFMI excels in zero-shot imputation of time-series data, effectively generalising to held-out data, and trains faster than the previous state-of-the-art diffusion-based method while achieving comparable accuracy, making it overall more efficient.

5 Conclusion

Despite many advances in machine learning, missing data remains a significant challenge, limiting its effectiveness in key areas of application. Missing data imputation is a common solution; deep learning methods often excel in high-dimensional settings, whereas traditional approaches remain more effective in low- to moderate-dimensional cases. Additionally, methods for tabular and time-series data—both crucial in scientific domains—are often not interchangeable, highlighting the need for a more versatile approach.

In this paper, we proposed a conditional continuous normalising flow (CNF) model for missing data imputation, efficiently trained via flow matching. The model builds on prior work by specifying a shared conditional model for all patterns of missingness, while leveraging the strengths of CNFs and flow matching for modelling capability and training efficiency. Our extensive evaluations demonstrated strong performance across both tabular (low- to moderate-dimensional) and time-series (high-dimensional) datasets, achieving comparable or better results than traditional or diffusion-based methods with faster training.

A scalable and general-purpose imputation method, like CFMI, that effectively handles incomplete data across modalities and dimensionalities is fundamental for building foundation models in scientific domains. We anticipate that future efforts to develop foundation models in these crucial domains will greatly benefit from the generality, scalability, and efficiency of imputation methods such as CFMI.

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A Experiment details

A.1 CSDI implementation details

Our implementation of CSDI closely follows the original approach by Tashiro et al. (2021), with a few modifications outlined below.

The primary difference lies in how we normalise the training objective. In our implementation, the objective is normalised by the number of target variables for each input, ∥*s*t∥ −1 , equivalent to eq. (9). In contrast, the original implementation normalises the mini-batch loss by the total number of target variables across the entire mini-batch. Importantly, this per-data-point normalisation does not alter the optimum of the objective. We observe that the per-data-point normalisation typically improves the method's performance by better balancing the learning of conditional distributions across different patterns of missingness.

Additionally, we identified and remedied an inconsistency in the original paper. Specifically, Figure 2 of the paper suggests that the noisy target inputs to the denoising network are set to zero everywhere except where the splitting mask, s^t in our paper, assumes the targets are observed. Conversely, Figure 5 implies that inputs correspond to 1 − s^c are set to zero, with the remaining non-zero inputs being either noisy targets or pure noise in dimensions that are neither in s^c nor in *s*t. This is because 1 − s^c does not correspond to s^t when training on incomplete data. In the released code by the original authors, the latter approach is implemented. While this discrepancy appears to have minimal impact on model performance, we adopted the approach described in Figure 2 of the paper. This decision ensures architectural consistency between CFMI and CSDI.

A.2 Synthetic 2D data sets

The experiments in section 4.1 are based on five synthetic data distributions from Wenliang et al. (2019). To generate the training data set, we sampled 20K samples from each distribution and then masked 50% of the data values as missing uniformly at random (MCAR).

For CFMI we have used the I-CFM form of flow-matching from Tong et al. (2023) with σ = 0, using the torchcfm package. And for CSDI, following Tashiro et al. (2021) we have used a quadratic noise scheduler with a total of 50 steps and a minimum noise level of 10⁻⁴ and a maximum of 0*.*5.

For the vector field and denoising models, we have used a residual network with 4 residual blocks, 256 hidden dimensions, and ReLU activations. Similar to the existing literature, the neural network takes a sinusoidal time embedding, a zero-padded observed data vector, a zero-padded target (missing) variable vector, and a binary missingness mask vector as input, and predicts the vector field. To optimise the model parameters we have used Adam optimiser (Kingma & Ba, 2014) with a learning rate of 10⁻³ and gradient norm clipping with a maximum magnitude of 2 for a total of 1M gradient steps.

A.3 Tabular UCI data sets

A.3.1 Datasets

We use the same UCI (Dua & Graff, 2017) data sets as Muzellec et al. (2020), with the addition of the banknote dataset (Lohweg, 2012), for a total of 24 diverse data sets.

Table 2: UCI datasets

Dataset	Reference	Num. samples	Num. dimensions (D)	Outcome
airfoil self noise	(Brooks et al., 1989)	1503	5	continuous
banknote authentication	(Lohweg, 2012)	1272	4	binary
blood transfusion	(Yeh, 2008)	748	4	binary
breast cancer diagnostic	(Wolberg et al., 1993)	569	30	binary
california housing	(Pace & Barry, 1997)	20640	8	continuous
climate model crashes	(Lucas et al., 2013)	540	18	binary
concrete compression	(Yeh, 1998)	1030	7	continuous
concrete slump	(Yeh, 2007)	103	7	continuous
connectionist bench sonar	(Sejnowski & Gorman, 1988)	208	60	binary
connectionist bench vowel	(Deterding et al., 1988)	990	10	binary
ecoli	(Nakai, 1996)	336	7	categorical
glass	(German, 1987)	214	9	categorical
ionosphere	(Sigillito et al., 1989)	351	34	binary
iris	(Fisher, 1936)	150	4	categorical
libras	(Dias et al., 2009)	360	90	categorical
parkinsons	(Little, 2007)	195	23	binary
planning relax	(Bhatt, 2004)	182	12	binary
qsar biodegradation	(Mansouri et al., 2013)	1055	41	binary
seeds	(Charytanowicz et al., 2010)	210	7	categorical
wine	(Aeberhard & Forina, 1992)	178	13	categorical
wine quality red	(Cortez et al., 2009)	1599	10	integer
wine quality white	(Cortez et al., 2009)	4898	11	integer
yacht hydrodynamics	(Gerritsma et al., 1981)	308	6	continuous
yeast	(Nakai, 1991)	1484	8	categorical

For each of the 24 data sets, 3 different incomplete data sets were generated using MCAR and MAR mechanisms. Then, for each of the incomplete data sets 5 imputations were generated using each of the imputation methods outlined in the next section. These imputations were then evaluated using a range of metrics outlined in appendix A.3.3.

A.3.2 Imputation method details

Here we outline the details of the imputation methods used in section 4.2. We start with the baselines, which closely follow the defaults from Muzellec et al. (2020):

• OT imputer (Muzellec et al., 2020). We use the default implementation provided by the original authors with 3000 optimisation steps.
• SoftImpute (Hastie et al., 2015). We use the implementation provided by Muzellec et al. (2020) with 3000 optimisation steps.
• MICE (van Buuren & Oudshoorn, 2000). We use the IterativeImputer implementation with BayesianRidge model from scikit-learn (Pedregosa et al., 2011) with 2000 iterations, each iteration consisting of a full scan of all dimensions.
• MissForest (Stekhoven & Bühlmann, 2012). We use the IterativeImputer implementation with RandomForestRegressor model from scikit-learn (Pedregosa et al., 2011) with 200 iterations, each iteration consisting of a full scan of all dimensions.

• ForestDiffusion (Jolicoeur-Martineau et al., 2024). We use the ForestDiffusion package released by the authors with the default hyperparameters, using the RePaint method (Lugmayr et al., 2022) for imputation.

• • GAIN (Yoon et al., 2018). We use the implementation provided in the hyperimpute (Jarrett et al., 2022) package with 25000 iterations and default hyperparameters.

• HyperImpute (Jarrett et al., 2022). We use the hyperimpute package released by the authors with 60 iterations, each iteration consisting of a full scan of all dimensions.

For CFMI we have used the I-CFM form of flow-matching from Tong et al. (2023) with σ = 0, using the torchcfm package. And for CSDI, following Tashiro et al. (2021) we have used a quadratic noise scheduler with a total of 50 steps and a minimum noise level of 10−⁴ and a maximum of 0*.*5.

For the vector field and denoising models, we have used a residual network with 4 residual blocks, 256 hidden dimensions, and ReLU activations. Similar to the existing literature, the neural network takes a sinusoidal time embedding, a zero-padded observed data vector, a zero-padded target (missing) variable vector, and a binary missingness mask vector as input, and predicts the vector field. To optimise the model parameters we have used AdamW optimiser (Loshchilov & Hutter, 2017) with a learning rate of 10−³ and a cosine learning rate schedule, batch size of 64, weight decay of 10−⁵ , and gradient norm clipping with a maximum magnitude of 2 for a total of 5000 to 75000 gradient steps.

For imputation using a trained CFMI model we have approximated the flow equation in eq. (6) using 100 steps of Euler integration.

A.3.3 Evaluation metrics

Here we summarise the main evaluation metrics used for the tabular data imputation in section 4.2:

• Wasserstein-2 distance (W2). The Wasserstein-2 distance quantifies the discrepancy between the ground-truth complete training dataset and an imputed dataset. This metric is computed using the POT Python package (Flamary et al., 2021).2 We show the average W2 distance of 5 imputed data sets.
• Average RMSE. The root-mean-squared-error (RMSE) is computed between the imputed missing values and their ground-truth counterparts, averaged across multiple imputations. While this metric does not capture sample diversity and achieves an optimum when the imputation method predicts the conditional mean of the missing values (van Buuren, 2018), it remains widely used and is included for comparison.
• Continuous ranked probability score (CRPS). CRPS (Matheson & Winkler, 1976) evaluates the compatibility of a predicted probability distribution (or its samples) with true observed data. It generalises the pinball (or quantile) loss across all quantiles. CRPS is computed independently for each dimension of the imputed data, and the scores are averaged. We use the properscoring Python package to calculate CRPS.3
• Maximum mean discrepancy (MMD). MMD (Gretton et al., 2012) is a sample-based metric measuring discrepancies between two distributions. It is zero if and only if the distributions are identical. In our evaluation, we use kernelised MMD with Laplacian, Gaussian, and energy-distance kernels to characterize the distance between samples. MMD scores are computed using the geomloss Python package (Feydy et al., 2019).4
• Area Under the Receiver Operating Characteristic Curve (AUROC). For datasets intended for classification tasks, we evaluate AUROC by training a classifier on the imputed data. Higher-quality imputations lead to higher classification performance and thus better AUROC scores. For consistency with other metrics, we report 1-AUROC, where smaller values indicate better performance.

²https://github.com/PythonOT/POT

³https://github.com/properscoring/properscoring

⁴https://www.kernel-operations.io/geomloss/

• • Prediction CRPS. To address potential biases from averaging CRPS across dimensions, we compute a one-dimensional CRPS focused on the target variable of each dataset. Specifically, a linear (or logistic) regression model is first fitted to the ground-truth data, and CRPS is calculated between predicted distributions for the ground-truth and imputed datasets. Low CRPS values indicate high accuracy in the imputed distributions.
• Downstream statistical inference accuracy. Imputation is often used for statistical inference of downstream model parameters (van Buuren, 2018, Section 2.5.2). For UCI datasets, where downstream tasks typically involve regression, we evaluate the accuracy of imputations using three metrics: percent-bias (PB), coverage rate (CR), and average width (AW).
  • – Percent-bias (PB): Measures the bias in model parameters estimated from imputed data relative to ground-truth data.
  • – Coverage rate (CR): Assesses the proportion of confidence intervals for model parameters that include the ground-truth values.
  • – Average width (AW): Reflects the statistical efficiency of the imputation method. An ideal AW is small but not so small that the coverage rate falls below a nominal level (van Buuren, 2018, Section 2.5.2).

Using these diverse metrics, we provide a comprehensive evaluation of the accuracy of the imputed data for various downstream tasks.

A.4 Time series data sets

A.4.1 Imputation method details

For CFMI we have used the I-CFM form of flow-matching from Tong et al. (2023) with σ = 0, using the torchcfm package. And for CSDI, following Tashiro et al. (2021) we have used a quadratic noise scheduler with a total of 50 steps and a minimum noise level of 10⁻⁴ and a maximum of 0*.*5.

For the vector field and denoising models we use the same transformer architecture from Tashiro et al. (2021, Section 5), which is based on DiffWave by Kong et al. (2021). The transformer uses 4 TransformerEncoder layers from PyTorch (Paszke et al., 2019), which is composed of a multi-head attention layer with 8 heads, fully-connected layers and layer normalisation. To encode the time-step of the time series we use 128 dimensional temporal embeddings (Tashiro et al., 2021, Appendix E.1). Each feature is similarly encoded using a learnable 16-dimensional encoding. And to optimise the model parameters we use AdamW optimiser (Loshchilov & Hutter, 2017) with a learning rate of 10⁻³ and a multi-step learning rate schedule, batch size of 16, weight decay of 10⁻⁶ , and gradient norm clipping with a maximum magnitude of 2 for a total of 200 epochs steps.

For imputation using a trained CFMI model we have approximated the flow equation in eq. (6) using 100 steps of Euler integration.

A.4.2 Evaluation metrics

Here we summarise the main evaluation metrics used for the time-series imputation in section 4.3:

• Continuous ranked probability score (CRPS). CRPS (Matheson & Winkler, 1976) is a proper scoring metric for one-dimensional distributions, assessing the compatibility of a predicted distribution (or its samples) with true observed data. It generalises the pinball (or quantile) loss over a continuum of quantiles. CRPS is computed independently for each dimension, and the results are averaged to produce a single score. We calculate CRPS using the properscoring Python package.5
• Magnitude-normalised CRPS. In Tashiro et al. (2021), to assess the accuracy of probabilistic imputation, the authors use a related metric. Rather than averaging the CRPS over different

⁵https://github.com/properscoring/properscoring

dimensions, the authors sum the CRPS scores of all dimensions and then normalise the sum-score by the magnitude of the data set. While the metric is similar to CRPS, we observe it to exhibit slightly different properties to the regular CRPS.

• Mean absolute error (MAE). To compute MAE, we first derive the median imputation by taking the element-wise median of imputations for each data point. The absolute error between these median imputations and the ground truth values is then averaged across the dataset. This metric achieves its optimum when the median imputation corresponds to the median of the conditional distribution for the missing variables.
• Root mean squared error (RMSE). RMSE is calculated by first obtaining the mean imputation for each data point as the mean of imputations. The squared error between these mean imputations and the ground truth values is then averaged across the dataset, with the square root of the result providing the RMSE. The metric is optimal when the mean imputation matches the mean of the conditional distribution of the missing variables.

Using these metrics we evaluate the both distributional and point-estimate accuracy of imputations in timeseries data.

B Additional figures

B.1 Synthetic 2D data

Figure 4: Imputation using CFMI on five synthetic 2D data sets. In each figure, the left-most column shows the kernel density estimate (KDE) of the ground truth distribution (top) and the KDE of the imputed data distribution (bottom). The red-tinted rows and columns illustrate that some data-points may be missing one (or both) of the dimensions, and thus imputation requires sampling the corresponding (conditional) data distribution. CFMI learns a model of all conditional distributions that are needed for imputation (illustrated in the grey box on the right, corresponding to the red-tinted rows and columns). This model is then used to impute the data by sampling the appropriate conditional distribution for each incomplete data point. A sample of imputations is shown as red dots in the bottom-left, and the KDE of the imputed data closely matches that of the ground truth distribution.

Figure 5: Comparison of CFMI and CSDI across varying training budgets (10 to 1M steps) on five synthetic data sets. Contour plots represent kernel density estimates of imputed data sets. Both methods approximate the ground truth distributions well when trained for 1M steps. However, CFMI learns faster on the more complex data sets (panels 1 and 4), particularly evident at smaller training budgets of 10–100 steps.

Figure 6: Comparison of CFMI and CSDI across varying training budgets (10 to 1M steps) on five synthetic data sets. Contour plots represent the difference between kernel density estimates of imputed data sets and the ground truth: blue and red mean over- and under-estimating, respectively. Both methods approximate the ground truth distributions well when trained for 1M steps. However, CFMI generally learns faster, particularly evident at smaller training budgets of 10–100 steps.

B.2 Tabular UCI data sets

B.2.1 Raw results tables

Table 3: Wasserstein-2 results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.315(0.003)	0.197(0.002)	0.281(0.012)	0.233(0.001)	0.060(0.003)	0.081(0.001)		0.294(0.024) 0.051(0.002) 0.052(0.000) 0.052(0.001) 0.112(0.012)				0.114(0.010)
banknote	0.199(0.010)	0.030(0.001)	0.099(0.003)		0.074(0.001) 0.018(0.001)	0.037(0.001)	0.141(0.021)	0.024(0.000)	0.022(0.000)	0.021(0.000)	0.035(0.001)	0.038(0.001)
blood	0.278(0.004)	0.067(0.004)	0.093(0.005)		0.083(0.005) 0.044(0.005)	0.086(0.006)	0.163(0.014)	0.057(0.002)		0.046(0.002) 0.039(0.005) 0.052(0.003)		0.058(0.003)
cancer	3.649(0.132)		1.001(0.067) 0.619(0.042) 0.896(0.024)		0.773(0.037)	1.470(0.043)	1.689(0.087)	1.105(0.064)	0.845(0.047)	0.877(0.052)	1.573(0.237)	1.324(0.039)
california	0.766(0.108)	0.471(0.055)	0.395(0.050)		0.463(0.013) 0.238(0.035)	1.651(0.289)	0.733(0.104)	0.304(0.044)		0.901(0.356) 0.238(0.027) 3.849(0.690)		0.292(0.027)
climate	2.278(0.021) 2.891(0.025)		2.319(0.022)	4.181(0.011)	2.495(0.025)	3.440(0.011)	2.604(0.042)	2.358(0.024)	2.582(0.031)	2.871(0.039)	3.688(0.020)	4.189(0.032)
concrete	0.662(0.006)	0.616(0.006)	0.569(0.045)	0.401(0.005)	0.159(0.011)	0.282(0.006)	0.569(0.032)	0.196(0.006)		0.162(0.009) 0.151(0.005) 0.313(0.006)		0.285(0.003)
slump	0.762(0.023)	0.648(0.020)	0.606(0.038)	0.658(0.013)	0.554(0.012)	0.649(0.018)	0.653(0.021)	0.563(0.007)	0.563(0.022)		0.589(0.022) 0.540(0.013)	0.556(0.015)
sonar	7.591(0.145)		3.892(0.037) 2.875(0.028) 5.073(0.047)		3.010(0.049)	5.110(0.073)	26.608(0.193)	3.707(0.048)	4.254(0.060)	3.588(0.049)	5.051(0.064)	4.998(0.181)
vowel	1.054(0.015)	0.357(0.009)	0.779(0.020)	0.959(0.009)	0.307(0.009)	0.453(0.012)	0.907(0.027)	0.389(0.014)		0.261(0.006) 0.208(0.002) 0.632(0.005)		0.492(0.001)
ecoli			0.664(0.187) 0.530(0.134) 0.533(0.174) 0.727(0.115) 0.551(0.123)			4.971(4.203)	0.797(0.151)			1.146(0.447) 0.448(0.163) 0.425(0.163) 1.591(1.057)		0.463(0.145)
glass	0.989(0.119)	0.659(0.089)	0.605(0.097)		0.694(0.060) 0.564(0.078)	0.675(0.088)		0.932(0.105) 0.578(0.099) 0.541(0.101) 0.504(0.095) 2.016(1.111)				0.565(0.093)
ionosphere	4.010(0.059)	2.495(0.030)	2.420(0.066)		4.282(0.093) 2.194(0.075)	2.855(0.034)		3.990(0.089) 2.243(0.068) 2.247(0.088) 2.273(0.066)			3.152(0.133)	3.215(0.039)
iris	0.286(0.016)	0.059(0.004)	0.078(0.003)		0.081(0.005) 0.056(0.002)	0.071(0.008)		0.121(0.012) 0.057(0.002) 0.075(0.006)		0.074(0.007)	0.062(0.006)	0.074(0.006)
libras	11.314(0.104)	4.122(0.076)		0.589(0.238) 0.050(0.002) 0.153(0.005)		—	6.897(0.393)	1.046(0.033)	0.682(0.021)	0.463(0.015)	2.760(0.107)	2.413(0.018)
parkinsons	2.777(0.095)	0.642(0.025)	0.634(0.022)		1.014(0.019) 0.600(0.020)	1.087(0.016)	2.170(0.021)	0.740(0.019)	0.694(0.030)	0.629(0.020)	0.879(0.060)	0.824(0.009)
planning	1.467(0.022)		0.763(0.003) 0.436(0.033) 0.552(0.034)		0.604(0.037)	1.086(0.060)	0.985(0.008)	0.838(0.025)	0.809(0.029)	0.849(0.050)	0.713(0.009)	0.807(0.013)
qsar	4.867(0.123)	2.110(0.101)	2.055(0.115)		3.349(0.052) 1.244(0.057)	3.167(0.099)	4.970(0.107)	—	1.605(0.057)	1.367(0.060)	17.045(4.009)	2.566(0.075)
seeds	0.682(0.005)	0.123(0.004)			0.127(0.009) 0.107(0.004) 0.099(0.009)	0.150(0.004)	0.213(0.028)	0.135(0.011)	0.119(0.004)	0.117(0.007)	0.116(0.004)	0.148(0.002)
wine		1.593(0.062) 0.840(0.028) 0.970(0.048)			1.516(0.027) 0.840(0.026)	1.086(0.002)		1.431(0.038) 0.842(0.032) 0.938(0.049)		0.894(0.034)	0.962(0.015)	1.016(0.027)
wine (red)	0.915(0.022)	0.558(0.015)	0.701(0.016)		0.928(0.006) 0.504(0.012)	0.682(0.011)		0.918(0.014) 0.504(0.008) 0.551(0.009)		0.516(0.011)	0.830(0.049)	0.680(0.009)
	wine (white) 1.052(0.015)	0.655(0.016)	0.734(0.011)		0.926(0.011) 0.535(0.012)	0.827(0.016)	0.967(0.010)	0.592(0.007)	0.668(0.014)	0.637(0.012)	1.013(0.103)	0.829(0.010)
yacht	0.633(0.014)	0.469(0.010)	0.552(0.025)	0.323(0.015)	0.219(0.016)	0.284(0.002)	0.505(0.012)			0.358(0.006) 0.174(0.005) 0.162(0.014) 0.261(0.007)		0.237(0.002)
yeast	0.689(0.064)	0.536(0.015)	0.660(0.067)			0.804(0.025) 0.449(0.017) 0.452(0.052) 0.697(0.084)		0.487(0.036)		0.483(0.046) 0.463(0.051) 32.746(9.396)		0.508(0.079)

Table 4: Wasserstein-2 results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.605(0.004)	0.397(0.004)	0.573(0.004)	0.491(0.002)	0.138(0.005)	0.146(0.007)	0.535(0.035)		0.118(0.004) 0.107(0.002) 0.112(0.002)		0.192(0.020)	0.228(0.024)
banknote	0.384(0.001)	0.065(0.002)	0.201(0.005)		0.137(0.004) 0.045(0.002)	0.076(0.002)	0.259(0.006)	0.057(0.007)	0.049(0.001)	0.049(0.001)	0.085(0.003)	0.077(0.002)
blood	0.481(0.001)	0.122(0.005)	0.168(0.006)	0.135(0.012)	0.114(0.014)	0.156(0.003)	0.303(0.019)		0.133(0.011) 0.075(0.008) 0.088(0.009)		0.211(0.064)	0.119(0.010)
cancer	7.149(0.176)		2.601(0.114) 1.906(0.096) 2.670(0.035)		2.404(0.097)	4.025(0.126)	5.711(0.224)	2.564(0.086)	2.740(0.110)	2.605(0.146)	9.708(1.656)	4.031(0.046)
california	1.125(0.056)	1.008(0.099)	0.795(0.052)		0.923(0.052) 0.618(0.110)	1.805(0.228)	1.288(0.061)	0.648(0.111)		1.130(0.210) 0.535(0.095)	5.977(1.554)	0.662(0.094)
climate	4.476(0.007) 5.386(0.004)		4.552(0.018)	6.233(0.018)	5.281(0.024)	5.156(0.052)	4.841(0.047)	4.843(0.034)	4.922(0.009)	4.996(0.013)	5.494(0.034)	6.190(0.066)
concrete	1.237(0.005)	1.114(0.009)	1.063(0.055)		0.811(0.008) 0.430(0.017)	0.630(0.025)	1.096(0.027)		0.701(0.027) 0.434(0.012) 0.437(0.005)		0.679(0.023)	0.639(0.025)
slump	1.383(0.014)	1.197(0.013)	1.237(0.034)	1.540(0.025)	1.122(0.068)	1.248(0.017)	1.410(0.088)	1.152(0.039)	1.089(0.035)	1.152(0.041)	1.056(0.014)	1.078(0.031)
sonar	15.230(0.082)		8.676(0.018) 7.886(0.036) 14.571(0.072)		8.640(0.025)	14.596(0.191)	66.120(0.466)	8.422(0.022)	9.766(0.132)	9.105(0.089)	22.408(0.957)	13.558(0.061)
vowel	1.919(0.012)	0.800(0.009)	1.482(0.013)	1.634(0.006)	0.851(0.008)	1.142(0.010)	1.965(0.067)	1.161(0.011)		0.809(0.001) 0.744(0.007)	1.269(0.012)	1.131(0.015)
ecoli	1.394(0.123)	1.131(0.080)	1.197(0.127)	1.418(0.071)	1.110(0.048)	36.232(8.817)	2.038(0.467)		3.259(1.911) 0.925(0.073) 0.886(0.080)		5.880(3.024)	0.889(0.087)
glass	1.704(0.109)	1.271(0.125)	1.329(0.122)	1.811(0.053)	1.191(0.132)	1.410(0.163)			2.124(0.040) 1.113(0.107) 1.076(0.081) 1.045(0.095)		3.597(2.383)	1.162(0.050)
ionosphere	7.865(0.032)	5.186(0.091)	5.267(0.072)	8.786(0.012)	5.402(0.138)	5.958(0.057)		9.608(0.134) 4.713(0.008) 4.878(0.079)		4.907(0.020)	8.697(0.241)	6.932(0.073)
iris	0.499(0.012)	0.118(0.014)	0.140(0.003)	0.155(0.007)	0.107(0.018)	0.122(0.007)		0.265(0.027) 0.094(0.006) 0.103(0.011)		0.106(0.013)	0.104(0.011)	0.120(0.009)
libras	22.772(0.079)	7.745(0.075)		0.790(0.235) 0.350(0.026) 0.915(0.019)		—	82.024(0.443)	2.836(0.012)	3.319(0.036)	2.455(0.149)	38.186(5.292)	10.493(0.339)
parkinsons	5.410(0.164)		1.753(0.033) 1.665(0.030) 2.203(0.050)		1.704(0.078)	3.328(0.089)	6.274(0.382)	1.753(0.039)	2.297(0.195)	2.109(0.148)	10.579(4.396)	2.465(0.034)
planning	2.777(0.049)		1.955(0.038) 1.428(0.066) 1.687(0.008)		1.732(0.029)	2.466(0.016)	2.405(0.105)	1.884(0.055)	2.010(0.058)	2.074(0.065)	1.964(0.079)	2.091(0.060)
qsar	9.305(0.149)	4.761(0.090)	4.981(0.098)	8.200(0.094)	4.297(0.078)	7.785(0.073)	13.529(0.834)	4.158(nan)			4.878(0.204) 3.979(0.154) 210.994(105.439)	7.231(0.178)
seeds	1.288(0.021)	0.253(0.006)		0.275(0.012) 0.201(0.001) 0.228(0.005)		0.320(0.027)	0.570(0.076)	0.264(0.009)	0.290(0.028)	0.264(0.019)	0.244(0.010)	0.292(0.011)
wine	2.957(0.019)	1.810(0.048)	1.955(0.040)	2.680(0.029)	1.907(0.026)	2.149(0.073)		2.862(0.041) 1.700(0.054) 1.849(0.021)		1.898(0.058)	1.909(0.015)	2.019(0.046)
wine (red)	1.639(0.015)	1.049(0.017)	1.320(0.004)	1.515(0.009)	1.069(0.011)	1.130(0.012)	1.815(0.101)	1.018(0.015)		0.979(0.017) 0.949(0.013)	2.591(0.701)	1.174(0.006)
	wine (white) 1.896(0.006)	1.273(0.012)	1.425(0.020)		1.472(0.008) 1.115(0.015)	1.229(0.010)	2.297(0.186)	1.232(0.013)		1.169(0.040) 1.115(0.017)	2.351(0.469)	1.379(0.022)
yacht	1.149(0.027)	0.889(0.034)	1.030(0.067)		0.633(0.023) 0.393(0.033)	0.691(0.008)	0.994(0.051)	0.676(0.020)	0.573(0.031)	0.615(0.012)	0.612(0.034)	0.655(0.003)
yeast	1.245(0.095)	0.870(0.024)	1.172(0.091)	1.480(0.052)	0.968(0.198)	0.692(0.092) 1.541(0.132)		2.668(0.593)			0.800(0.108) 0.724(0.158) 54.744(21.336)	0.793(0.077)

Table 5: Wasserstein-2 results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.846(0.012)	0.687(0.017)	0.795(0.032)	1.051(0.003)	0.310(0.029)	0.382(0.056)	0.775(0.049)	0.497(0.027)	0.248(0.017)	0.255(0.030)	0.384(0.022)	0.389(0.056)
banknote	0.510(0.027)	0.137(0.011)	0.293(0.021)	0.204(0.004)	0.145(0.013)	0.197(0.026)	0.521(0.099)	0.177(0.008)	0.126(0.009)	0.129(0.005)	0.418(0.241)	0.160(0.013)
blood		0.556(0.059) 0.224(0.017)	0.290(0.023)	0.225(0.011) 0.289(0.022)		0.402(0.032)		0.438(0.069) 0.228(0.013)	0.242(0.039)	0.244(0.044)	7.431(5.977)	0.223(0.015)
cancer	10.342(0.096)	4.855(0.129)	4.152(0.073)	5.283(0.023)	4.816(0.033)	7.710(0.064)	9.930(0.674)	4.438(0.041)	5.917(0.191)	5.391(0.163)	1133.831(120.235)	7.792(0.109)
california	2.216(0.313)	1.884(0.300)	1.708(0.288)		1.345(0.269) 1.096(0.277)	1.612(0.353)		8.120(2.763) 1.138(0.235)	2.156(0.436)	0.918(0.249)	159.032(43.211)	1.344(0.310)
climate	6.136(0.023)	6.820(0.043)	6.124(0.010)	6.821(0.034)	7.210(0.074)	5.961(0.025)	7.830(0.101)	6.898(0.027)	5.935(0.052)	5.903(0.042)	55.578(7.254)	6.866(0.071)
concrete	1.629(0.017)	1.458(0.013)	1.439(0.046)	1.267(0.025)	0.957(0.010)	1.127(0.021)	1.717(0.081)	1.074(0.029)	0.880(0.015)	0.874(0.041)	1.075(0.033)	1.147(0.084)
slump		1.873(0.081) 1.632(0.113)	1.804(0.083)	2.193(0.085)	2.359(0.121)	2.078(0.129)	2.823(0.083)	1.972(0.148)	1.607(0.087)	1.593(0.082)	1.660(0.074)	1.686(0.028)
sonar	22.777(0.120)		18.090(0.177) 16.151(0.142) 27.390(0.280)		21.893(0.373)	28.026(0.105)	94.824(2.545)	16.559(0.107)	19.481(0.288)	19.625(0.185)	2945.524(658.457)	26.132(0.219)
vowel		2.747(0.007) 1.537(0.041)	2.359(0.041)	2.244(0.043)	1.803(0.055)	2.037(0.016)	4.328(0.287)	1.768(0.067)	1.602(0.005)	1.586(0.011)	2.783(0.530)	1.845(0.029)
ecoli		1.790(0.329) 1.402(0.330)	1.581(0.313)	1.895(0.251)	1.847(0.428)	130.121(23.051)	2.791(0.319)	1.674(0.620)	1.280(0.322)	1.232(0.352)	6.841(5.364)	1.256(0.312)
glass		2.722(0.253) 2.216(0.239)	2.280(0.265)	3.086(0.134)	2.596(0.295)	175.178(147.300)		6.512(0.881) 2.284(0.225)	2.155(0.285)	2.060(0.256)	5.242(2.933)	2.287(0.387)
ionosphere	11.687(0.042)	8.433(0.034)	8.799(0.157)	13.467(0.231)	10.295(0.119)	12.733(0.413)		22.423(0.523) 8.079(0.054)	8.459(0.111)	8.419(0.067)	446.835(88.893)	11.287(0.047)
iris		0.635(0.058) 0.236(0.049)	0.288(0.061)	0.363(0.024)	0.278(0.048)	0.273(0.071)		0.855(0.021) 0.221(0.034)	0.308(0.040)	0.289(0.032)	0.291(0.043)	0.292(0.047)
libras	34.445(0.073)	14.270(0.038)	8.794(1.893)		6.779(0.073) 6.589(0.217)	—	148.193(1.426)	8.708(0.291)	40.346(22.691)	17.488(0.672)	39115.493(6298.722)	29.207(0.288)
parkinsons	7.788(0.128)	3.732(0.105)	3.378(0.125)	4.753(0.253)	3.821(0.250)	7.136(0.187)		20.139(1.389) 3.342(0.133)	4.709(0.180)	4.870(0.187)	343.704(141.247)	5.071(0.067)
planning	3.873(0.054)	3.538(0.144)	2.811(0.097)	4.134(0.154)	3.830(0.301)	4.062(0.048)	6.246(0.572)	3.473(0.206)	3.149(0.095)	3.232(0.085)	9.430(2.264)	3.381(0.140)
qsar		13.748(0.286) 8.953(0.360)	9.498(0.345)	15.126(0.245)	11.958(0.230)	17.698(1.016)	47.465(11.634)	10.200(0.388)	3582.086(3559.520)	9.561(0.313)	8464.414(1857.287)	15.400(0.244)
seeds	1.787(0.131)	0.501(0.038)	0.481(0.031)	0.406(0.008) 0.457(0.022)		0.673(0.074)	1.758(0.297)	0.487(0.042)	0.564(0.065)	0.529(0.049)	0.536(0.042)	0.571(0.036)
wine		4.229(0.063) 2.792(0.015)	3.097(0.111)	3.898(0.077)	3.340(0.123)	3.087(0.055)		6.465(0.177) 2.764(0.087)	2.792(0.088)	2.784(0.058)	3.299(0.277)	2.941(0.087)
wine (red)	2.411(0.022)	1.703(0.031)	2.036(0.024)	2.105(0.021)	1.841(0.029)	1.906(0.061)	4.235(0.458)	1.718(0.038)	1.622(0.064)	1.532(0.066)	98.660(41.538)	2.077(0.126)
	wine (white) 2.769(0.010)	2.040(0.019)	2.255(0.029)	1.878(0.021)	1.826(0.042)	1.816(0.043)	4.079(0.021)	1.825(0.018)	1.860(0.161)	1.588(0.022)	73.247(13.149)	2.258(0.117)
yacht	1.534(0.049)	1.292(0.054)	1.349(0.075)		1.353(0.067) 0.932(0.022)	1.201(0.082)	1.666(0.029)	1.115(0.013)	1.020(0.017)	1.069(0.032)	0.964(0.032)	1.097(0.045)
yeast	1.764(0.203)	1.208(0.114)	1.693(0.194)	2.023(0.038)	1.863(0.242)	27.983(13.467)	3.229(0.516)	2.592(0.645)	1.004(0.098)	1.083(0.185)	181.188(50.349)	1.091(0.195)

Table 6: Wasserstein-2 results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.391(0.020)	0.258(0.012)	0.365(0.012)	0.303(0.002)	0.080(0.005)	0.093(0.012)	0.316(0.008)		0.070(0.001) 0.059(0.003) 0.062(0.006)		0.117(0.013)	0.127(0.016)
banknote	0.242(0.003)	0.044(0.006)	0.129(0.005)		0.093(0.008) 0.023(0.003)	0.056(0.010)	0.133(0.013)	0.063(0.034)	0.026(0.003)	0.025(0.002)	0.041(0.002)	0.040(0.001)
blood	0.226(0.018)	0.064(0.008)	0.094(0.020)	0.064(0.010)	0.040(0.006)	0.085(0.009)	0.149(0.018)		0.071(0.008) 0.033(0.006) 0.035(0.006)		0.037(0.005)	0.050(0.003)
cancer	5.740(0.022)		1.874(0.047) 1.145(0.004) 1.360(0.073)		1.526(0.041)	3.257(0.044)	5.293(0.428)	1.782(0.041)	1.766(0.049)	1.740(0.001)	3.229(0.967)	1.980(0.067)
california	1.023(0.198)	0.904(0.170)	0.810(0.189)	0.911(0.116)	0.658(0.118)	1.712(0.408)	1.145(0.056)	0.745(0.135)		1.290(0.443) 0.527(0.076) 2.886(1.019)		0.687(0.115)
climate	2.347(0.037) 2.840(0.028)		2.384(0.041)	4.071(0.001)	2.698(0.023)	3.427(0.066)	2.967(0.164)	2.474(0.044)	2.707(0.056)	2.952(0.037)	3.591(0.019)	3.748(0.013)
concrete	0.760(0.025)	0.690(0.032)	0.608(0.011)		0.435(0.025) 0.207(0.025)	0.360(0.028)	0.613(0.046)			0.288(0.022) 0.208(0.022) 0.208(0.019) 0.342(0.012)		0.322(0.014)
slump	0.817(0.099)	0.670(0.065)			0.755(0.113) 0.607(0.054) 0.602(0.049)	0.682(0.112)		0.819(0.027) 0.611(0.057) 0.584(0.052) 0.631(0.061)			0.573(0.058)	0.571(0.044)
sonar	9.599(0.194)		5.428(0.183) 4.238(0.118) 6.746(0.091)		4.479(0.109)	8.653(0.210)	30.249(0.665)	5.045(0.205)	6.184(0.226)	5.366(0.249)	7.806(0.467)	6.939(0.215)
vowel	1.152(0.008)	0.427(0.011)	0.843(0.015)	0.983(0.002)	0.413(0.022)	0.582(0.019)	0.902(0.029)	0.585(0.015)		0.332(0.004) 0.293(0.004) 0.682(0.009)		0.537(0.003)
ecoli	1.045(0.035)	0.780(0.024)	0.888(0.055)	0.969(0.062)	0.769(0.022)	79.820(24.128)	1.360(0.148)	0.810(0.062)		0.761(0.014) 0.737(0.001) 0.777(0.024)		0.812(0.065)
glass	0.994(0.121)	0.666(0.043)	0.614(0.083)	0.766(0.092)	0.602(0.074)	0.810(0.047)	1.017(0.292)	0.628(0.045)	0.594(0.029)	0.667(0.019)	0.554(0.036)	0.625(0.029)
ionosphere	3.993(0.065)	2.254(0.146)	2.205(0.082)		4.021(0.108) 2.002(0.117)	2.799(0.144)		4.148(0.198) 2.094(0.083) 2.039(0.123) 2.034(0.105) 2.719(0.119)				2.866(0.162)
iris		0.325(0.030) 0.057(0.025) 0.079(0.019)			0.070(0.013) 0.053(0.024)			0.056(0.012) 0.121(0.042) 0.055(0.024) 0.056(0.013) 0.057(0.014)			0.048(0.014)	0.058(0.013)
libras	11.508(0.173)	5.486(0.165)		0.453(0.253) 0.058(0.007) 0.188(0.020)		—	18.502(0.263)	1.107(0.036)	0.717(0.084)	0.611(0.035)	2.437(0.056)	2.714(0.145)
parkinsons	5.064(0.028)		1.881(0.133) 1.428(0.077) 1.601(0.138) 1.456(0.123)			3.838(0.084)	4.732(0.204)	1.662(0.078)	2.436(0.142)	2.415(0.081)	1.607(0.104)	1.571(0.027)
planning	1.724(0.020)		0.919(0.065) 0.616(0.047) 0.708(0.045)		0.711(0.044)	1.342(0.010)	1.468(0.178)	1.004(0.002)	1.005(0.048)	1.159(0.061)	0.833(0.031)	0.963(0.051)
qsar	6.541(0.367)	3.661(0.220)	3.492(0.165)	4.814(0.153)	3.279(0.052)	5.338(0.136)	12.055(0.392)	—		2.984(0.237) 2.829(0.021) 27.516(11.101)		4.147(0.027)
seeds	1.027(0.042)	0.169(0.021)		0.192(0.013) 0.130(0.008) 0.150(0.018)		0.190(0.021)	0.385(0.068)	0.179(0.020)	0.165(0.022)	0.164(0.011)	0.149(0.024)	0.178(0.014)
wine		1.787(0.068) 0.944(0.115) 1.061(0.106)			1.651(0.076) 0.911(0.063)	1.265(0.110)		1.958(0.019) 0.938(0.074) 1.018(0.090)		1.055(0.069)	1.028(0.092)	1.134(0.094)
wine (red)	1.061(0.016)	0.691(0.022)	0.821(0.018)	1.067(0.008)	0.727(0.036)	0.780(0.016)	1.067(0.039)	0.696(0.031)		0.638(0.010) 0.617(0.016) 0.810(0.016)		0.753(0.021)
	wine (white) 1.256(0.008)	0.787(0.013)	0.870(0.022)		1.011(0.016) 0.657(0.012)	0.893(0.030)	1.107(0.036)	0.770(0.032)	0.735(0.005)	0.704(0.011)	0.934(0.040)	0.843(0.007)
yacht	0.739(0.026)	0.523(0.013)	0.576(0.014)		0.391(0.010) 0.264(0.034)	0.371(0.010)	0.586(0.037)		0.423(0.034) 0.279(0.028) 0.310(0.031)		0.300(0.008)	0.304(0.015)
yeast	0.928(0.033)	0.682(0.043)	0.892(0.042)		0.977(0.032) 0.619(0.085) 30.165(12.158)		1.339(0.247)				0.801(0.062) 0.633(0.023) 0.604(0.048) 61.606(33.201)	0.642(0.021)

Table 7: Average RMSE results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	1.009(0.008)	0.832(0.003)	0.954(0.022)	1.212(0.003)	0.535(0.010)	0.980(0.003)		0.977(0.020) 0.513(0.021) 0.765(0.006)		0.719(0.014)	1.074(0.011)	1.049(0.004)
banknote	0.979(0.022)	0.470(0.013)	0.711(0.009)		0.765(0.007) 0.412(0.010)	0.657(0.009)	0.831(0.047)	0.448(0.014)	0.498(0.001)	0.489(0.004)	0.595(0.012)	0.606(0.006)
blood	1.035(0.007)	0.689(0.014)	0.712(0.013)		0.905(0.015) 0.599(0.007)	0.910(0.004)	0.960(0.019)	0.633(0.018)	0.731(0.016)	0.701(0.013)	0.803(0.024)	0.804(0.015)
cancer	0.985(0.009)		0.514(0.014) 0.399(0.010) 0.484(0.002)		0.449(0.007)	0.623(0.005)	0.657(0.013)	0.537(0.011)	0.470(0.009)	0.481(0.010)	0.644(0.048)	0.592(0.005)
california	1.075(0.066)	0.933(0.073)	0.861(0.040)		1.060(0.018) 0.786(0.049)	1.645(0.086)		1.103(0.092) 0.829(0.077) 1.300(0.150)		0.922(0.037)	2.889(0.310)	0.998(0.033)
climate	1.005(0.005) 1.133(0.005)		1.015(0.005)	1.413(0.000)	1.051(0.006)	1.251(0.007)	1.076(0.008)	1.022(0.005)	1.067(0.006)	1.131(0.011)	1.306(0.005)	1.412(0.007)
concrete	1.003(0.005)	0.998(0.005)	0.912(0.045)		0.857(0.004) 0.486(0.011)	0.823(0.008)	0.937(0.036)	0.571(0.015)	0.588(0.012)	0.547(0.010)	0.824(0.016)	0.773(0.013)
slump	0.967(0.005)	0.893(0.008)	0.850(0.027)		0.956(0.028) 0.818(0.005)	1.010(0.026)	0.926(0.009)	0.827(0.002)	0.886(0.043)	0.904(0.021)	0.876(0.014)	0.879(0.032)
sonar	1.009(0.011)		0.722(0.003) 0.620(0.003) 0.822(0.003)		0.634(0.003)	0.827(0.006)	1.864(0.008)	0.704(0.005)	0.756(0.008)	0.693(0.004)	0.818(0.005)	0.817(0.014)
vowel	0.999(0.003)	0.544(0.008)	0.835(0.008)	1.048(0.007)	0.502(0.010)	0.657(0.012)	0.919(0.011)	0.589(0.015)		0.472(0.005) 0.415(0.004) 0.804(0.005)		0.687(0.002)
ecoli		0.968(0.072) 0.856(0.027) 0.831(0.073) 1.131(0.031) 0.828(0.048)				1.886(0.862)	1.065(0.079)		1.126(0.223) 0.824(0.065) 0.790(0.080) 1.162(0.221)			0.901(0.052)
glass	1.043(0.061)		0.814(0.042) 0.760(0.067) 0.838(0.024) 0.726(0.058)			0.863(0.045)		1.008(0.053) 0.721(0.055) 0.710(0.075) 0.696(0.091) 1.371(0.424)				0.764(0.065)
ionosphere	1.001(0.005)	0.786(0.004)	0.773(0.005)		1.031(0.006) 0.739(0.006)	0.845(0.004)		0.989(0.011) 0.745(0.005) 0.748(0.009)		0.753(0.004)	0.879(0.013)	0.896(0.000)
iris	1.007(0.022)	0.507(0.007)	0.559(0.021)	0.702(0.019)	0.490(0.012)	0.607(0.019)		0.721(0.071) 0.460(0.012) 0.568(0.008)		0.570(0.015)	0.561(0.016)	0.607(0.015)
libras	1.003(0.004)	0.603(0.005)		0.193(0.036) 0.065(0.001) 0.115(0.003)		—	0.759(0.024)	0.304(0.005)	0.243(0.003)	0.200(0.003)	0.490(0.010)	0.460(0.002)
parkinsons	1.000(0.007)		0.477(0.001) 0.471(0.016) 0.614(0.010) 0.462(0.013)			0.638(0.005)	0.871(0.018)	0.510(0.010)		0.497(0.004) 0.475(0.005) 0.561(0.011)		0.546(0.014)
planning	1.000(0.026)		0.704(0.008) 0.505(0.015) 0.570(0.020)		0.613(0.016)	0.864(0.034)	0.795(0.018)	0.741(0.011)	0.728(0.020)	0.738(0.030)	0.681(0.002)	0.727(0.010)
qsar	0.999(0.009)	0.657(0.013)	0.642(0.017)		0.826(0.016) 0.495(0.016)	0.823(0.011)	1.030(0.009)	—	0.599(0.019)	0.551(0.010)	1.806(0.191)	0.749(0.009)
seeds	1.009(0.017)	0.438(0.015)	0.427(0.031)		0.469(0.020) 0.377(0.024)	0.532(0.026)	0.602(0.048)	0.438(0.032)	0.456(0.016)	0.450(0.022)	0.466(0.014)	0.520(0.009)
wine		1.006(0.031) 0.716(0.019) 0.775(0.028)			1.014(0.009) 0.721(0.005)	0.832(0.011)		0.947(0.023) 0.723(0.021) 0.768(0.029)		0.740(0.022)	0.784(0.012)	0.791(0.012)
wine (red)	0.997(0.007)	0.782(0.001)	0.847(0.002)		1.118(0.004) 0.734(0.015)	0.977(0.002)		1.004(0.013) 0.737(0.011) 0.870(0.002)		0.809(0.013)	1.072(0.016)	1.036(0.010)
	wine (white) 1.003(0.015)	0.804(0.019)	0.826(0.013)		1.081(0.008) 0.714(0.014)	1.069(0.011)	0.968(0.005)	0.776(0.008)	0.946(0.017)	0.918(0.013)	1.100(0.011)	1.107(0.013)
yacht	1.020(0.015)	0.886(0.016)	0.924(0.025)		0.947(0.021) 0.708(0.026)	1.000(0.017)		0.889(0.015) 0.733(0.010) 0.832(0.020)		0.796(0.029)	0.886(0.014)	0.861(0.009)
yeast	1.061(0.057)	1.050(0.049)	1.030(0.060)	1.381(0.038)	0.979(0.065)	1.080(0.049)		1.120(0.066) 0.914(0.012) 1.185(0.052)		1.084(0.056)	4.330(0.413)	1.248(0.040)

Table 8: Average RMSE results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	1.000(0.003)	0.881(0.003)	0.979(0.003)	1.363(0.001)	0.749(0.012)	1.058(0.003)		0.995(0.018) 0.726(0.008) 0.967(0.007)		0.932(0.009)	1.155(0.007)	1.223(0.009)
banknote		0.996(0.009) 0.608(0.018) 0.781(0.013)			0.895(0.009) 0.605(0.012)	0.800(0.013)	0.875(0.008)	0.638(0.004)	0.699(0.018)	0.692(0.020)	0.813(0.026)	0.812(0.019)
blood	1.023(0.021)	0.746(0.014)	0.740(0.007)	0.963(0.004)	0.767(0.040)	0.965(0.013)		0.970(0.047) 0.684(0.011) 0.830(0.016)		0.858(0.020)	1.187(0.135)	0.983(0.024)
cancer	1.005(0.010)		0.603(0.016) 0.506(0.017) 0.620(0.002)		0.578(0.013)	0.762(0.011)	0.899(0.014)	0.594(0.014)	0.620(0.017)	0.602(0.021)	1.172(0.065)	0.775(0.005)
california	1.086(0.039)		1.004(0.042) 0.951(0.046) 1.170(0.030) 0.966(0.043)			1.444(0.022)		1.119(0.027) 0.980(0.030) 1.793(0.285)		1.092(0.046)	3.529(0.333)	1.149(0.030)
climate	1.002(0.000) 1.171(0.005)		1.011(0.001)	1.411(0.004)	1.165(0.002)	1.158(0.007)	1.057(0.008)	1.061(0.003)	1.117(0.003)	1.126(0.006)	1.262(0.001)	1.399(0.015)
concrete	1.002(0.002)	0.992(0.004)	0.930(0.021)		1.110(0.009) 0.773(0.008)	1.003(0.002)	0.994(0.030)	0.832(0.008)	0.896(0.012)	0.855(0.011)	1.072(0.012)	1.070(0.018)
slump			1.004(0.010) 0.954(0.009) 0.949(0.019) 1.339(0.007)		1.027(0.013)	1.166(0.011)	1.119(0.038)	0.969(0.004)	1.075(0.015)	1.090(0.011)	1.076(0.015)	1.097(0.007)
sonar	1.008(0.001)		0.760(0.002) 0.723(0.004) 0.984(0.004)		0.755(0.002)	0.986(0.004)	2.185(0.006)	0.748(0.002)	0.807(0.003)	0.780(0.005)	1.224(0.026)	0.949(0.003)
vowel		1.003(0.003) 0.645(0.005) 0.878(0.005)		1.170(0.004)	0.732(0.007)	0.869(0.007)	1.035(0.016)	0.803(0.008)	0.736(0.003)	0.688(0.005)	0.994(0.004)	0.931(0.009)
ecoli	1.016(0.029)	0.957(0.037)	0.936(0.033)		1.224(0.018) 0.932(0.022)	4.665(0.820)	1.274(0.099)		1.300(0.378) 0.903(0.056) 0.884(0.050) 1.742(0.401)			0.982(0.055)
glass	0.976(0.020)	0.871(0.024)	0.844(0.024)	1.114(0.010)	0.859(0.053)	1.002(0.053)		1.157(0.011) 0.778(0.032) 0.818(0.021) 0.787(0.032) 1.402(0.457)				0.907(0.013)
ionosphere	0.998(0.004)	0.805(0.008)	0.811(0.008)	1.073(0.004)	0.822(0.008)	0.870(0.003)		1.115(0.010) 0.767(0.002) 0.781(0.008)		0.781(0.005)	1.067(0.019)	0.942(0.009)
iris	1.000(0.010)	0.577(0.051)	0.603(0.016)	0.747(0.017)	0.574(0.074)	0.637(0.017)		0.794(0.048) 0.493(0.032) 0.562(0.018)		0.566(0.014)	0.590(0.029)	0.637(0.016)
libras	1.007(0.002)	0.586(0.002)		0.173(0.021) 0.123(0.004) 0.200(0.002)		—	1.945(0.005)	0.354(0.001)	0.382(0.002)	0.329(0.010)	1.288(0.089)	0.681(0.011)
parkinsons	1.022(0.013)		0.594(0.002) 0.561(0.005) 0.701(0.011)		0.590(0.012)	0.837(0.012)	1.170(0.038)	0.584(0.007)	0.669(0.022)	0.642(0.019)	1.326(0.287)	0.714(0.003)
planning	1.006(0.011)		0.889(0.004) 0.710(0.017) 0.919(0.015)		0.863(0.014)	1.038(0.006)	0.974(0.023)	0.852(0.008)	0.922(0.011)	0.915(0.008)	0.962(0.020)	0.977(0.019)
qsar	0.990(0.005)	0.716(0.006)	0.716(0.007)	0.945(0.001)	0.689(0.007)	0.960(0.004)		1.236(0.036) 0.668(nan)	0.800(0.017)	0.691(0.010)	4.364(1.026)	0.958(0.005)
seeds	0.991(0.013)		0.502(0.006) 0.493(0.016) 0.576(0.015) 0.493(0.006)			0.619(0.021)		0.721(0.049) 0.488(0.012) 0.539(0.024)		0.517(0.012)	0.555(0.016)	0.598(0.006)
wine	1.012(0.007)	0.805(0.007)	0.815(0.008)	1.071(0.003)	0.852(0.013)	0.913(0.018)		1.044(0.006) 0.770(0.010) 0.817(0.004)		0.815(0.012)	0.853(0.001)	0.886(0.008)
wine (red)		0.989(0.006) 0.865(0.006) 0.890(0.006)		1.199(0.001)	0.956(0.005)	1.030(0.007)	1.059(0.037)	0.969(0.011)	1.024(0.005)	0.980(0.004)	1.432(0.089)	1.202(0.016)
	wine (white) 1.004(0.004) 0.877(0.006) 0.884(0.008)			1.190(0.005)	0.940(0.009)	1.078(0.001)	1.129(0.032)	1.057(0.010)	1.110(0.008)	1.108(0.012)	1.368(0.064)	1.253(0.008)
yacht	1.017(0.017)	0.937(0.023)	0.953(0.034)		1.135(0.002) 0.863(0.022)	1.146(0.014)		0.997(0.034) 0.865(0.020) 1.045(0.013)		1.053(0.020)	1.047(0.014)	1.087(0.016)
yeast			1.014(0.033) 0.997(0.032) 0.986(0.033) 1.397(0.016)		1.121(0.102)	1.031(0.039)	1.179(0.043)	1.357(0.081)	1.121(0.018)	1.080(0.027)	4.655(0.631)	1.225(0.017)

Table 9: Average RMSE results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil		1.004(0.005) 0.957(0.006) 0.985(0.012)		1.610(0.006)	1.000(0.005)	1.127(0.011)	1.049(0.032)	0.968(0.005)	1.159(0.004)	1.100(0.011)	1.264(0.033)	1.336(0.027)
banknote		0.991(0.006) 0.733(0.022) 0.841(0.017)		1.008(0.010)	0.812(0.020)	0.901(0.023)	1.090(0.087)	0.800(0.018)	0.832(0.022)	0.837(0.013)	1.005(0.052)	0.940(0.012)
blood	1.015(0.027)		0.889(0.030) 0.836(0.013) 1.079(0.020)		1.063(0.057)	1.236(0.070)		1.068(0.031) 0.843(0.015) 1.073(0.078)		1.000(0.020)	2.989(0.715)	1.092(0.052)
cancer	1.006(0.004)		0.734(0.008) 0.647(0.006) 0.833(0.000)		0.759(0.003)	0.925(0.006)	1.101(0.032)	0.696(0.004)	0.825(0.013)	0.764(0.007)	10.231(0.562)	1.024(0.005)
california			1.063(0.023) 1.025(0.024) 1.004(0.026) 1.237(0.022)		1.103(0.023)	1.211(0.039)	1.737(0.211)	1.131(0.015)	1.485(0.115)	1.218(0.028)	8.813(1.050)	1.322(0.039)
climate	1.003(0.001) 1.190(0.001)		1.016(0.001)	1.416(0.001)	1.319(0.008)	1.135(0.003)	1.211(0.013)	1.174(0.002)	1.146(0.009)	1.106(0.002)	3.144(0.182)	1.385(0.008)
concrete	0.998(0.004)		0.983(0.004) 0.959(0.013) 1.273(0.007)		1.079(0.012)	1.132(0.005)	1.117(0.039)	1.012(0.015)	1.092(0.015)	1.067(0.008)	1.203(0.024)	1.294(0.033)
slump			0.997(0.002) 0.985(0.004) 0.983(0.003) 1.396(0.016)		1.306(0.008)	1.247(0.017)	1.470(0.016)	1.137(0.016)	1.101(0.023)	1.099(0.013)	1.144(0.015)	1.176(0.005)
sonar	1.008(0.003)		0.914(0.008) 0.849(0.005) 1.164(0.004)		1.038(0.010)	1.148(0.002)	2.220(0.028)	0.875(0.003)	0.951(0.007)	0.948(0.003)	11.477(1.305)	1.128(0.004)
vowel		1.004(0.003) 0.818(0.008) 0.953(0.007)		1.287(0.009)	1.025(0.017)	1.069(0.007)	1.302(0.035)	0.950(0.013)	0.954(0.007)	0.930(0.005)	1.319(0.070)	1.142(0.013)
ecoli	0.970(0.046)		0.940(0.091) 0.919(0.043) 1.289(0.052)		1.117(0.108)	7.372(0.445)	1.322(0.016)		0.977(0.151) 0.874(0.055) 0.858(0.067) 1.813(0.762)			0.966(0.044)
glass	1.024(0.028)		1.001(0.009) 0.952(0.030) 1.289(0.014)		1.125(0.042)	5.949(3.458)	1.620(0.081)	0.988(0.027)		0.983(0.034) 0.958(0.036) 1.556(0.169)		1.093(0.053)
ionosphere	1.001(0.001)	0.900(0.006)	0.882(0.005)	1.200(0.012)	1.030(0.005)	1.116(0.025)		1.556(0.021) 0.873(0.001) 0.905(0.007)		0.879(0.002)	6.221(0.572)	1.095(0.002)
iris	0.984(0.012)	0.654(0.027)	0.724(0.044)	0.872(0.019)	0.714(0.036)	0.706(0.018)		1.232(0.035) 0.626(0.014) 0.682(0.021)		0.662(0.013)	0.684(0.010)	0.702(0.002)
libras	1.013(0.001)	0.654(0.001)		0.487(0.047) 0.449(0.002) 0.443(0.008)		—	2.220(0.010)	0.512(0.007)	0.900(0.145)	0.723(0.013)	33.842(2.719)	0.944(0.004)
parkinsons	1.018(0.005)		0.771(0.016) 0.693(0.011) 0.943(0.027)		0.828(0.032)	1.063(0.003)	1.824(0.058)	0.735(0.020)	0.851(0.009)	0.842(0.005)	6.300(1.089)	0.959(0.016)
planning	1.019(0.004)		1.088(0.021) 0.918(0.012) 1.363(0.013)		1.252(0.039)	1.213(0.014)	1.499(0.038)	1.099(0.019)	1.053(0.022)	1.053(0.012)	1.808(0.007)	1.182(0.020)
qsar	0.999(0.011)		0.869(0.013) 0.847(0.014) 1.162(0.008)		1.048(0.007)	1.233(0.027)	1.817(0.147)	0.940(0.021)	8.595(7.209)	0.967(0.007)	23.244(2.478)	1.249(0.010)
seeds		1.005(0.007) 0.605(0.006) 0.634(0.011)		0.792(0.014)	0.633(0.009)	0.740(0.031)		1.135(0.077) 0.609(0.014) 0.629(0.029)		0.626(0.021)	0.659(0.020)	0.705(0.018)
wine	1.020(0.005)	0.869(0.005)	0.894(0.007)	1.200(0.017)	1.041(0.017)	0.984(0.009)	1.517(0.021)		0.892(0.013) 0.855(0.012) 0.852(0.012) 1.008(0.030)			0.967(0.007)
wine (red)	0.992(0.003)		0.956(0.003) 0.942(0.003) 1.309(0.007)		1.175(0.005)	1.081(0.018)	1.326(0.066)	1.016(0.005)	1.184(0.012)	1.140(0.004)	6.180(1.348)	1.423(0.028)
	wine (white) 1.004(0.003) 0.939(0.005) 0.942(0.005) 1.289(0.002)				1.167(0.010)	1.084(0.006)	1.242(0.007)	1.225(0.005)	1.401(0.136)	1.261(0.011)	5.008(0.353)	1.462(0.024)
yacht			1.010(0.011) 0.963(0.017) 0.955(0.018) 1.325(0.006)		1.088(0.026)	1.179(0.048)	1.147(0.010)	1.040(0.012)	1.038(0.031)	1.050(0.016)	1.070(0.016)	1.192(0.023)
yeast			0.993(0.025) 1.002(0.021) 0.982(0.024) 1.417(0.006)		1.282(0.060)	3.207(0.678)	1.354(0.102)	1.172(0.112)	1.169(0.014)	1.134(0.029)	8.580(0.702)	1.309(0.054)

Table 10: Average RMSE results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.989(0.008)	0.840(0.007)	0.961(0.013)		1.263(0.013) 0.582(0.012)	0.985(0.031)		0.930(0.022) 0.567(0.017) 0.804(0.007)		0.733(0.007)	1.055(0.003)	1.034(0.015)
banknote	1.014(0.030)	0.503(0.038)	0.750(0.025)		0.829(0.012) 0.443(0.046)	0.712(0.029)	0.765(0.015)	0.622(0.100)	0.517(0.040)	0.515(0.038)	0.637(0.008)	0.612(0.033)
blood	1.081(0.035)		0.735(0.048) 0.715(0.059) 0.924(0.068) 0.665(0.054)			1.038(0.103)		0.994(0.076) 0.703(0.044) 0.787(0.035)		0.786(0.043)	1.011(0.120)	0.883(0.029)
cancer	1.016(0.004)		0.560(0.009) 0.441(0.006) 0.490(0.010)		0.502(0.005)	0.737(0.006)	0.932(0.039)	0.560(0.008)	0.529(0.008)	0.529(0.004)	0.681(0.072)	0.604(0.008)
california	1.033(0.037)		0.915(0.046) 0.885(0.038) 1.106(0.018) 0.833(0.053)			1.485(0.179)		1.146(0.016) 0.860(0.053) 1.180(0.082)		0.942(0.047)	3.228(1.162)	1.016(0.054)
climate	1.003(0.010) 1.108(0.004)		1.013(0.011)	1.415(0.003)	1.075(0.007)	1.257(0.010)	1.144(0.037)	1.030(0.010)	1.080(0.013)	1.137(0.008)	1.306(0.005)	1.339(0.009)
concrete	1.011(0.009)	0.998(0.017)	0.890(0.028)		0.888(0.013) 0.545(0.026)	0.841(0.014)	0.930(0.043)	0.662(0.029)	0.622(0.026)	0.596(0.012)	0.835(0.012)	0.808(0.021)
slump	0.968(0.048)	0.889(0.037)	0.928(0.061)	0.954(0.036)	0.897(0.027)	0.960(0.074)		1.045(0.016) 0.847(0.033) 0.867(0.036) 0.897(0.050)			0.895(0.039)	0.860(0.027)
sonar	1.018(0.004)		0.759(0.003) 0.653(0.002) 0.842(0.002)		0.668(0.003)	0.926(0.003)	1.824(0.023)	0.727(0.004)	0.802(0.003)	0.739(0.006)	0.860(0.020)	0.855(0.011)
vowel	1.000(0.014)	0.578(0.011)	0.833(0.009)	1.059(0.006)	0.585(0.020)	0.718(0.023)	0.877(0.005)	0.719(0.027)		0.519(0.010) 0.471(0.008) 0.840(0.016)		0.714(0.008)
ecoli		1.059(0.007) 0.891(0.021) 0.940(0.018)			1.173(0.029) 0.875(0.039)	7.423(0.806)		1.174(0.066) 0.865(0.034) 0.912(0.026) 0.873(0.010) 1.005(0.060)				1.005(0.075)
glass	0.943(0.043)		0.797(0.027) 0.721(0.026) 0.878(0.015) 0.746(0.009)			0.918(0.036)		0.972(0.084) 0.734(0.013) 0.758(0.018)		0.785(0.029)	0.784(0.017)	0.814(0.011)
ionosphere	1.007(0.004)	0.773(0.012)	0.762(0.008)		1.025(0.010) 0.729(0.007)	0.843(0.015)	1.010(0.017)	0.745(0.008)	0.742(0.013)	0.741(0.014)	0.855(0.008)	0.895(0.007)
iris		1.057(0.030) 0.460(0.091) 0.557(0.050)			0.641(0.098) 0.450(0.103)	0.536(0.063)		0.647(0.133) 0.436(0.092) 0.499(0.051) 0.503(0.059) 0.499(0.079)				0.536(0.073)
libras	1.024(0.002)	0.660(0.008)		0.141(0.036) 0.066(0.003) 0.120(0.004)		—	1.065(0.007)	0.307(0.000)	0.228(0.011)	0.214(0.003)	0.425(0.002)	0.470(0.013)
parkinsons	1.029(0.007)		0.616(0.026) 0.561(0.025) 0.656(0.037) 0.566(0.036)			0.881(0.018)	1.036(0.024)	0.597(0.020)	0.679(0.010)	0.678(0.002)	0.607(0.032)	0.614(0.013)
planning	1.019(0.011)		0.709(0.015) 0.563(0.008) 0.622(0.021)		0.618(0.003)	0.908(0.016)	0.910(0.050)	0.753(0.017)	0.763(0.021)	0.798(0.007)	0.698(0.008)	0.747(0.017)
qsar	1.005(0.009)	0.705(0.009)	0.691(0.013)		0.840(0.026) 0.634(0.017)	0.896(0.022)	1.241(0.016)	—		0.654(0.008) 0.634(0.025) 1.659(0.372)		0.800(0.007)
seeds			1.061(0.010) 0.507(0.039) 0.491(0.022) 0.563(0.032) 0.489(0.057)			0.584(0.066)		0.695(0.034) 0.492(0.033) 0.518(0.062) 0.510(0.029) 0.522(0.049)				0.591(0.055)
wine		1.025(0.021) 0.767(0.027) 0.794(0.020)			1.056(0.019) 0.752(0.016)	0.890(0.025)		1.078(0.014) 0.758(0.019) 0.789(0.033)		0.802(0.013)	0.800(0.016)	0.853(0.026)
wine (red)		1.009(0.021) 0.824(0.034) 0.872(0.024)			1.159(0.012) 0.824(0.012)	1.009(0.013)	1.000(0.010)	0.842(0.020)	0.897(0.021)	0.859(0.023)	1.086(0.023)	1.055(0.021)
	wine (white) 1.015(0.013)	0.814(0.018)	0.834(0.020)		1.075(0.019) 0.753(0.014)	1.047(0.012)	0.953(0.006)	0.838(0.030)	0.950(0.019)	0.933(0.020)	1.068(0.027)	1.077(0.017)
yacht	1.018(0.008)	0.882(0.001)	0.875(0.003)		1.042(0.033) 0.679(0.078)	1.025(0.019)	0.921(0.042)	0.772(0.038)	0.844(0.071)	0.845(0.063)	0.889(0.041)	0.915(0.041)
yeast	1.041(0.027)		1.044(0.034) 1.015(0.039) 1.370(0.028)		1.041(0.084)	4.256(0.579)		1.309(0.110) 0.988(0.036) 1.143(0.037)		1.119(0.052)	5.827(1.590)	1.241(0.056)

Table 11: CRPS reults: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.827(0.006) 0.604(0.005) 0.727(0.021)		0.546(0.007)	0.279(0.006)	0.400(0.004)		0.566(0.030) 0.248(0.010) 0.283(0.006)		0.277(0.008)	0.462(0.007)	0.446(0.014)
banknote		0.796(0.013) 0.301(0.005) 0.586(0.007)		0.366(0.005)	0.265(0.006)	0.292(0.005)	0.512(0.035)	0.269(0.003)		0.236(0.002) 0.227(0.004) 0.284(0.005)		0.277(0.004)
blood		0.756(0.008) 0.353(0.005) 0.429(0.009)		0.344(0.004)	0.326(0.010)	0.364(0.009)	0.425(0.002)		0.302(0.006) 0.258(0.006) 0.264(0.012)		0.290(0.006)	0.293(0.008)
cancer				0.748(0.008) 0.308(0.006) 0.241(0.004) 0.187(0.003) 0.248(0.003)		0.280(0.004)	0.348(0.005)	0.329(0.004)	0.254(0.004)	0.263(0.004)	0.252(0.008)	0.243(0.002)
california		0.603(0.001) 0.343(0.001) 0.454(0.001)		0.385(0.006)	0.294(0.001)	0.479(0.015)	0.514(0.018)	0.289(0.003)		0.282(0.001) 0.271(0.002) 0.336(0.003)		0.310(0.002)
climate				0.873(0.005) 0.887(0.005) 0.871(0.005) 0.702(0.007) 0.846(0.007)		0.741(0.001)	0.807(0.009)	0.854(0.007)	0.841(0.007)		0.878(0.008) 0.707(0.002)	0.707(0.003)
concrete		0.837(0.002) 0.713(0.004) 0.710(0.059)			0.432(0.004) 0.268(0.006)	0.341(0.003)	0.628(0.030)			0.319(0.007) 0.267(0.008) 0.265(0.007) 0.377(0.009)		0.358(0.003)
slump				0.838(0.013) 0.727(0.017) 0.681(0.021) 0.466(0.019) 0.607(0.011)		0.516(0.015)	0.588(0.011)	0.649(0.011)	0.594(0.038)	0.643(0.018)	0.534(0.016)	0.560(0.023)
sonar				0.791(0.007) 0.525(0.003) 0.447(0.002) 0.371(0.003) 0.408(0.005)		0.436(0.006)	0.820(0.010)	0.497(0.002)	0.534(0.006)	0.463(0.003)	0.418(0.003)	0.425(0.004)
vowel		0.815(0.001) 0.400(0.004) 0.656(0.004)		0.514(0.004)	0.342(0.005)	0.309(0.009)	0.643(0.006)	0.347(0.008)		0.270(0.003) 0.239(0.004) 0.401(0.003)		0.336(0.002)
ecoli		0.663(0.013) 0.428(0.014) 0.532(0.011)		0.476(0.015)	0.398(0.009)	0.409(0.054)	0.567(0.046)			0.470(0.054) 0.335(0.015) 0.342(0.018) 0.331(0.008)		0.330(0.012)
glass		0.700(0.018) 0.431(0.016) 0.496(0.020)		0.378(0.003)	0.381(0.023)	0.363(0.008) 0.474(0.016)				0.424(0.021) 0.346(0.027) 0.342(0.037) 0.336(0.039)		0.337(0.025)
ionosphere		0.780(0.003) 0.468(0.002) 0.496(0.006)		0.458(0.005)	0.382(0.007)	0.352(0.005) 0.538(0.008)		0.441(0.007)	0.404(0.008)	0.396(0.005)	0.379(0.011)	0.398(0.004)
iris		0.861(0.024) 0.294(0.004) 0.416(0.018)		0.284(0.001)	0.314(0.002)	0.259(0.004)	0.334(0.039)	0.300(0.013)	0.300(0.008)		0.312(0.007) 0.249(0.009)	0.280(0.007)
libras				0.836(0.004) 0.459(0.003) 0.097(0.021) 0.025(0.000) 0.063(0.001)		—	0.278(0.008)	0.190(0.003)	0.143(0.004)	0.117(0.002)	0.237(0.006)	0.185(0.001)
parkinsons				0.743(0.010) 0.276(0.006) 0.286(0.007) 0.229(0.006) 0.242(0.003)		0.277(0.005)	0.380(0.009)	0.314(0.005)	0.279(0.009)	0.257(0.006)	0.248(0.010)	0.240(0.002)
planning				0.777(0.005) 0.496(0.005) 0.356(0.012) 0.238(0.010) 0.412(0.008)		0.476(0.020)	0.464(0.004)	0.524(0.006)	0.492(0.017)	0.509(0.025)	0.363(0.007)	0.396(0.009)
qsar		0.591(0.003) 0.279(0.004) 0.316(0.002)		0.313(0.001)	0.169(0.000)	0.259(0.004)	0.344(0.003)	—	0.166(0.001) 0.169(0.002)		0.296(0.015)	0.243(0.002)
seeds				0.851(0.010) 0.250(0.009) 0.271(0.016) 0.172(0.012) 0.227(0.012)		0.253(0.010)	0.295(0.043)	0.286(0.017)	0.248(0.007)	0.247(0.011)	0.204(0.009)	0.222(0.007)
wine		0.816(0.011) 0.488(0.004) 0.581(0.008)		0.467(0.004)	0.493(0.014)	0.442(0.005) 0.554(0.016)		0.504(0.003)	0.505(0.016)	0.481(0.005)	0.452(0.006)	0.446(0.005)
wine (red)		0.724(0.006) 0.465(0.004) 0.600(0.008)		0.504(0.001)	0.460(0.003)	0.444(0.005)	0.612(0.014)			0.419(0.006) 0.415(0.002) 0.414(0.005) 0.442(0.002)		0.435(0.002)
	wine (white) 0.761(0.001) 0.493(0.001) 0.596(0.001)			0.481(0.002)	0.449(0.004)	0.478(0.004)		0.615(0.005) 0.412(0.001) 0.419(0.002)		0.413(0.003)	0.470(0.002)	0.461(0.002)
yacht		0.764(0.009) 0.590(0.009) 0.631(0.026)		0.456(0.016)	0.307(0.012)	0.403(0.006)	0.564(0.016)		0.521(0.005) 0.284(0.011) 0.327(0.025)		0.385(0.005)	0.376(0.006)
yeast		0.617(0.005) 0.515(0.005) 0.581(0.011)		0.573(0.005)	0.494(0.008)	0.433(0.003) 0.563(0.002)				0.442(0.002) 0.429(0.004) 0.432(0.003) 0.480(0.007)		0.437(0.005)

Table 12: CRPS results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.822(0.003) 0.661(0.002) 0.771(0.004)		0.600(0.003)	0.418(0.002)	0.463(0.002)		0.621(0.038) 0.373(0.002) 0.416(0.003)		0.409(0.000)	0.527(0.004)	0.527(0.003)
banknote		0.797(0.005) 0.389(0.013) 0.637(0.008)		0.434(0.009)	0.377(0.007)	0.375(0.007)	0.545(0.018)			0.359(0.003) 0.331(0.007) 0.331(0.008) 0.382(0.014)		0.366(0.008)
blood		0.749(0.008) 0.402(0.003) 0.482(0.005)		0.385(0.004)	0.394(0.015)	0.413(0.004)	0.478(0.008)		0.437(0.001) 0.322(0.003) 0.346(0.003)		0.359(0.012)	0.369(0.006)
cancer				0.752(0.003) 0.359(0.005) 0.309(0.002) 0.248(0.003) 0.311(0.003)		0.367(0.006)	0.412(0.002)	0.356(0.003)	0.351(0.003)	0.339(0.004)	0.425(0.009)	0.330(0.002)
california		0.603(0.001) 0.421(0.001) 0.505(0.001)		0.430(0.007)	0.377(0.002)	0.459(0.016)	0.463(0.015)	0.357(0.004)		0.349(0.000) 0.334(0.002) 0.410(0.009)		0.366(0.002)
climate				0.869(0.000) 0.831(0.003) 0.868(0.005) 0.697(0.004) 0.803(0.012)		0.750(0.004)	0.783(0.004)	0.840(0.004)	0.805(0.001)	0.814(0.003)	0.707(0.002)	0.727(0.002)
concrete		0.842(0.004) 0.726(0.003) 0.736(0.015)			0.547(0.007) 0.444(0.009)	0.494(0.004)	0.675(0.015)		0.601(0.007) 0.452(0.011) 0.456(0.008)		0.512(0.004)	0.497(0.004)
slump				0.853(0.011) 0.766(0.015) 0.755(0.019) 0.650(0.020) 0.690(0.029)		0.657(0.016)		0.666(0.017) 0.724(0.020)	0.739(0.011)		0.778(0.020) 0.650(0.019)	0.672(0.010)
sonar				0.792(0.004) 0.539(0.003) 0.522(0.003) 0.450(0.002) 0.475(0.002)		0.562(0.007)	1.050(0.005)	0.523(0.002)	0.557(0.007)	0.523(0.003)	0.564(0.008)	0.478(0.001)
vowel		0.818(0.002) 0.470(0.003) 0.714(0.002)		0.571(0.003)	0.487(0.007)	0.462(0.001)	0.667(0.014)	0.604(0.004)		0.429(0.004) 0.401(0.004) 0.498(0.001)		0.452(0.002)
ecoli		0.665(0.001) 0.437(0.009) 0.567(0.008)		0.481(0.004)	0.398(0.012)	0.558(0.033)	0.563(0.036)				0.445(0.055) 0.372(0.007) 0.384(0.006) 0.368(0.007)	0.372(0.004)
glass		0.692(0.008) 0.477(0.009) 0.554(0.010)		0.483(0.008)	0.430(0.025)	0.436(0.028)	0.541(0.008)			0.434(0.014) 0.410(0.018) 0.401(0.025) 0.409(0.018)		0.397(0.018)
ionosphere		0.774(0.003) 0.445(0.004) 0.539(0.003)				0.479(0.003) 0.421(0.008) 0.424(0.007)	0.570(0.009)	0.451(0.002)		0.432(0.004) 0.424(0.001) 0.485(0.014)		0.426(0.003)
iris		0.845(0.015) 0.342(0.011) 0.470(0.018)		0.311(0.005)	0.350(0.034)	0.302(0.010)	0.376(0.019)	0.308(0.014)	0.329(0.018)		0.335(0.012) 0.282(0.010)	0.300(0.002)
libras				0.840(0.002) 0.442(0.003) 0.087(0.006) 0.044(0.001) 0.106(0.000)		—	0.902(0.007)	0.225(0.000)	0.231(0.001)	0.199(0.003)	0.650(0.053)	0.278(0.003)
parkinsons				0.746(0.005) 0.339(0.002) 0.339(0.005) 0.264(0.002) 0.303(0.004)		0.411(0.005)	0.461(0.009)	0.347(0.002)	0.382(0.014)	0.370(0.011)	0.431(0.030)	0.329(0.001)
planning				0.782(0.006) 0.606(0.006) 0.514(0.015) 0.403(0.005) 0.547(0.013)		0.608(0.007)	0.560(0.005)	0.582(0.012)	0.585(0.017)	0.610(0.013)	0.516(0.013)	0.532(0.012)
qsar		0.592(0.001) 0.311(0.003) 0.374(0.003)		0.367(0.003)	0.246(0.004)	0.351(0.002)	0.426(0.003)	0.210(nan)	0.262(0.001)	0.246(0.003)	0.588(0.071)	0.335(0.002)
seeds				0.834(0.011) 0.295(0.010) 0.339(0.007) 0.220(0.007) 0.284(0.006)		0.313(0.014)	0.332(0.017)	0.313(0.004)	0.333(0.020)	0.308(0.008)	0.278(0.004)	0.276(0.001)
wine				0.819(0.008) 0.527(0.005) 0.622(0.009) 0.491(0.001) 0.536(0.020)		0.499(0.010)	0.609(0.020)	0.523(0.009)	0.539(0.005)	0.539(0.011)	0.504(0.004)	0.486(0.005)
wine (red)		0.721(0.003) 0.529(0.001) 0.633(0.008)		0.539(0.002)	0.575(0.004)	0.499(0.001)	0.614(0.006)	0.502(0.005)		0.489(0.002) 0.486(0.001) 0.508(0.001)		0.500(0.001)
	wine (white) 0.762(0.001) 0.560(0.001) 0.648(0.001)			0.536(0.001)	0.564(0.006)	0.516(0.001)	0.602(0.006)			0.533(0.006) 0.500(0.003) 0.503(0.003) 0.533(0.002)		0.527(0.001)
yacht		0.754(0.006) 0.611(0.015) 0.666(0.028)			0.539(0.004) 0.375(0.021)	0.545(0.011)	0.576(0.012)	0.577(0.009)	0.514(0.006)	0.602(0.013)	0.505(0.010)	0.545(0.005)
yeast		0.611(0.005) 0.499(0.003) 0.587(0.005)		0.584(0.003)	0.504(0.005)	0.446(0.003)	0.528(0.012)		0.467(0.007) 0.427(0.004) 0.437(0.004)		0.497(0.007)	0.443(0.002)

Table 13: CRPS results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.822(0.004)	0.750(0.007)	0.778(0.019)	0.687(0.002)	0.543(0.008)	0.564(0.009)	0.622(0.006)	0.635(0.008)		0.531(0.003) 0.518(0.004) 0.574(0.004)		0.586(0.008)
banknote	0.796(0.002)	0.471(0.013)	0.673(0.013)	0.478(0.004)	0.476(0.012)	0.472(0.022)	0.577(0.018)			0.548(0.017) 0.429(0.017) 0.428(0.008) 0.462(0.017)		0.440(0.013)
blood	0.749(0.006)	0.496(0.005)		0.567(0.015) 0.445(0.005) 0.519(0.024)		0.592(0.041)	0.499(0.010)	0.481(0.015)	0.475(0.004)	0.473(0.010)	0.537(0.039)	0.461(0.011)
cancer	0.749(0.001)	0.423(0.003)		0.417(0.001) 0.345(0.002) 0.382(0.001)		0.474(0.003)	0.448(0.005)	0.413(0.003)	0.454(0.007)	0.424(0.003)	2.270(0.135)	0.438(0.003)
california	0.603(0.001)	0.497(0.001)	0.546(0.002)	0.455(0.001)	0.428(0.004)	0.434(0.006)	0.550(0.039)	0.420(0.002)		0.402(0.003) 0.398(0.002) 0.655(0.019)		0.428(0.003)
climate	0.868(0.001)	0.806(0.004)		0.850(0.004) 0.699(0.001) 0.790(0.005)		0.760(0.003)	0.709(0.008)	0.823(0.002)	0.757(0.002)	0.771(0.002)	0.912(0.025)	0.721(0.005)
concrete	0.838(0.003)	0.736(0.003)	0.766(0.007)	0.613(0.005)	0.599(0.011)	0.605(0.002)	0.661(0.003)			0.695(0.011) 0.593(0.007) 0.596(0.005) 0.589(0.009)		0.609(0.011)
slump	0.841(0.012)	0.802(0.005)		0.772(0.013) 0.677(0.011) 0.771(0.028)		0.764(0.018)	0.703(0.008)	0.762(0.020)	0.767(0.020)	0.770(0.020)	0.719(0.015)	0.713(0.002)
sonar	0.790(0.002)	0.591(0.005)		0.615(0.011) 0.542(0.002) 0.571(0.002)		0.658(0.002)	1.083(0.001)	0.592(0.002)	0.592(0.005)	0.607(0.004)	3.594(0.406)	0.559(0.002)
vowel	0.819(0.002)	0.595(0.004)	0.743(0.004)	0.620(0.005)	0.589(0.013)	0.599(0.001)	0.668(0.014)	0.665(0.009)		0.567(0.006) 0.553(0.004) 0.594(0.010)		0.556(0.002)
ecoli	0.664(0.006)	0.437(0.017)	0.584(0.009)	0.517(0.017)	0.456(0.011)	0.801(0.039)	0.516(0.009)			0.446(0.039) 0.411(0.004) 0.408(0.013) 0.423(0.005)		0.413(0.008)
glass	0.697(0.018)	0.532(0.003)	0.630(0.010)	0.555(0.010)	0.526(0.018)	0.829(0.201)	0.625(0.014)	0.514(0.022)		0.492(0.007) 0.483(0.004) 0.506(0.003)		0.488(0.012)
ionosphere		0.778(0.000) 0.475(0.006) 0.609(0.010)		0.546(0.003)	0.491(0.008)	0.612(0.015)	0.680(0.014)	0.519(0.004)	0.517(0.010)	0.515(0.011)	1.542(0.101)	0.506(0.006)
iris	0.844(0.019)	0.422(0.016)		0.529(0.032) 0.346(0.006) 0.387(0.003)		0.362(0.008)	0.453(0.015)	0.353(0.010)	0.431(0.014)	0.399(0.008)	0.381(0.009)	0.366(0.006)
libras	0.845(0.001)	0.467(0.003)		0.293(0.027) 0.178(0.000) 0.224(0.002)		—	1.163(0.008)	0.322(0.004)	0.396(0.004)	0.401(0.011)	11.817(0.966)	0.384(0.002)
parkinsons	0.741(0.001)	0.458(0.006)		0.452(0.009) 0.402(0.011) 0.434(0.012)		0.557(0.007)	0.755(0.005)	0.443(0.011)	0.485(0.004)	0.506(0.006)	1.064(0.118)	0.446(0.010)
planning	0.797(0.004)	0.711(0.013)		0.676(0.002) 0.645(0.006) 0.715(0.024)		0.714(0.013)	0.693(0.012)	0.707(0.015)	0.683(0.010)	0.684(0.007)	0.699(0.013)	0.658(0.008)
qsar	0.593(0.002)	0.383(0.002)	0.455(0.004)		0.466(0.005) 0.358(0.003)	0.493(0.013)	0.546(0.021)	0.392(0.002)	0.520(0.099)	0.376(0.002)	4.953(0.822)	0.469(0.004)
seeds	0.854(0.007)	0.379(0.012)		0.466(0.010) 0.346(0.013) 0.359(0.004)		0.410(0.015)	0.488(0.029)	0.388(0.007)	0.418(0.015)	0.389(0.012)	0.379(0.004)	0.360(0.007)
wine	0.826(0.006)	0.559(0.008)	0.677(0.007)	0.559(0.011)	0.592(0.022)	0.543(0.005)	0.712(0.016)	0.586(0.005)	0.573(0.015)	0.564(0.013)	0.560(0.007)	0.524(0.001)
wine (red)	0.723(0.001)	0.597(0.005)	0.676(0.002)	0.584(0.003)	0.615(0.007)	0.583(0.011)	0.618(0.008)	0.639(0.004)		0.564(0.000) 0.558(0.003) 0.794(0.050)		0.581(0.007)
wine (white) 0.763(0.000)		0.633(0.002)	0.696(0.004)	0.589(0.002)	0.613(0.001)	0.582(0.004) 0.608(0.003)				0.605(0.001) 0.577(0.005) 0.579(0.003) 0.760(0.016)		0.606(0.005)
yacht	0.734(0.008)	0.638(0.014)	0.662(0.018)		0.629(0.005) 0.568(0.009)	0.629(0.020)	0.581(0.010)	0.659(0.015)	0.604(0.009)		0.653(0.006) 0.573(0.004)	0.634(0.004)
yeast	0.614(0.003)	0.515(0.003)	0.589(0.004)	0.594(0.013)	0.520(0.014)	0.539(0.023)	0.540(0.010)		0.533(0.021) 0.456(0.002) 0.463(0.007)		0.643(0.025)	0.474(0.004)

Table 14: CRPS results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.840(0.020)	0.650(0.022)	0.774(0.021)	0.559(0.007)	0.323(0.021)	0.401(0.023)		0.538(0.016) 0.276(0.013) 0.306(0.009)		0.293(0.012)	0.463(0.013)	0.444(0.017)
banknote	0.852(0.041)	0.336(0.024)	0.617(0.024)	0.413(0.003)	0.294(0.029)	0.347(0.029)	0.467(0.029)			0.405(0.083) 0.251(0.016) 0.253(0.013) 0.308(0.007)		0.290(0.014)
blood	0.844(0.067)	0.448(0.041)	0.534(0.067)	0.380(0.029)	0.390(0.030)	0.517(0.077)	0.473(0.030)			0.469(0.040) 0.313(0.024) 0.336(0.025) 0.323(0.015)		0.355(0.013)
cancer	0.897(0.002)	0.413(0.004)		0.314(0.003) 0.231(0.004) 0.341(0.005)		0.462(0.003)	0.507(0.029)	0.402(0.005)	0.355(0.005)	0.354(0.002)	0.317(0.016)	0.298(0.003)
california	0.596(0.028)	0.386(0.037)	0.462(0.036)	0.403(0.026)	0.329(0.031)	0.453(0.032)	0.482(0.038)	0.315(0.026)		0.312(0.041) 0.284(0.026) 0.370(0.061)		0.323(0.029)
climate	0.867(0.006)	0.877(0.006)		0.866(0.009) 0.695(0.004) 0.843(0.004)		0.746(0.013)	0.797(0.011)	0.836(0.010)	0.824(0.009)	0.820(0.003)	0.703(0.005)	0.717(0.006)
concrete	0.854(0.009)	0.729(0.016)	0.678(0.019)		0.461(0.020) 0.312(0.010)	0.389(0.013)	0.561(0.022)			0.383(0.045) 0.309(0.015) 0.318(0.009) 0.404(0.009)		0.387(0.012)
slump	0.853(0.055)	0.742(0.040)		0.779(0.077) 0.482(0.027) 0.691(0.003)		0.567(0.051)	0.607(0.009)	0.695(0.035)	0.626(0.019)	0.690(0.044)	0.561(0.019)	0.563(0.020)
sonar	0.853(0.011)	0.605(0.009)		0.529(0.007) 0.439(0.005) 0.493(0.008)		0.613(0.014)	0.945(0.008)	0.559(0.008)	0.633(0.013)	0.559(0.013)	0.496(0.005)	0.500(0.003)
vowel	0.828(0.004)	0.439(0.006)	0.681(0.007)	0.532(0.002)	0.415(0.014)	0.380(0.014)	0.627(0.001)	0.439(0.015)		0.316(0.004) 0.292(0.004) 0.430(0.005)		0.360(0.003)
ecoli	0.661(0.034)	0.420(0.025)	0.561(0.021)	0.464(0.015)	0.405(0.008)	0.934(0.192)	0.496(0.022)			0.410(0.016) 0.377(0.033) 0.367(0.037) 0.365(0.018)		0.369(0.015)
glass	0.692(0.045)	0.445(0.022)	0.526(0.043)	0.415(0.024)	0.415(0.019)	0.428(0.010)	0.467(0.046)	0.470(0.020)	0.393(0.015)		0.420(0.018) 0.330(0.011)	0.368(0.015)
ionosphere	0.785(0.013)	0.440(0.021)	0.474(0.007)	0.427(0.009)	0.348(0.011)	0.361(0.014)	0.471(0.019)	0.421(0.011)	0.377(0.018)		0.364(0.013) 0.333(0.009)	0.355(0.014)
iris				0.948(0.035) 0.286(0.066) 0.457(0.057) 0.296(0.054) 0.313(0.074)				0.259(0.046) 0.272(0.046) 0.303(0.068) 0.315(0.042)			0.319(0.044) 0.260(0.061)	0.270(0.053)
libras	0.851(0.005)	0.539(0.008)		0.064(0.012) 0.027(0.001) 0.070(0.002)		—	0.482(0.002)	0.196(0.002)	0.149(0.006)	0.137(0.003)	0.206(0.003)	0.199(0.004)
parkinsons	0.912(0.015)	0.446(0.015)		0.409(0.019) 0.290(0.013) 0.369(0.020)		0.581(0.016)	0.547(0.015)	0.436(0.021)	0.495(0.008)	0.474(0.006)	0.335(0.019)	0.328(0.007)
planning	0.827(0.013)	0.526(0.004)		0.415(0.006) 0.284(0.008) 0.435(0.002)		0.543(0.013)	0.473(0.033)	0.563(0.014)	0.533(0.017)	0.568(0.002)	0.392(0.005)	0.432(0.007)
qsar	0.621(0.025)	0.344(0.016)	0.374(0.011)	0.359(0.006)	0.253(0.003)	0.345(0.011)	0.459(0.026)	—		0.218(0.014) 0.227(0.007) 0.368(0.029)		0.294(0.005)
seeds	0.993(0.007)	0.294(0.038)		0.340(0.031) 0.199(0.016) 0.279(0.035)		0.287(0.038)	0.268(0.007)	0.320(0.033)	0.284(0.037)	0.285(0.023)	0.237(0.035)	0.248(0.025)
wine	0.836(0.009)	0.498(0.026)	0.592(0.019)	0.477(0.018)	0.490(0.014)	0.469(0.021)	0.576(0.005)	0.504(0.012)	0.509(0.016)		0.509(0.010) 0.447(0.016)	0.455(0.014)
wine (red)	0.742(0.010)	0.514(0.007)	0.635(0.007)	0.541(0.006)	0.559(0.010)	0.479(0.006)	0.616(0.010)			0.486(0.003) 0.442(0.004) 0.445(0.007) 0.475(0.003)		0.460(0.005)
	wine (white) 0.767(0.003)	0.509(0.004)	0.595(0.006)	0.488(0.007)	0.473(0.003)	0.482(0.009)	0.587(0.014)			0.462(0.015) 0.428(0.005) 0.424(0.007) 0.470(0.004)		0.461(0.006)
yacht	0.761(0.023)	0.574(0.004)	0.581(0.015)		0.495(0.024) 0.279(0.039)	0.437(0.016)	0.518(0.007)	0.526(0.034)	0.329(0.015)	0.400(0.016)	0.392(0.013)	0.408(0.014)
yeast	0.569(0.007)	0.493(0.008)	0.549(0.003)	0.556(0.005)	0.486(0.026)	0.521(0.038)	0.503(0.014)		0.446(0.020) 0.399(0.003) 0.411(0.010)		0.462(0.005)	0.416(0.004)

Table 15: Energy MMD results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.016(0.000) 0.010(0.000) 0.014(0.001)		0.002(0.000)	0.001(0.000)	0.001(0.000)	0.019(0.002)		0.001(0.000) 0.000(0.000) 0.000(0.000)		0.001(0.000)	0.001(0.000)
banknote		0.015(0.000) 0.001(0.000) 0.009(0.000)		0.001(0.000)	0.000(0.000)	0.001(0.000)	0.013(0.004)	0.001(0.000)	0.000(0.000)	0.000(0.000) 0.001(0.000)		0.001(0.000)
blood		0.017(0.000) 0.003(0.000) 0.007(0.000)		0.002(0.000)	0.001(0.000)	0.002(0.000)	0.017(0.003)		0.002(0.000) 0.001(0.000) 0.001(0.000)		0.001(0.000)	0.001(0.000)
cancer				0.033(0.000) 0.005(0.000) 0.004(0.000) 0.002(0.000) 0.003(0.000)		0.005(0.000)	0.009(0.000)	0.007(0.000)	0.004(0.000)	0.004(0.000)	0.004(0.000)	0.003(0.000)
california		0.014(0.000) 0.005(0.000) 0.008(0.000)		0.002(0.000)	0.001(0.000)	0.017(0.003)	0.022(0.006)	0.001(0.000)	0.000(0.000)	0.000(0.000) 0.000(0.000)		0.000(0.000)
climate				0.021(0.000) 0.014(0.000) 0.020(0.000) 0.005(0.000) 0.018(0.000)		0.011(0.001)	0.018(0.000)	0.020(0.000)	0.016(0.000)	0.012(0.000)	0.007(0.000)	0.006(0.000)
concrete		0.018(0.001) 0.015(0.001) 0.016(0.002)		0.002(0.000)	0.001(0.000)	0.002(0.000)	0.016(0.003)		0.002(0.000) 0.001(0.000) 0.001(0.000) 0.002(0.000)			0.002(0.000)
slump				0.025(0.001) 0.021(0.001) 0.021(0.002) 0.011(0.001) 0.014(0.001)		0.014(0.002)	0.020(0.001)	0.019(0.001)	0.013(0.001)	0.014(0.001)	0.012(0.001)	0.014(0.001)
sonar		0.053(0.002) 0.027(0.001) 0.018(0.000)		0.016(0.000)	0.014(0.000)	0.025(0.001)	0.132(0.004)	0.026(0.001)	0.029(0.001)	0.019(0.000)	0.017(0.000)	0.016(0.000)
vowel		0.017(0.000) 0.004(0.000) 0.013(0.000)		0.003(0.000)	0.003(0.000)	0.002(0.000)	0.015(0.002)	0.002(0.000)	0.001(0.000)	0.001(0.000) 0.002(0.000)		0.001(0.000)
ecoli		0.021(0.001) 0.005(0.000) 0.014(0.000)		0.007(0.001)	0.004(0.000)	0.004(0.001)	0.018(0.002)	0.023(0.014)	0.003(0.000)		0.003(0.000) 0.003(0.000)	0.003(0.000)
glass		0.022(0.001) 0.007(0.001) 0.013(0.001)		0.008(0.000)	0.005(0.001)	0.006(0.001)	0.014(0.000)		0.008(0.001) 0.005(0.000) 0.004(0.000) 0.004(0.000)			0.005(0.000)
ionosphere		0.038(0.001) 0.010(0.001) 0.012(0.000)		0.015(0.001)	0.007(0.000)	0.007(0.000)	0.018(0.001)	0.011(0.000)	0.007(0.000)	0.007(0.000)	0.007(0.000)	0.007(0.000)
iris		0.030(0.001) 0.003(0.000) 0.009(0.000)		0.004(0.000)	0.003(0.000)	0.004(0.001)	0.015(0.003)	0.004(0.000)	0.003(0.000)	0.003(0.000)	0.003(0.000)	0.003(0.000)
libras				0.066(0.001) 0.029(0.000) 0.005(0.002) 0.001(0.000) 0.001(0.000)		—	0.018(0.001)	0.007(0.000)	0.004(0.000)	0.003(0.000)	0.017(0.001)	0.009(0.000)
parkinsons		0.035(0.001) 0.008(0.000) 0.009(0.000)		0.007(0.000)	0.006(0.000)	0.009(0.000)	0.016(0.000)	0.010(0.000)	0.008(0.000)		0.007(0.000) 0.007(0.000)	0.007(0.000)
planning				0.025(0.001) 0.010(0.000) 0.008(0.000) 0.005(0.000) 0.007(0.001)		0.012(0.001)	0.016(0.001)	0.014(0.000)	0.011(0.000)	0.010(0.000)	0.007(0.000)	0.008(0.000)
qsar		0.025(0.001) 0.007(0.000) 0.008(0.001)		0.005(0.000)	0.002(0.000)	0.005(0.000)	0.008(0.000)	—	0.002(0.000)	0.002(0.000) 0.004(0.000)		0.003(0.000)
seeds				0.033(0.001) 0.003(0.000) 0.005(0.001) 0.002(0.000) 0.002(0.000)		0.003(0.000)	0.009(0.001)	0.004(0.001)	0.003(0.000)	0.002(0.000)	0.002(0.000)	0.003(0.000)
wine		0.030(0.001) 0.010(0.001) 0.016(0.001)		0.010(0.000)	0.009(0.000)	0.009(0.001) 0.018(0.001)		0.012(0.001)	0.011(0.001)	0.010(0.001)	0.009(0.000)	0.009(0.001)
wine (red)		0.015(0.000) 0.005(0.000) 0.011(0.000)		0.002(0.000)	0.003(0.000)	0.002(0.000)	0.016(0.003)	0.002(0.000)	0.002(0.000)	0.002(0.000)	0.001(0.000)	0.001(0.000)
	wine (white) 0.016(0.000) 0.007(0.000) 0.010(0.000)			0.001(0.000)	0.003(0.000)	0.001(0.000)	0.016(0.002)	0.001(0.000)	0.001(0.000)	0.000(0.000) 0.001(0.000)		0.001(0.000)
yacht		0.020(0.001) 0.013(0.000) 0.018(0.001)		0.004(0.000)	0.003(0.000)	0.004(0.000)	0.017(0.001)		0.011(0.000) 0.002(0.000) 0.002(0.001) 0.004(0.000)			0.003(0.000)
yeast		0.014(0.000) 0.005(0.000) 0.013(0.000)		0.004(0.000)	0.004(0.000)	0.001(0.000)	0.013(0.000)	0.006(0.000)	0.001(0.000)	0.001(0.000)	0.002(0.000)	0.001(0.000)

Table 16: Energy MMD results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.064(0.001) 0.046(0.001) 0.061(0.001)		0.010(0.000)	0.006(0.000)	0.005(0.001)	0.062(0.009)	0.006(0.000)	0.002(0.000)	0.002(0.000) 0.004(0.001)		0.004(0.000)
banknote		0.056(0.001) 0.008(0.000) 0.038(0.001)		0.005(0.000)	0.002(0.000)	0.003(0.000)	0.044(0.003)	0.004(0.001)	0.001(0.000)	0.001(0.000) 0.003(0.001)		0.002(0.000)
blood		0.056(0.001) 0.010(0.000) 0.030(0.001)		0.006(0.000)	0.007(0.001)	0.006(0.001)	0.066(0.008)		0.016(0.001) 0.002(0.000) 0.003(0.001)		0.003(0.000)	0.004(0.001)
cancer				0.133(0.002) 0.018(0.001) 0.022(0.002) 0.006(0.000) 0.009(0.000)		0.025(0.002)	0.144(0.025)	0.024(0.001)	0.018(0.001)	0.018(0.001)	0.043(0.005)	0.015(0.001)
california		0.066(0.001) 0.037(0.001) 0.045(0.000)		0.007(0.001)	0.004(0.000)	0.025(0.006)	0.095(0.005)	0.005(0.000)	0.000(0.000)	0.000(0.000) 0.001(0.000)		0.001(0.000)
climate				0.090(0.000) 0.042(0.000) 0.087(0.000) 0.009(0.000) 0.043(0.002)		0.053(0.001)	0.096(0.005)	0.071(0.000)	0.049(0.000)	0.045(0.000)	0.020(0.001)	0.012(0.001)
concrete		0.073(0.001) 0.059(0.001) 0.067(0.004)		0.008(0.000)	0.009(0.001)	0.014(0.002)	0.061(0.006)	0.031(0.001)	0.007(0.001)	0.007(0.000)	0.008(0.001)	0.006(0.000)
slump				0.089(0.004) 0.071(0.005) 0.079(0.003) 0.033(0.000) 0.051(0.005)		0.052(0.002)	0.070(0.006)	0.059(0.002)	0.040(0.003)	0.049(0.006)	0.034(0.002)	0.036(0.001)
sonar				0.189(0.004) 0.077(0.002) 0.065(0.002) 0.035(0.000) 0.034(0.001)		0.111(0.007)	0.662(0.007)	0.078(0.002)	0.092(0.004)	0.065(0.001)	0.074(0.005)	0.036(0.001)
vowel		0.080(0.000) 0.020(0.000) 0.064(0.001)		0.007(0.000)	0.011(0.000)	0.010(0.001)	0.098(0.008)	0.033(0.000)	0.008(0.000)	0.007(0.000)	0.007(0.000)	0.004(0.000)
ecoli		0.087(0.002) 0.019(0.002) 0.069(0.004)		0.015(0.001)	0.014(0.001)	0.020(0.002)	0.133(0.029)	0.034(0.006)	0.010(0.001)	0.011(0.002)	0.008(0.000)	0.007(0.000)
glass		0.076(0.001) 0.022(0.001) 0.060(0.003)		0.033(0.003)	0.016(0.002)	0.015(0.002)	0.078(0.003)	0.026(0.001)	0.016(0.002)	0.017(0.002)	0.013(0.002)	0.012(0.001)
ionosphere		0.131(0.002) 0.024(0.001) 0.043(0.001)		0.036(0.001)	0.017(0.001)	0.026(0.002)	0.082(0.001)	0.031(0.001)	0.021(0.001)	0.020(0.001)	0.038(0.003)	0.018(0.001)
iris		0.100(0.004) 0.013(0.002) 0.031(0.003)		0.011(0.000)	0.014(0.005)	0.012(0.002)	0.053(0.005)	0.011(0.003)	0.011(0.003)	0.011(0.002)	0.009(0.002)	0.008(0.001)
libras				0.245(0.002) 0.085(0.001) 0.009(0.003) 0.002(0.000) 0.004(0.000)		—	0.687(0.013)	0.022(0.001)	0.015(0.000)	0.010(0.000)	0.375(0.060)	0.046(0.001)
parkinsons				0.124(0.002) 0.025(0.001) 0.036(0.001) 0.015(0.000) 0.017(0.001)		0.044(0.005)	0.116(0.018)	0.030(0.001)	0.032(0.003)	0.035(0.003)	0.029(0.003)	0.019(0.001)
planning				0.091(0.002) 0.037(0.001) 0.041(0.002) 0.015(0.002) 0.027(0.001)		0.048(0.006)	0.065(0.009)	0.046(0.001)	0.036(0.001)	0.047(0.005)	0.025(0.000)	0.029(0.002)
qsar		0.114(0.002) 0.031(0.001) 0.046(0.001)		0.014(0.000)	0.011(0.000)	0.025(0.001)	0.230(0.073)	0.022(nan)		0.009(0.000) 0.008(0.001) 0.079(0.022)		0.013(0.001)
seeds				0.108(0.005) 0.011(0.000) 0.023(0.002) 0.005(0.000) 0.008(0.000)		0.012(0.003)	0.058(0.015)	0.012(0.001)	0.015(0.004)	0.013(0.003)	0.009(0.001)	0.007(0.001)
wine				0.110(0.004) 0.030(0.003) 0.058(0.004) 0.021(0.001) 0.025(0.002)		0.031(0.004)	0.061(0.004)	0.035(0.003)	0.036(0.004)	0.042(0.005)	0.030(0.001)	0.025(0.002)
wine (red)		0.068(0.001) 0.024(0.001) 0.055(0.001)		0.007(0.000)	0.016(0.001)	0.014(0.002)	0.093(0.009)	0.007(0.001)	0.005(0.000)	0.006(0.000)	0.004(0.000)	0.004(0.001)
	wine (white) 0.076(0.000) 0.041(0.000) 0.055(0.001)			0.003(0.000)	0.009(0.000)	0.008(0.001)	0.140(0.022)	0.003(0.000)	0.002(0.000)	0.001(0.000) 0.002(0.000)		0.003(0.000)
yacht				0.081(0.004) 0.054(0.004) 0.074(0.004) 0.010(0.001) 0.014(0.002)		0.022(0.002)	0.079(0.008)	0.041(0.001)	0.016(0.002)	0.019(0.001)	0.016(0.001)	0.015(0.001)
yeast		0.063(0.001) 0.024(0.001) 0.059(0.001)		0.015(0.003)	0.014(0.001)	0.006(0.000)	0.109(0.029)	0.043(0.012)	0.005(0.000)	0.005(0.000)	0.004(0.000)	0.003(0.000)

Table 17: Energy MMD results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.141(0.001) 0.119(0.004) 0.137(0.004)		0.040(0.001)	0.020(0.003)	0.044(0.017)	0.131(0.008)		0.050(0.001) 0.010(0.001) 0.009(0.002) 0.015(0.000)			0.013(0.003)
banknote		0.109(0.001) 0.023(0.001) 0.084(0.004)		0.008(0.000)	0.009(0.001)	0.021(0.008)	0.132(0.027)	0.024(0.002)		0.006(0.000) 0.005(0.000) 0.008(0.001)		0.006(0.000)
blood				0.108(0.003) 0.024(0.003) 0.072(0.004) 0.012(0.001) 0.026(0.002)		0.077(0.032)	0.122(0.039)	0.035(0.002)	0.023(0.001)	0.017(0.005)	0.015(0.002)	0.012(0.003)
cancer				0.355(0.004) 0.056(0.002) 0.097(0.002) 0.019(0.000) 0.031(0.002)		0.083(0.004)	0.240(0.049)	0.057(0.003)	0.076(0.003)	0.083(0.002)	3.372(0.144)	0.060(0.007)
california		0.171(0.000) 0.128(0.000) 0.142(0.002)		0.012(0.000)	0.017(0.001)	0.019(0.001)	0.874(0.268)		0.009(0.001) 0.002(0.000) 0.002(0.000) 0.043(0.002)			0.006(0.000)
climate				0.283(0.001) 0.113(0.005) 0.257(0.001) 0.017(0.002) 0.075(0.002)		0.171(0.012)	0.497(0.011)	0.127(0.003)	0.119(0.006)	0.162(0.003)	0.187(0.032)	0.039(0.001)
concrete				0.183(0.002) 0.148(0.001) 0.171(0.004) 0.018(0.001) 0.040(0.006)		0.054(0.006)	0.182(0.007)	0.069(0.001)	0.037(0.003)	0.032(0.003)	0.025(0.002)	0.024(0.002)
slump				0.197(0.007) 0.150(0.007) 0.179(0.007) 0.055(0.002) 0.147(0.013)		0.168(0.015)	0.180(0.013)	0.139(0.006)	0.098(0.009)	0.100(0.008)	0.087(0.008)	0.070(0.007)
sonar				0.528(0.007) 0.192(0.008) 0.271(0.014) 0.084(0.001) 0.132(0.002)		0.275(0.017)	1.699(0.043)	0.198(0.005)	0.241(0.011)	0.275(0.013)	13.829(1.999)	0.096(0.003)
vowel				0.222(0.001) 0.075(0.002) 0.197(0.003) 0.016(0.001) 0.043(0.005)		0.029(0.002)	0.539(0.071)	0.068(0.002)	0.040(0.002)	0.039(0.005)	0.023(0.002)	0.017(0.001)
ecoli		0.205(0.002) 0.058(0.006) 0.169(0.006)			0.042(0.002) 0.040(0.004)	0.066(0.006)	0.449(0.019)	0.066(0.008)	0.047(0.014)	0.043(0.008)	0.044(0.013)	0.034(0.007)
glass		0.203(0.013) 0.080(0.004) 0.154(0.011)		0.072(0.001)	0.068(0.001)	0.134(0.051)	0.654(0.034)	0.073(0.007)	0.068(0.012)		0.073(0.011) 0.057(0.009)	0.052(0.009)
ionosphere		0.359(0.004) 0.062(0.004) 0.162(0.019)			0.078(0.001) 0.050(0.003)	0.144(0.013)	0.457(0.040)	0.081(0.003)	0.094(0.006)	0.114(0.012)	2.248(0.242)	0.065(0.001)
iris				0.176(0.005) 0.049(0.009) 0.079(0.015) 0.027(0.003) 0.038(0.007)		0.031(0.005)		0.231(0.020) 0.027(0.001) 0.055(0.007)		0.048(0.006)	0.039(0.008)	0.031(0.004)
libras				0.682(0.008) 0.146(0.001) 0.159(0.047) 0.016(0.000) 0.019(0.001)		—	1.986(0.029)	0.061(0.001)	0.095(0.006)	0.118(0.009)	71.196(6.348)	0.101(0.001)
parkinsons				0.327(0.001) 0.098(0.004) 0.125(0.004) 0.047(0.004) 0.077(0.012)		0.134(0.003)	0.663(0.039)	0.084(0.002)	0.120(0.005)	0.165(0.006)	0.466(0.115)	0.066(0.006)
planning				0.264(0.011) 0.132(0.015) 0.175(0.009) 0.061(0.011) 0.118(0.025)		0.124(0.008)	0.316(0.074)	0.122(0.013)	0.126(0.015)	0.142(0.009)	0.075(0.013)	0.079(0.013)
qsar		0.348(0.004) 0.113(0.005) 0.191(0.006)			0.048(0.004) 0.043(0.002)	0.101(0.026)	2.045(0.400)	0.084(0.002)	0.064(0.015)	0.051(0.005)	14.435(2.802)	0.081(0.008)
seeds				0.232(0.003) 0.047(0.001) 0.078(0.007) 0.018(0.003) 0.029(0.002)		0.047(0.009)	0.238(0.042)	0.040(0.005)	0.050(0.007)	0.045(0.007)	0.036(0.004)	0.029(0.003)
wine				0.300(0.008) 0.099(0.011) 0.186(0.017) 0.047(0.004) 0.087(0.008)		0.075(0.006)	0.213(0.023)	0.090(0.010)	0.125(0.021)	0.127(0.014)	0.091(0.010)	0.071(0.008)
wine (red)				0.197(0.001) 0.089(0.004) 0.173(0.003) 0.015(0.001) 0.035(0.002)		0.092(0.006)	0.512(0.050)	0.062(0.001)	0.023(0.002)	0.021(0.001)	0.107(0.035)	0.029(0.005)
				wine (white) 0.224(0.001) 0.152(0.002) 0.189(0.002) 0.006(0.000) 0.023(0.001)		0.065(0.003)	0.496(0.010)	0.010(0.000)	0.011(0.003)	0.008(0.001)	0.072(0.010)	0.024(0.003)
yacht				0.186(0.008) 0.141(0.011) 0.162(0.012) 0.024(0.003) 0.047(0.001)		0.087(0.022)	0.239(0.017)	0.075(0.005)	0.056(0.002)	0.054(0.004)	0.041(0.000)	0.032(0.003)
yeast		0.175(0.003) 0.066(0.001) 0.166(0.001)		0.031(0.008)	0.039(0.001)	0.030(0.012)	0.471(0.110)	0.080(0.009)	0.018(0.003)	0.015(0.002)	0.031(0.007)	0.008(0.000)

Table 18: Energy MMD results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
		0.031(0.004) 0.021(0.001) 0.027(0.002)		0.006(0.001)	0.002(0.000)	0.002(0.001)	0.027(0.003)		0.002(0.000) 0.001(0.000) 0.001(0.000)		0.002(0.001)	0.002(0.000)
airfoil		0.036(0.003) 0.004(0.001) 0.016(0.001)		0.004(0.001)	0.001(0.000)	0.004(0.002)	0.015(0.002)		0.007(0.006) 0.001(0.000) 0.001(0.000) 0.001(0.000)			0.001(0.000)
banknote		0.032(0.004) 0.007(0.002) 0.013(0.003)		0.003(0.000)	0.003(0.000)	0.010(0.004)	0.027(0.007)		0.007(0.001) 0.001(0.000) 0.001(0.000) 0.001(0.000)			0.001(0.000)
blood
cancer				0.074(0.003) 0.010(0.000) 0.007(0.000) 0.002(0.000) 0.005(0.000)		0.031(0.001)	0.027(0.005)	0.010(0.000)	0.010(0.001)	0.010(0.000)	0.005(0.001)	0.004(0.000)
california		0.023(0.003) 0.010(0.002) 0.011(0.002)		0.003(0.001)	0.002(0.000)	0.016(0.007)	0.027(0.002)	0.002(0.001)	0.000(0.000)	0.000(0.000) 0.001(0.000)		0.001(0.000)
climate				0.024(0.001) 0.018(0.001) 0.024(0.001) 0.006(0.000) 0.019(0.001)		0.012(0.001)	0.021(0.001)	0.023(0.001)	0.017(0.001)	0.014(0.001)	0.008(0.001)	0.008(0.000)
concrete		0.028(0.002) 0.023(0.002) 0.021(0.002)		0.003(0.000)	0.003(0.000)	0.006(0.001)	0.023(0.006)		0.007(0.002) 0.002(0.000) 0.002(0.000)		0.003(0.000)	0.003(0.000)
slump				0.036(0.003) 0.027(0.003) 0.032(0.004) 0.012(0.001) 0.020(0.002)		0.020(0.005)	0.026(0.003)	0.025(0.003)	0.017(0.003)	0.022(0.002)	0.016(0.002)	0.017(0.002)
sonar		0.072(0.003) 0.037(0.002) 0.025(0.001)		0.019(0.001)	0.018(0.001)	0.072(0.004)	0.143(0.007)	0.033(0.002)	0.043(0.002)	0.030(0.002)	0.022(0.001)	0.021(0.002)
vowel		0.025(0.001) 0.006(0.001) 0.017(0.001)		0.004(0.000)	0.004(0.001)	0.004(0.000)	0.016(0.000)	0.006(0.001)	0.003(0.000)		0.002(0.000) 0.003(0.001)	0.002(0.000)
ecoli		0.026(0.003) 0.006(0.000) 0.017(0.001)		0.006(0.002)	0.005(0.000)	0.017(0.005)	0.035(0.008)	0.008(0.001)	0.003(0.000)	0.003(0.000)	0.003(0.000)	0.004(0.000)
glass		0.033(0.002) 0.011(0.002) 0.020(0.002)		0.013(0.001)	0.010(0.002)	0.014(0.003)	0.026(0.001)	0.016(0.001)	0.008(0.001)	0.011(0.002)	0.007(0.001)	0.008(0.001)
ionosphere		0.074(0.001) 0.018(0.001) 0.018(0.001)		0.021(0.000)	0.007(0.001)	0.015(0.001)	0.031(0.004)	0.014(0.001)	0.009(0.001)	0.008(0.000)	0.007(0.001)	0.008(0.000)
iris				0.065(0.013) 0.006(0.003) 0.014(0.005) 0.005(0.002) 0.006(0.003)		0.004(0.001) 0.018(0.007)			0.006(0.003) 0.005(0.001) 0.004(0.001) 0.004(0.002)			0.004(0.002)
libras				0.116(0.003) 0.074(0.002) 0.005(0.003) 0.001(0.000) 0.002(0.000)		—	0.082(0.004)	0.009(0.000)	0.006(0.001)	0.005(0.000)	0.014(0.001)	0.011(0.000)
parkinsons				0.075(0.006) 0.018(0.002) 0.015(0.002) 0.010(0.001) 0.012(0.001)		0.043(0.003)	0.030(0.001)	0.016(0.002)	0.024(0.001)	0.023(0.000)	0.013(0.001)	0.011(0.000)
planning				0.047(0.007) 0.016(0.001) 0.012(0.001) 0.006(0.001) 0.010(0.002)		0.026(0.002)	0.029(0.003)	0.023(0.002)	0.019(0.001)	0.024(0.001)	0.010(0.001)	0.013(0.000)
qsar		0.032(0.002) 0.011(0.001) 0.011(0.001)		0.005(0.000)	0.004(0.000)	0.015(0.002)	0.019(0.003)	—	0.004(0.000)		0.004(0.000) 0.007(0.002)	0.004(0.000)
seeds				0.083(0.003) 0.007(0.002) 0.011(0.002) 0.002(0.000) 0.005(0.001)		0.006(0.001)	0.029(0.004)	0.007(0.002)	0.007(0.002)	0.007(0.001)	0.006(0.002)	0.005(0.001)
wine				0.050(0.004) 0.013(0.002) 0.021(0.003) 0.011(0.001) 0.011(0.001)		0.016(0.001)	0.028(0.002)	0.015(0.002)	0.015(0.001)	0.016(0.001)	0.011(0.002)	0.011(0.001)
wine (red)		0.019(0.001) 0.008(0.001) 0.013(0.001)		0.003(0.000)	0.007(0.000)	0.004(0.001)	0.019(0.002)	0.003(0.000)	0.002(0.000)	0.003(0.000)	0.002(0.000)	0.002(0.000)
	wine (white) 0.026(0.003) 0.009(0.000) 0.013(0.001)			0.001(0.000)	0.003(0.000)	0.006(0.001)	0.019(0.003)		0.006(0.001) 0.001(0.000) 0.001(0.000) 0.001(0.000)			0.001(0.000)
yacht				0.036(0.003) 0.022(0.002) 0.028(0.002) 0.006(0.000) 0.006(0.002)		0.007(0.001)	0.034(0.006)		0.019(0.003) 0.006(0.002) 0.008(0.002)		0.005(0.001)	0.007(0.001)
yeast		0.018(0.001) 0.008(0.000) 0.016(0.001)		0.006(0.000)	0.005(0.000)	0.002(0.000)	0.053(0.017)	0.009(0.002)	0.002(0.000)	0.002(0.000)	0.002(0.000)	0.002(0.000)

Table 19: Gaussian MMD results: mean and standard error (×10³ ). MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.088(0.007)	0.095(0.009)	0.076(0.007)	0.013(0.001)	0.010(0.002)	0.023(0.002)	0.397(0.101)	0.013(0.004)		0.005(0.001) 0.004(0.000) 0.014(0.003)		0.020(0.006)
banknote	0.237(0.013)	0.033(0.002)	0.151(0.007)		0.010(0.002) 0.005(0.002)	0.012(0.001)	0.376(0.179)	0.008(0.001)		0.007(0.000) 0.005(0.000) 0.012(0.003)		0.010(0.001)
blood	0.612(0.037)	0.074(0.002)	0.181(0.011)	0.050(0.004)	0.039(0.003)	0.053(0.006)	1.243(0.249)			0.088(0.011) 0.021(0.002) 0.028(0.006) 0.022(0.001)		0.031(0.004)
cancer	0.010(0.000)		0.002(0.000) 0.001(0.000) 0.001(0.000)		0.001(0.000)	0.007(0.001)	0.004(0.000)	0.002(0.000)	0.002(0.000)	0.003(0.001)	0.004(0.001)	0.002(0.000)
california	0.066(0.001)	0.064(0.001)	0.053(0.001)	0.005(0.002)	0.008(0.000)	0.289(0.135)	0.454(0.275)	0.006(0.000)		0.001(0.000) 0.000(0.000) 0.003(0.000)		0.002(0.000)
climate	0.007(0.000)		0.007(0.001) 0.006(0.000) 0.007(0.000)		0.007(0.001)	0.019(0.003)	0.010(0.001)			0.007(0.001) 0.006(0.001) 0.005(0.001) 0.008(0.001)		0.010(0.001)
concrete	0.036(0.002)	0.039(0.004)	0.037(0.005)	0.007(0.002)	0.009(0.001)	0.014(0.004)	0.103(0.071)			0.013(0.003) 0.005(0.001) 0.005(0.001) 0.010(0.001)		0.007(0.003)
slump			0.092(0.014) 0.078(0.014) 0.082(0.021) 0.069(0.022) 0.061(0.018)			0.114(0.043)		0.115(0.026) 0.078(0.014) 0.073(0.024) 0.079(0.024) 0.059(0.020)				0.074(0.022)
sonar	0.006(0.001)		0.003(0.000) 0.001(0.000) 0.003(0.000)		0.002(0.000)	0.008(0.001)	0.021(0.002)	0.002(0.001)	0.003(0.001)	0.002(0.000)	0.003(0.000)	0.003(0.000)
vowel	0.016(0.001)	0.004(0.000)	0.015(0.002)	0.005(0.000)	0.003(0.000)	0.003(0.000)	0.039(0.022)	0.006(0.001)		0.002(0.000) 0.001(0.000) 0.004(0.001)		0.003(0.000)
ecoli	0.097(0.007)	0.055(0.001)	0.121(0.015)	0.053(0.009)	0.058(0.016)	0.041(0.014)	0.099(0.006)	0.760(0.624)		0.034(0.007) 0.031(0.012) 0.031(0.002)		0.034(0.004)
glass	0.095(0.014)	0.045(0.016)			0.061(0.002) 0.033(0.001) 0.033(0.010)	0.042(0.012)	0.075(0.013)			0.040(0.011) 0.031(0.006) 0.029(0.007) 0.033(0.007)		0.029(0.005)
ionosphere	0.005(0.001)		0.003(0.001) 0.002(0.000) 0.005(0.001) 0.002(0.001)			0.004(0.000)		0.007(0.001) 0.003(0.001) 0.003(0.001) 0.003(0.001) 0.004(0.000)				0.003(0.000)
iris	0.348(0.017)	0.044(0.014)	0.084(0.010)		0.036(0.004) 0.020(0.009)	0.051(0.018)		0.333(0.137) 0.024(0.004) 0.034(0.008)			0.031(0.004) 0.025(0.003)	0.030(0.000)
libras	0.002(0.000)	0.001(0.000)		0.000(0.000) 0.000(0.000) 0.000(0.000)		—	0.001(0.000)	0.000(0.000)	0.000(0.000)	0.000(0.000)	0.000(0.000)	0.000(0.000)
parkinsons	0.044(0.003)		0.006(0.000) 0.005(0.001) 0.007(0.001)		0.007(0.002)	0.018(0.001)	0.025(0.001)	0.007(0.001)		0.007(0.000) 0.005(0.001) 0.008(0.001)		0.007(0.000)
planning	0.039(0.004)		0.021(0.006) 0.011(0.003) 0.012(0.003) 0.020(0.007)			0.028(0.002)	0.052(0.009)	0.023(0.006)	0.017(0.003)	0.018(0.002)	0.017(0.003)	0.017(0.004)
qsar	0.005(0.000)	0.002(0.000)	0.001(0.000)		0.001(0.000) 0.001(0.000)	0.005(0.000)	0.005(0.000)	—	0.001(0.000)	0.001(0.000)	0.008(0.002)	0.002(0.000)
seeds	0.177(0.003)	0.009(0.002)	0.017(0.004)		0.008(0.001) 0.004(0.002)	0.014(0.004)	0.073(0.015)	0.011(0.004)	0.007(0.001)	0.006(0.001)	0.007(0.002)	0.008(0.002)
wine	0.021(0.003)	0.009(0.002)	0.009(0.002)		0.014(0.001) 0.007(0.001)	0.010(0.002)		0.026(0.004) 0.007(0.001) 0.010(0.001)		0.008(0.002)	0.008(0.001)	0.009(0.002)
wine (red)	0.025(0.000)	0.025(0.001)		0.020(0.001) 0.005(0.000) 0.011(0.002)		0.015(0.001)		0.091(0.069) 0.004(0.001) 0.008(0.001)			0.007(0.001) 0.005(0.001)	0.005(0.002)
	wine (white) 0.014(0.001)	0.019(0.002)	0.009(0.000)	0.001(0.000)	0.004(0.000)	0.005(0.001)	0.079(0.029)	0.002(0.000)		0.002(0.000) 0.001(0.000) 0.002(0.001)		0.003(0.001)
yacht	0.057(0.006)	0.055(0.004)	0.072(0.006)	0.030(0.008)	0.037(0.009)	0.026(0.010) 0.110(0.003)				0.039(0.010) 0.026(0.005) 0.019(0.009) 0.032(0.008)		0.025(0.008)
yeast	0.085(0.008)	0.075(0.008)	0.078(0.005)	0.021(0.004)	0.036(0.001)	0.017(0.003) 0.093(0.017)		0.042(0.004)		0.020(0.002) 0.018(0.003) 0.022(0.002)		0.015(0.004)

Table 20: Gaussian MMD results: mean and standard error (×10³ ). MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.298(0.014)	0.429(0.013)	0.281(0.013)	0.081(0.006)	0.076(0.007)	0.093(0.015)	0.888(0.213)	0.074(0.015)		0.033(0.015) 0.022(0.004) 0.049(0.006)		0.090(0.019)
banknote	0.817(0.023)	0.191(0.025)	0.581(0.024)	0.054(0.012)	0.033(0.013)	0.079(0.032)	1.137(0.243)	0.089(0.024)		0.033(0.013) 0.022(0.009) 0.103(0.039)		0.054(0.019)
blood	1.926(0.026)	0.390(0.066)	0.763(0.058)	0.154(0.042)	0.325(0.135)	0.356(0.127)	4.707(0.872)		0.456(0.049) 0.090(0.028) 0.142(0.053)		0.123(0.024)	0.167(0.052)
cancer	0.035(0.002)		0.009(0.002) 0.005(0.001) 0.006(0.001)		0.008(0.001)	0.043(0.003)	0.385(0.099)	0.008(0.001)	0.015(0.003)	0.016(0.004)	0.131(0.008)	0.020(0.004)
california	0.255(0.004)	0.333(0.004)	0.234(0.005)	0.017(0.003)	0.066(0.019)	0.299(0.093)	1.799(0.197)	0.033(0.006)		0.006(0.001) 0.003(0.000) 0.012(0.001)		0.014(0.002)
climate	0.017(0.002)		0.022(0.002) 0.012(0.001) 0.017(0.001)		0.022(0.006)	0.084(0.009)	0.110(0.023)	0.016(0.001)	0.017(0.001)	0.018(0.001)	0.019(0.004)	0.027(0.001)
concrete	0.113(0.001)	0.116(0.007)		0.126(0.012) 0.025(0.002) 0.047(0.006)		0.105(0.032)	0.251(0.094)	0.064(0.004)	0.037(0.004)	0.033(0.003)	0.036(0.006)	0.030(0.004)
slump	0.264(0.020)	0.243(0.016)	0.237(0.051)	0.244(0.028)	0.348(0.048)	0.588(0.074)	0.523(0.172)	0.239(0.042)	0.278(0.079)		0.441(0.108) 0.180(0.034)	0.264(0.055)
sonar	0.016(0.002)		0.008(0.001) 0.003(0.000) 0.009(0.000)		0.009(0.000)	0.051(0.006)	0.285(0.001)	0.007(0.001)	0.010(0.001)	0.011(0.001)	0.060(0.007)	0.012(0.001)
vowel	0.068(0.008)	0.017(0.002)	0.074(0.006)	0.022(0.003)	0.022(0.002)	0.041(0.010)	0.508(0.107)	0.035(0.005)		0.017(0.002) 0.012(0.000) 0.025(0.005)		0.017(0.003)
ecoli	0.331(0.028)	0.271(0.122)	0.381(0.030)	0.133(0.053)	0.152(0.022)	0.367(0.020)	2.235(0.874)			0.735(0.298) 0.101(0.022) 0.119(0.021) 0.121(0.019)		0.102(0.016)
glass	0.285(0.049)	0.204(0.020)	0.225(0.017)		0.126(0.017) 0.104(0.015)	0.178(0.038)	0.866(0.092)	0.115(0.032)	0.109(0.007)	0.113(0.008)	0.135(0.048)	0.098(0.010)
ionosphere	0.014(0.003)		0.011(0.000) 0.005(0.000) 0.014(0.001)		0.009(0.001)	0.012(0.002)	0.059(0.004)	0.008(0.001)	0.009(0.000)	0.007(0.000)	0.042(0.001)	0.015(0.003)
iris	1.291(0.136)	0.206(0.102)	0.360(0.075)	0.149(0.026)	0.344(0.258)	0.165(0.054)	1.223(0.124)	0.163(0.105)	0.119(0.057)	0.119(0.045)	0.137(0.046)	0.098(0.012)
libras	0.005(0.000)	0.002(0.000)		0.000(0.000) 0.000(0.000) 0.000(0.000)		—	0.142(0.007)	0.001(0.000)	0.001(0.000)	0.001(0.000)	0.033(0.008)	0.002(0.000)
parkinsons	0.147(0.011)		0.033(0.002) 0.019(0.001) 0.029(0.003)		0.024(0.004)	0.150(0.028)	0.650(0.137)	0.024(0.004)	0.049(0.014)	0.059(0.017)	0.185(0.056)	0.046(0.008)
planning	0.112(0.026)		0.090(0.025) 0.035(0.005) 0.058(0.020)		0.091(0.021)	0.180(0.085)	0.295(0.111)	0.095(0.029)	0.092(0.015)	0.128(0.042)	0.091(0.018)	0.102(0.027)
qsar	0.016(0.001)		0.011(0.000) 0.005(0.000) 0.005(0.000) 0.007(0.001)			0.033(0.003)	0.539(0.211)	0.008(nan)	0.012(0.001)	0.008(0.001)	0.255(0.093)	0.012(0.001)
seeds	0.591(0.055)	0.064(0.018)		0.107(0.026) 0.029(0.004) 0.034(0.006)		0.122(0.062)	0.940(0.433)	0.049(0.007)	0.157(0.072)	0.122(0.045)	0.090(0.017)	0.071(0.029)
wine			0.073(0.010) 0.040(0.009) 0.032(0.010) 0.044(0.005)		0.045(0.009)	0.070(0.022)		0.109(0.009) 0.033(0.004) 0.052(0.008)		0.074(0.026)	0.067(0.014)	0.051(0.009)
wine (red)	0.096(0.004)	0.112(0.005)		0.082(0.001) 0.016(0.001) 0.177(0.019)		0.107(0.024)	0.725(0.222)	0.051(0.012)	0.036(0.007)	0.032(0.001)	0.029(0.007)	0.039(0.016)
	wine (white) 0.051(0.002)	0.101(0.001)		0.039(0.002) 0.003(0.001) 0.025(0.005)		0.038(0.004)	1.223(0.293)	0.005(0.001)	0.009(0.001)	0.007(0.002)	0.012(0.003)	0.019(0.002)
yacht	0.214(0.030)	0.207(0.034)		0.244(0.024) 0.086(0.023) 0.161(0.046)		0.251(0.081)	0.921(0.242)	0.176(0.026)	0.171(0.046)	0.159(0.025)	0.120(0.011)	0.121(0.017)
yeast	0.304(0.006)	0.270(0.012)	0.284(0.004)	0.088(0.016)	0.121(0.016)	0.090(0.022)	2.225(1.078)	1.656(0.785)	0.072(0.018)	0.067(0.013)	0.073(0.003)	0.048(0.003)

Table 21: Gaussian MMD results: mean and standard error (×10³ ). MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		0.556(0.024) 1.078(0.047)	0.531(0.020)	0.379(0.021)	0.297(0.071)	1.168(0.542)	1.503(0.176)				0.238(0.007) 0.191(0.058) 0.181(0.083) 0.259(0.030)	0.380(0.110)
banknote		1.533(0.029) 0.518(0.019)		1.184(0.076) 0.095(0.027) 0.128(0.005)		0.931(0.469)	5.887(1.980)	0.325(0.031)	0.177(0.036)	0.153(0.035)	0.273(0.037)	0.176(0.017)
blood		3.543(0.341) 1.001(0.040)		2.074(0.321) 0.412(0.097) 1.282(0.308)		6.898(2.709)	9.169(4.484)	1.076(0.054)	1.544(0.228)	1.130(0.443)	0.691(0.219)	0.518(0.066)
cancer			0.079(0.006) 0.036(0.006) 0.016(0.001) 0.023(0.001)		0.062(0.010)	0.173(0.018)	0.720(0.189)	0.031(0.001)	0.109(0.033)	0.077(0.012)	26.618(1.470)	0.167(0.043)
california		0.553(0.013) 0.637(0.003)	0.554(0.015)	0.033(0.003)	0.331(0.026)	0.528(0.017)	41.075(17.530)		0.062(0.012) 0.018(0.005) 0.035(0.007)		1.154(0.035)	0.159(0.019)
climate			0.055(0.009) 0.096(0.021) 0.023(0.002) 0.052(0.009)		0.136(0.007)	0.237(0.027)	1.403(0.058)	0.065(0.017)	0.088(0.014)	0.125(0.015)	1.365(0.255)	0.134(0.007)
concrete		0.266(0.021) 0.268(0.012)		0.270(0.021) 0.095(0.014) 0.308(0.098)		0.529(0.088)	1.549(0.194)	0.236(0.033)	0.326(0.064)	0.236(0.010)	0.175(0.025)	0.296(0.030)
slump			0.480(0.064) 0.461(0.055) 0.370(0.033) 0.494(0.043)		1.307(0.359)	2.387(0.229)	2.772(0.394)	0.870(0.075)	0.710(0.195)	0.784(0.163)	0.632(0.224)	0.545(0.230)
sonar			0.042(0.003) 0.036(0.003) 0.007(0.001) 0.039(0.001)		0.075(0.002)	0.136(0.009)	1.544(0.061)	0.034(0.002)	0.027(0.003)	0.051(0.003)	32.704(7.500)	0.045(0.003)
vowel		0.161(0.007) 0.097(0.008)		0.169(0.014) 0.074(0.017) 0.238(0.043)		0.148(0.036)	5.060(0.854)	0.142(0.006)	0.116(0.025)	0.118(0.012)	0.210(0.049)	0.113(0.017)
ecoli		0.870(0.141) 0.865(0.348)		0.666(0.099) 0.529(0.058) 0.445(0.199)		1.012(0.113)	10.443(0.675)		0.883(0.336) 0.554(0.218) 0.512(0.099)		0.829(0.269)	0.529(0.107)
glass		0.903(0.176) 0.807(0.066)		0.578(0.084) 0.451(0.098) 0.664(0.046)		1.682(0.406)	16.277(1.727)		0.551(0.072) 0.544(0.218) 0.551(0.185)		0.726(0.165)	0.703(0.277)
ionosphere			0.047(0.007) 0.027(0.004) 0.010(0.000) 0.038(0.006)		0.051(0.006)	0.147(0.035)	0.717(0.091)	0.028(0.004)	0.035(0.003)	0.038(0.006)	7.167(1.233)	0.057(0.004)
iris		2.310(0.287) 0.940(0.301)	1.051(0.505)	0.452(0.057)	0.821(0.424)	0.522(0.110)		10.359(2.339) 0.411(0.023) 1.206(0.079)		1.009(0.072)	0.818(0.165)	0.671(0.020)
libras		0.015(0.002) 0.007(0.000)		0.002(0.000) 0.002(0.000) 0.002(0.000)		—	0.771(0.017)	0.004(0.000)	0.014(0.001)	0.009(0.000)	198.992(25.539)	0.010(0.001)
parkinsons			0.325(0.031) 0.190(0.018) 0.068(0.003) 0.140(0.009)		0.314(0.108)	0.555(0.027)	4.256(0.348)	0.130(0.011)	0.231(0.024)	0.256(0.039)	5.483(1.486)	0.352(0.082)
planning			0.291(0.086) 0.389(0.117) 0.079(0.006) 0.341(0.100)		0.684(0.242)	0.613(0.055)	2.790(0.822)	0.346(0.102)	0.250(0.107)	0.316(0.095)	0.378(0.119)	0.358(0.150)
qsar			0.042(0.001) 0.048(0.004) 0.014(0.001) 0.026(0.003)		0.060(0.008)	0.187(0.072)	6.996(2.338)	0.048(0.003)	0.113(0.027)	0.060(0.006)	106.536(23.668)	0.112(0.018)
seeds		1.058(0.083) 0.414(0.026)		0.342(0.058) 0.232(0.096) 0.206(0.060)		0.652(0.175)	5.342(1.428)	0.274(0.105)	0.345(0.106)	0.372(0.104)	0.380(0.127)	0.360(0.070)
wine			0.189(0.002) 0.178(0.030) 0.071(0.017) 0.157(0.030)		0.256(0.066)	0.293(0.026)	1.080(0.184)	0.134(0.016)	0.120(0.021)	0.144(0.013)	0.227(0.022)	0.224(0.013)
wine (red)		0.204(0.005) 0.307(0.033)		0.177(0.003) 0.037(0.002) 0.368(0.023)		0.671(0.109)	7.492(1.321)	0.149(0.005)	0.227(0.040)	0.166(0.011)	2.293(0.641)	0.490(0.143)
	wine (white) 0.107(0.003) 0.240(0.004)			0.096(0.003) 0.012(0.002) 0.121(0.010)		0.247(0.035)	4.423(0.246)	0.019(0.001)	0.101(0.040)	0.065(0.019)	1.250(0.185)	0.301(0.029)
yacht		0.378(0.054) 0.560(0.095)		0.422(0.052) 0.171(0.055) 0.447(0.070)		1.194(0.363)	4.126(0.565)	0.272(0.058)	0.395(0.079)	0.278(0.054)	0.294(0.056)	0.300(0.116)
yeast		0.625(0.007) 0.615(0.070)		0.553(0.006) 0.213(0.035) 0.422(0.071)		1.018(0.667)	12.825(3.914)	0.972(0.520)	0.314(0.092)	0.278(0.095)	0.791(0.203)	0.201(0.013)

Table 22: Gaussian MMD results: mean and standard error (×10³ ). MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.420(0.160)	0.346(0.028)	0.287(0.079)	0.055(0.005)	0.026(0.012)	0.061(0.020)	0.551(0.114)			0.016(0.005) 0.006(0.002) 0.008(0.004) 0.028(0.013)		0.021(0.005)
banknote	1.192(0.261)	0.122(0.040)	0.254(0.039)	0.139(0.052)	0.020(0.010)	0.213(0.127)	0.424(0.046)			0.395(0.346) 0.019(0.005) 0.018(0.003) 0.050(0.017)		0.026(0.007)
blood	2.132(0.533)	0.412(0.144)	0.546(0.150)	0.143(0.072)	0.150(0.036)	0.993(0.433)	2.074(0.735)			0.340(0.086) 0.080(0.036) 0.077(0.032) 0.076(0.016)		0.079(0.015)
cancer	0.219(0.017)	0.016(0.001)	0.007(0.000)	0.002(0.000)	0.008(0.001)	0.129(0.005)	0.098(0.021)	0.014(0.001)	0.028(0.003)	0.028(0.002)	0.015(0.005)	0.008(0.001)
california	0.237(0.102)	0.158(0.037)	0.091(0.027)	0.016(0.005)	0.029(0.006)	0.391(0.243)	0.539(0.065)	0.043(0.019)		0.006(0.002) 0.003(0.000) 0.012(0.004)		0.012(0.003)
climate	0.006(0.001)		0.010(0.002) 0.005(0.001)	0.009(0.001)	0.008(0.001)	0.024(0.002)	0.022(0.003)	0.007(0.001)	0.007(0.002)	0.009(0.002)	0.011(0.002)	0.015(0.003)
concrete	0.174(0.033)	0.151(0.030)	0.088(0.021)	0.015(0.004)	0.023(0.007)	0.070(0.016)	0.223(0.099)		0.103(0.031) 0.010(0.003) 0.014(0.005)		0.020(0.006)	0.023(0.006)
slump	0.215(0.008)	0.165(0.023)	0.146(0.009)	0.096(0.016)	0.134(0.007)	0.183(0.046)	0.220(0.017)		0.131(0.026) 0.103(0.002) 0.179(0.018)		0.115(0.014)	0.114(0.012)
sonar	0.026(0.002)		0.008(0.002) 0.004(0.001)	0.006(0.000)	0.004(0.000)	0.055(0.004)	0.060(0.003)	0.007(0.002)	0.011(0.001)	0.009(0.001)	0.009(0.001)	0.008(0.002)
vowel		0.066(0.013) 0.009(0.001) 0.033(0.002)		0.011(0.003)	0.008(0.004)	0.016(0.001)	0.037(0.005)			0.046(0.010) 0.009(0.002) 0.008(0.000) 0.011(0.006)		0.007(0.002)
ecoli	0.180(0.028)	0.047(0.008)	0.114(0.016)	0.062(0.037)	0.049(0.013)	0.231(0.063)	0.781(0.368)			0.054(0.009) 0.032(0.002) 0.031(0.005) 0.029(0.004)		0.056(0.022)
glass	0.266(0.020)	0.125(0.023)	0.137(0.041)	0.117(0.031)	0.088(0.026)	0.247(0.076)	0.309(0.064)	0.117(0.025)	0.090(0.021)		0.139(0.043) 0.064(0.011)	0.085(0.027)
ionosphere	0.037(0.002)		0.010(0.000) 0.004(0.001)	0.006(0.002)	0.003(0.001)	0.014(0.003)	0.024(0.006)			0.005(0.001) 0.004(0.001) 0.003(0.001) 0.004(0.001)		0.005(0.001)
iris	2.149(0.582)	0.121(0.074)	0.285(0.110)	0.094(0.071)	0.122(0.083)	0.056(0.030) 0.591(0.377)				0.098(0.073) 0.070(0.033) 0.061(0.030) 0.083(0.070)		0.077(0.052)
libras	0.016(0.001)	0.014(0.001)		0.000(0.000) −0.000(0.000) 0.000(0.000)		—	0.012(0.001)	0.001(0.000)	0.001(0.000)	0.001(0.000)	0.001(0.000)	0.001(0.000)
parkinsons	0.499(0.046)	0.082(0.010)	0.051(0.003)	0.037(0.018)	0.035(0.002)	0.395(0.012)	0.190(0.008)	0.055(0.003)	0.159(0.015)	0.155(0.009)	0.062(0.006)	0.046(0.004)
planning	0.215(0.069)		0.058(0.008) 0.022(0.007) 0.022(0.014)		0.027(0.013)	0.158(0.022)	0.169(0.039)	0.088(0.015)	0.081(0.012)	0.137(0.007)	0.040(0.007)	0.054(0.004)
qsar	0.033(0.004)		0.010(0.001) 0.005(0.001)	0.006(0.001)	0.007(0.001)	0.037(0.005)	0.054(0.011)	—	0.008(0.001)	0.007(0.001)	0.025(0.010)	0.007(0.000)
seeds	1.607(0.102)	0.045(0.020)	0.090(0.021)	0.013(0.003)	0.037(0.015)	0.063(0.022)	0.534(0.087)	0.051(0.017)	0.057(0.033)	0.063(0.022)	0.066(0.049)	0.038(0.013)
wine	0.094(0.019)	0.017(0.004)	0.018(0.003)	0.019(0.003)	0.012(0.004)	0.047(0.002)		0.073(0.012) 0.013(0.003) 0.018(0.004)		0.037(0.003)	0.020(0.009)	0.020(0.003)
wine (red)	0.065(0.016)	0.047(0.006)	0.028(0.003)	0.015(0.002)	0.082(0.016)	0.037(0.012)	0.135(0.038)	0.023(0.003)	0.016(0.001)	0.014(0.001)	0.017(0.002)	0.010(0.001)
	wine (white) 0.101(0.025)	0.039(0.004)	0.028(0.003)	0.004(0.001)	0.010(0.003)	0.062(0.009)	0.111(0.031)	0.065(0.014)	0.005(0.002)	0.006(0.002)	0.006(0.001)	0.007(0.001)
yacht	0.186(0.041)	0.137(0.032)	0.157(0.009)	0.045(0.003)	0.056(0.012)	0.079(0.012)	0.434(0.122)	0.105(0.032)	0.049(0.033)		0.105(0.056) 0.033(0.012)	0.062(0.016)
yeast	0.172(0.042)	0.119(0.004)	0.134(0.025)	0.027(0.007)	0.076(0.018)	0.035(0.013)	1.563(0.686)	0.098(0.030)	0.041(0.002)		0.039(0.003) 0.033(0.004)	0.037(0.009)

Table 23: Laplacian MMD results: mean and standard error (×10³ ). MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		1.845(0.038) 1.146(0.043) 1.642(0.073)		0.264(0.013)	0.137(0.004)	0.115(0.012)	1.920(0.222)		0.118(0.009) 0.046(0.001) 0.048(0.003)		0.129(0.004)	0.122(0.002)
banknote		2.462(0.059) 0.217(0.009) 1.431(0.025)		0.226(0.010)	0.067(0.010)	0.098(0.004)	1.866(0.482)	0.092(0.003)	0.049(0.002)	0.044(0.002) 0.079(0.009)		0.080(0.004)
blood		3.578(0.077) 0.553(0.036) 1.643(0.056)		0.415(0.020)	0.317(0.014)	0.366(0.006)	2.939(0.432)		0.481(0.033) 0.130(0.019) 0.145(0.023) 0.191(0.015)			0.224(0.013)
cancer				0.784(0.004) 0.118(0.003) 0.103(0.002) 0.056(0.001) 0.068(0.001)		0.116(0.005)	0.215(0.006)	0.163(0.003)	0.087(0.002)	0.087(0.004)	0.085(0.004)	0.072(0.001)
california		1.562(0.017) 0.512(0.007) 0.843(0.004)		0.226(0.044)	0.089(0.002)	1.785(0.301)	2.050(0.491)	0.103(0.001)	0.010(0.000)	0.006(0.000) 0.023(0.001)		0.021(0.001)
climate				0.572(0.007) 0.383(0.011) 0.563(0.006) 0.145(0.000) 0.504(0.010)		0.288(0.014)	0.501(0.002)	0.557(0.007)	0.438(0.006)	0.345(0.012)	0.196(0.004)	0.163(0.004)
concrete		1.381(0.039) 1.150(0.037) 1.234(0.124)		0.188(0.008)	0.105(0.010)	0.164(0.008)	1.216(0.142)		0.133(0.016) 0.080(0.005) 0.080(0.005) 0.154(0.007)			0.130(0.008)
slump				1.888(0.072) 1.555(0.060) 1.568(0.102) 0.799(0.042) 1.070(0.037)		0.994(0.117)	1.413(0.024)	1.378(0.041)	0.964(0.049)	1.038(0.056)	0.901(0.044)	0.987(0.038)
sonar		0.496(0.020) 0.259(0.005) 0.168(0.003)		0.155(0.002)	0.134(0.002)	0.227(0.008)	1.241(0.035)	0.247(0.007)	0.277(0.009)	0.177(0.001)	0.156(0.001)	0.153(0.002)
vowel		0.928(0.000) 0.219(0.009) 0.690(0.019)		0.137(0.003)	0.139(0.010)	0.093(0.007)	0.792(0.077)	0.080(0.006)	0.078(0.005)	0.056(0.002) 0.110(0.005)		0.075(0.003)
ecoli		2.062(0.060) 0.461(0.014) 1.342(0.036)		0.638(0.085)	0.403(0.018)	0.281(0.026)	1.770(0.182)	1.643(0.900)	0.277(0.020)	0.261(0.027)	0.249(0.007)	0.261(0.004)
glass		1.807(0.097) 0.558(0.045) 1.068(0.051)		0.650(0.022)	0.399(0.045)	0.445(0.034)	1.150(0.028)	0.667(0.061)	0.368(0.016)		0.344(0.033) 0.334(0.019)	0.365(0.027)
ionosphere		0.630(0.016) 0.174(0.009) 0.203(0.007)		0.253(0.011)	0.112(0.005)	0.114(0.002)	0.303(0.017)	0.178(0.007)	0.119(0.006)	0.111(0.005)	0.107(0.002)	0.110(0.002)
iris		4.619(0.097) 0.526(0.036) 1.341(0.066)		0.564(0.051)	0.491(0.035)	0.548(0.089)	2.168(0.417)	0.571(0.019)	0.515(0.028)	0.537(0.040)	0.427(0.036)	0.470(0.019)
libras				0.378(0.006) 0.169(0.001) 0.031(0.010) 0.005(0.000) 0.008(0.000)		—	0.102(0.005)	0.041(0.001)	0.024(0.001)	0.017(0.000)	0.096(0.008)	0.049(0.001)
parkinsons		1.319(0.029) 0.289(0.003) 0.361(0.017)		0.281(0.007)	0.243(0.010)	0.344(0.013)	0.578(0.003)	0.383(0.015)	0.301(0.003)		0.252(0.002) 0.251(0.006)	0.252(0.007)
planning				1.139(0.029) 0.450(0.012) 0.366(0.014) 0.227(0.013) 0.335(0.019)		0.549(0.036)	0.713(0.055)	0.646(0.018)	0.500(0.024)	0.473(0.018)	0.342(0.011)	0.362(0.008)
qsar		0.468(0.023) 0.131(0.007) 0.156(0.012)		0.092(0.004)	0.040(0.002)	0.085(0.005)	0.147(0.005)	—	0.034(0.001)	0.030(0.001) 0.061(0.005)		0.047(0.000)
seeds				3.225(0.054) 0.314(0.034) 0.488(0.063) 0.184(0.005) 0.225(0.030)		0.281(0.024)	0.846(0.139)	0.394(0.051)	0.248(0.028)	0.236(0.026)	0.227(0.018)	0.253(0.012)
wine		1.177(0.052) 0.372(0.020) 0.618(0.046)		0.378(0.011)	0.355(0.014)	0.356(0.022) 0.686(0.037)		0.464(0.028)	0.430(0.036)	0.384(0.029)	0.359(0.019)	0.350(0.021)
wine (red)		0.955(0.018) 0.298(0.018) 0.693(0.018)		0.143(0.007)	0.168(0.007)	0.130(0.005)	0.963(0.171)	0.115(0.003)	0.095(0.004)	0.102(0.006)	0.077(0.002)	0.067(0.002)
	wine (white) 0.849(0.011) 0.347(0.008) 0.528(0.007)			0.047(0.000)	0.142(0.003)	0.059(0.005)	0.814(0.101)	0.076(0.013)	0.028(0.001)	0.021(0.001) 0.034(0.002)		0.029(0.003)
yacht		1.780(0.074) 1.195(0.030) 1.574(0.109)		0.329(0.027)	0.282(0.023)	0.317(0.031)	1.473(0.058)		0.993(0.022) 0.172(0.016) 0.185(0.039) 0.338(0.016)			0.297(0.026)
yeast		1.338(0.021) 0.505(0.016) 1.245(0.015)		0.394(0.028)	0.402(0.004)	0.117(0.004)	1.274(0.023)	0.628(0.022)	0.118(0.003)	0.120(0.006)	0.121(0.005)	0.096(0.006)

Table 24: Laplacian MMD results: mean and standard error (×10³ ). MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	7.561(0.133)		5.205(0.108) 7.249(0.132)	1.184(0.046)	0.701(0.048)	0.479(0.056)	6.858(0.969)	0.611(0.028)		0.207(0.042) 0.188(0.012) 0.425(0.064)		0.412(0.010)
banknote	9.117(0.123)		1.178(0.059) 6.191(0.168)	0.750(0.068)	0.315(0.011)	0.371(0.042)	6.783(0.333)	0.533(0.080)		0.194(0.027) 0.169(0.018) 0.409(0.069)		0.259(0.022)
blood		11.969(0.144) 2.139(0.038) 6.638(0.216)		1.214(0.075)	1.327(0.166)	1.182(0.131)	11.476(0.914)		3.407(0.060) 0.411(0.018) 0.476(0.064)		0.669(0.030)	0.699(0.045)
cancer	3.163(0.043)			0.425(0.029) 0.536(0.045) 0.143(0.004) 0.212(0.010)		0.574(0.048)	3.116(0.507)	0.570(0.034)	0.410(0.028)	0.429(0.031)	0.917(0.125)	0.337(0.015)
california	7.197(0.070)		3.811(0.044) 4.936(0.050)	0.787(0.104)	0.441(0.020)	2.637(0.652)	9.165(0.460)	0.483(0.024)		0.048(0.001) 0.032(0.003) 0.108(0.005)		0.108(0.004)
climate	2.511(0.008)			1.155(0.004) 2.439(0.007) 0.227(0.008) 1.183(0.053)		1.412(0.031)	2.600(0.127)	1.986(0.007)	1.359(0.006)	1.244(0.007)	0.548(0.015)	0.318(0.023)
concrete	5.732(0.101)		4.621(0.105) 5.216(0.276)	0.594(0.022)	0.712(0.055)	1.005(0.099)	4.643(0.410)	2.393(0.060)	0.504(0.038)	0.538(0.029)	0.599(0.041)	0.426(0.013)
slump	6.648(0.306)			5.238(0.423) 5.914(0.218) 2.366(0.031) 3.637(0.317)		3.477(0.110)	4.916(0.332)	4.342(0.191)	2.823(0.181)	3.434(0.359)	2.470(0.142)	2.548(0.070)
sonar	1.786(0.033)			0.731(0.021) 0.620(0.020) 0.321(0.002) 0.319(0.006)		1.015(0.057)	6.041(0.067)	0.734(0.018)	0.871(0.033)	0.609(0.007)	0.656(0.042)	0.331(0.007)
vowel	4.280(0.017)		1.053(0.017) 3.406(0.047)	0.361(0.003)	0.589(0.012)	0.505(0.036)	4.892(0.350)	1.763(0.021)	0.426(0.019)	0.364(0.015)	0.352(0.021)	0.224(0.008)
ecoli	8.554(0.220)		1.713(0.162) 6.741(0.368)	1.365(0.122)	1.312(0.035)	1.182(0.059)	11.597(2.143)	2.640(0.419)	0.912(0.111)	1.062(0.167)	0.708(0.031)	0.640(0.040)
glass	6.144(0.148)		1.673(0.084) 4.873(0.199)	2.637(0.199)	1.263(0.139)	1.144(0.141)	5.771(0.147)	2.086(0.113)	1.271(0.151)		1.308(0.152) 0.959(0.099)	0.963(0.087)
ionosphere	2.193(0.038)		0.393(0.009) 0.718(0.022)		0.588(0.019) 0.271(0.011)	0.420(0.039)	1.326(0.021)	0.519(0.015)	0.347(0.012)	0.329(0.015)	0.601(0.049)	0.287(0.014)
iris		14.952(0.484) 1.878(0.242) 4.725(0.392)		1.619(0.037)	1.885(0.657)	1.820(0.338)	7.435(0.779)	1.708(0.329)	1.665(0.415)		1.609(0.320) 1.293(0.237)	1.265(0.133)
libras	1.412(0.011)			0.488(0.008) 0.050(0.019) 0.012(0.000) 0.025(0.000)		—	3.835(0.070)	0.124(0.003)	0.083(0.003)	0.056(0.002)	2.132(0.338)	0.267(0.003)
parkinsons	4.660(0.066)			0.952(0.019) 1.352(0.043) 0.567(0.016) 0.626(0.022)		1.565(0.159)	3.883(0.584)	1.141(0.023)	1.198(0.112)	1.291(0.092)	0.913(0.055)	0.697(0.041)
planning	4.235(0.057)			1.673(0.025) 1.913(0.106) 0.675(0.056) 1.210(0.041)		2.138(0.232)	2.867(0.314)	2.138(0.024)	1.630(0.051)	2.130(0.238)	1.115(0.008)	1.281(0.058)
qsar	2.118(0.034)		0.574(0.013) 0.861(0.028)	0.266(0.004)	0.191(0.002)	0.441(0.018)	3.814(1.174)	0.395(nan)		0.148(0.006) 0.145(0.013) 1.194(0.295)		0.238(0.019)
seeds				10.403(0.433) 1.094(0.055) 2.279(0.177) 0.432(0.015) 0.783(0.031)		1.069(0.256)	4.989(1.188)	1.211(0.073)	1.390(0.308)	1.208(0.216)	0.823(0.064)	0.667(0.092)
wine	4.280(0.152)			1.160(0.115) 2.270(0.151) 0.781(0.023) 0.937(0.059)		1.156(0.129)	2.293(0.169)	1.356(0.126)	1.359(0.138)	1.596(0.188)	1.116(0.020)	0.953(0.058)
wine (red)	4.494(0.047)		1.535(0.061) 3.600(0.047)	0.451(0.013)	0.923(0.074)	0.834(0.101)	5.573(0.442)	0.427(0.063)	0.333(0.017)		0.350(0.005) 0.229(0.011)	0.250(0.049)
	wine (white) 4.123(0.029)		2.147(0.020) 2.974(0.011)	0.141(0.002)	0.450(0.015)	0.417(0.026)	6.590(0.935)	0.135(0.008)		0.091(0.004) 0.065(0.006) 0.104(0.008)		0.122(0.009)
yacht	7.318(0.318)			4.763(0.325) 6.583(0.376) 0.835(0.070) 1.164(0.179)		1.783(0.150)	6.540(0.544)	3.619(0.085)	1.363(0.171)	1.657(0.067)	1.385(0.127)	1.322(0.105)
yeast	6.299(0.133)		2.319(0.072) 5.887(0.153)	1.505(0.289)	1.313(0.067)	0.549(0.011)	9.348(2.050)	2.858(0.466)	0.416(0.040)	0.440(0.012)	0.306(0.033)	0.265(0.025)

Table 25: Laplacian MMD results: mean and standard error (×10³ ). MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		16.771(0.140) 13.523(0.413) 16.190(0.436)		4.579(0.139)	2.209(0.238)	4.332(1.503)	14.622(0.703)		5.911(0.172) 1.004(0.099) 0.955(0.216)		1.657(0.017)	1.287(0.219)
banknote	18.224(0.323)	3.630(0.277)	13.916(0.816)	1.395(0.028)	1.423(0.112)	2.946(1.002)	18.326(3.290)	3.997(0.307)		0.910(0.079) 0.778(0.053)	1.262(0.172)	0.879(0.040)
blood	23.310(0.261)	4.875(0.563)		15.893(0.841) 2.490(0.238) 4.819(0.349)		11.409(5.023)	20.851(5.092)	7.399(0.084)	3.760(0.405)	3.016(0.703)	2.646(0.374)	2.305(0.497)
cancer	8.490(0.091)	1.311(0.042)		2.326(0.054) 0.431(0.009) 0.691(0.032)		1.844(0.088)	5.140(1.013)	1.357(0.072)	1.743(0.050)	1.937(0.034)	49.929(2.173)	1.297(0.143)
california		18.940(0.099) 13.778(0.098) 15.714(0.352)		1.378(0.030)	1.642(0.115)	1.669(0.072)	57.738(12.064)		1.019(0.131) 0.160(0.017) 0.165(0.011)		1.528(0.009)	0.517(0.020)
climate	7.902(0.028)	3.072(0.119)		7.182(0.014) 0.422(0.043) 1.971(0.063)		4.594(0.306)	12.668(0.290)	3.491(0.078)	3.270(0.167)	4.424(0.085)	4.075(0.694)	0.984(0.028)
concrete				14.355(0.130) 11.593(0.034) 13.414(0.322) 1.356(0.046) 2.940(0.369)		3.889(0.359)	13.173(0.475)	5.336(0.079)	2.636(0.235)	2.346(0.271)	1.836(0.186)	1.631(0.138)
slump				15.115(0.579) 11.449(0.544) 13.845(0.517) 3.930(0.131) 10.494(0.690)		11.175(1.009)	11.840(0.794)	10.266(0.524)	7.128(0.573)	7.195(0.447)	6.308(0.461)	5.087(0.425)
sonar	5.000(0.061)	1.803(0.076)		2.582(0.133) 0.763(0.008) 1.190(0.019)		2.498(0.150)	14.779(0.360)	1.855(0.045)	2.279(0.100)	2.580(0.121)	105.276(13.317)	0.876(0.030)
vowel	11.939(0.063)	3.971(0.123)		10.589(0.125) 0.781(0.042) 2.153(0.247)		1.423(0.107)	24.839(3.055)	3.579(0.102)	2.047(0.125)	2.033(0.252)	1.054(0.047)	0.815(0.059)
ecoli	20.304(0.194)	5.232(0.498)	16.766(0.553)	3.880(0.195)	3.745(0.301)	4.070(0.098)	37.071(1.352)	6.026(0.546)	4.307(1.244)	4.013(0.744)	3.753(1.186)	3.055(0.682)
glass	16.647(0.984)	6.207(0.397)	12.768(0.967)	5.806(0.224)	5.240(0.206)	6.130(1.028)	41.962(2.985)	5.842(0.701)	5.360(0.847)	5.762(0.710)	4.188(0.737)	3.873(0.609)
ionosphere	6.012(0.058)	1.026(0.065)	2.720(0.316)		1.277(0.018) 0.803(0.049)	2.299(0.180)	7.069(0.599)	1.336(0.047)	1.546(0.096)	1.890(0.192)	31.921(2.971)	1.055(0.009)
iris	27.043(0.520)	7.185(1.251)		12.082(1.935) 3.930(0.218) 5.556(0.763)		4.704(0.767)		29.275(2.758) 4.106(0.339) 7.946(0.799)		6.928(0.501)	5.682(1.178)	4.563(0.415)
libras	3.929(0.044)	0.840(0.004)		0.917(0.269) 0.093(0.002) 0.107(0.003)		—	10.750(0.151)	0.352(0.003)	0.535(0.032)	0.673(0.051)	276.647(14.145)	0.574(0.006)
parkinsons	12.287(0.009)	3.617(0.124)		4.747(0.151) 1.688(0.133) 2.705(0.378)		4.686(0.113)	21.710(1.251)	3.115(0.068)	4.411(0.197)	6.112(0.215)	10.672(2.381)	2.219(0.174)
planning	12.328(0.391)	5.940(0.594)		8.282(0.484) 2.618(0.421) 5.004(0.974)		5.347(0.365)	12.618(2.842)	5.512(0.528)	5.780(0.607)	6.480(0.313)	3.177(0.498)	3.437(0.456)
qsar	6.462(0.080)	2.064(0.081)	3.548(0.114)		0.872(0.085) 0.744(0.039)	1.720(0.429)	31.922(5.417)	1.525(0.037)	1.064(0.256)	0.901(0.094)	146.750(21.593)	1.421(0.137)
seeds	21.949(0.442)	4.315(0.155)		7.375(0.703) 1.549(0.214) 2.696(0.181)		4.054(0.661)	18.735(2.692)	3.705(0.415)	4.610(0.594)	4.122(0.543)	3.201(0.277)	2.527(0.195)
wine	11.679(0.349)	3.748(0.423)		7.278(0.650) 1.708(0.148) 3.205(0.265)		2.683(0.209)	7.432(0.745)	3.450(0.403)	4.842(0.815)	4.875(0.543)	3.385(0.437)	2.588(0.334)
wine (red)	13.067(0.109)	5.680(0.249)		11.434(0.205) 0.992(0.055) 2.045(0.089)		5.601(0.346)	27.749(2.250)	4.025(0.033)	1.360(0.107)	1.246(0.033)	3.114(0.668)	1.491(0.235)
	wine (white) 12.137(0.060)	8.157(0.144)		10.224(0.236) 0.335(0.015) 1.155(0.043)		3.392(0.183)	23.487(0.250)	0.545(0.012)	0.484(0.118)	0.357(0.062)	2.446(0.333)	1.039(0.151)
yacht				16.821(0.761) 12.455(0.853) 14.514(1.079) 2.076(0.177) 3.943(0.085)		6.908(1.615)	18.446(1.402)	6.723(0.448)	4.844(0.320)	4.735(0.366)	3.544(0.041)	2.716(0.233)
yeast	17.631(0.228)	6.364(0.146)	16.817(0.116)	2.974(0.815)	3.640(0.101)	2.138(0.543)	37.438(7.716)	7.400(0.443)	1.628(0.264)	1.334(0.161)	1.085(0.177)	0.674(0.040)

Table 26: Laplacian MMD results: mean and standard error (×10³ ). MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
		3.467(0.303) 2.265(0.097) 3.067(0.123)		0.639(0.109)	0.266(0.047)	0.224(0.064)	2.813(0.273)		0.188(0.018) 0.064(0.010) 0.077(0.022)		0.206(0.067)	0.179(0.039)
airfoil		5.289(0.297) 0.604(0.164) 2.551(0.219)		0.599(0.125)	0.134(0.015)	0.545(0.282)	2.320(0.348)		0.905(0.667) 0.109(0.023) 0.103(0.012) 0.209(0.035)			0.142(0.011)
banknote		5.752(0.418) 1.203(0.260) 2.729(0.497)		0.515(0.026)	0.684(0.075)	1.387(0.462)	4.458(0.887)		1.466(0.239) 0.224(0.044) 0.248(0.042) 0.232(0.024)
blood												0.272(0.035)
cancer				1.581(0.051) 0.222(0.007) 0.156(0.004) 0.056(0.001) 0.109(0.006)		0.635(0.019)	0.567(0.091)	0.225(0.010)	0.220(0.011)	0.216(0.007)	0.114(0.009)	0.095(0.004)
california		2.344(0.243) 0.994(0.240) 1.200(0.207)		0.304(0.098)	0.161(0.028)	1.598(0.703)	2.553(0.160)	0.224(0.056)	0.020(0.003)		0.011(0.001) 0.047(0.011)	0.058(0.004)
climate				0.670(0.020) 0.492(0.025) 0.660(0.019) 0.148(0.005) 0.524(0.024)		0.328(0.019)	0.570(0.035)	0.631(0.020)	0.479(0.015)	0.383(0.017)	0.228(0.018)	0.221(0.006)
concrete		2.076(0.148) 1.699(0.112) 1.578(0.180)		0.255(0.008)	0.195(0.023)	0.424(0.049)	1.606(0.419)		0.500(0.132) 0.145(0.005) 0.162(0.009)		0.228(0.019)	0.213(0.030)
slump				2.624(0.262) 1.974(0.226) 2.327(0.334) 0.864(0.036) 1.457(0.152)		1.425(0.313)	1.853(0.220)	1.816(0.187)	1.215(0.227)	1.520(0.194)	1.120(0.175)	1.210(0.182)
sonar		0.663(0.028) 0.346(0.019) 0.231(0.009)		0.176(0.005)	0.167(0.005)	0.635(0.036)	1.311(0.062)	0.307(0.018)	0.398(0.016)	0.277(0.013)	0.205(0.008)	0.198(0.014)
vowel		1.283(0.070) 0.328(0.030) 0.884(0.033)		0.189(0.016)	0.188(0.032)	0.192(0.015)	0.829(0.023)	0.276(0.036)	0.133(0.008)		0.117(0.005) 0.154(0.023)	0.109(0.009)
ecoli		2.478(0.294) 0.528(0.016) 1.658(0.139)		0.574(0.162)	0.498(0.034)	0.424(0.050)	2.935(0.475)	0.736(0.049)	0.305(0.034)	0.309(0.044)	0.274(0.014)	0.313(0.003)
glass		2.539(0.159) 0.842(0.125) 1.569(0.172)		1.026(0.047)	0.773(0.125)	0.974(0.167)	1.936(0.067)	1.231(0.089)	0.631(0.062)	0.773(0.127)	0.524(0.066)	0.581(0.035)
ionosphere		1.211(0.015) 0.292(0.019) 0.298(0.010)		0.351(0.008)	0.115(0.009)	0.237(0.012)	0.504(0.057)	0.226(0.011)	0.154(0.010)	0.131(0.007)	0.119(0.008)	0.132(0.004)
iris				8.823(1.646) 0.825(0.424) 2.012(0.660) 0.704(0.208) 0.811(0.365)		0.535(0.192) 2.449(0.900)			0.880(0.456) 0.689(0.187) 0.668(0.197) 0.586(0.286)			0.579(0.235)
libras				0.657(0.017) 0.413(0.011) 0.029(0.015) 0.005(0.000) 0.009(0.000)		—	0.460(0.024)	0.052(0.001)	0.037(0.005)	0.028(0.001)	0.081(0.005)	0.061(0.003)
parkinsons				2.434(0.196) 0.608(0.056) 0.544(0.067) 0.368(0.032) 0.414(0.039)		1.322(0.086)	1.007(0.038)	0.586(0.057)	0.803(0.023)	0.731(0.007)	0.440(0.029)	0.388(0.005)
planning				2.069(0.252) 0.726(0.037) 0.551(0.048) 0.265(0.044) 0.442(0.085)		1.107(0.063)	1.231(0.112)	1.040(0.095)	0.817(0.029)	1.024(0.034)	0.458(0.039)	0.567(0.015)
qsar		0.567(0.039) 0.202(0.010) 0.207(0.010)		0.095(0.009)	0.067(0.000)	0.246(0.034)	0.303(0.041)	—	0.075(0.007)		0.064(0.007) 0.101(0.018)	0.065(0.002)
seeds				7.011(0.199) 0.643(0.148) 1.034(0.157) 0.239(0.034) 0.489(0.120)		0.534(0.120)	2.439(0.329)	0.696(0.153)	0.630(0.176)	0.625(0.119)	0.503(0.209)	0.430(0.053)
wine					1.870(0.136) 0.503(0.061) 0.829(0.130) 0.432(0.024) 0.435(0.048)	0.578(0.026)	1.027(0.061)	0.582(0.081)	0.577(0.051)	0.617(0.055)	0.431(0.070)	0.426(0.035)
wine (red)		1.241(0.080) 0.498(0.035) 0.851(0.036)		0.192(0.019)	0.430(0.018)	0.241(0.032)	1.134(0.085)	0.204(0.019)	0.144(0.014)	0.158(0.014)	0.126(0.007)	0.096(0.004)
	wine (white) 1.311(0.120) 0.484(0.023) 0.664(0.048)			0.072(0.008)	0.151(0.010)	0.249(0.024)	0.976(0.126)	0.301(0.060)	0.043(0.004)		0.039(0.003) 0.046(0.003)	0.046(0.005)
yacht				3.159(0.283) 1.924(0.159) 2.418(0.199) 0.497(0.036) 0.536(0.146)		0.553(0.057)	2.736(0.492)		1.691(0.213) 0.474(0.121) 0.664(0.157)		0.464(0.056)	0.560(0.088)
yeast		1.700(0.094) 0.788(0.023) 1.559(0.051)		0.558(0.027)	0.493(0.021)	0.163(0.013)	4.105(1.176)	0.890(0.224)	0.187(0.008)	0.199(0.009)	0.145(0.013)	0.151(0.011)

Table 27: Classifier 1-AUROC results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	—	—	—	—	—	—	—	—	—	—	—	—
banknote	0.054(0.005)	0.000(0.000)		0.015(0.003) 0.001(0.000)	0.000(0.000)	0.002(0.000)		0.011(0.002) 0.000(0.000) 0.000(0.000)		0.001(0.000)	0.001(0.000)	0.002(0.000)
blood	0.270(0.007)	0.213(0.004)		0.249(0.005) 0.252(0.002)	0.226(0.006)	0.239(0.005)	0.186(0.015) 0.227(0.005)		0.234(0.005)	0.234(0.006)	0.250(0.012)	0.240(0.010)
cancer	0.002(0.001)	0.002(0.000)		0.002(0.000) 0.002(0.000)	0.002(0.000)	0.000(0.000)	0.002(0.000)	0.002(0.000)	0.001(0.000)	0.001(0.000)	0.002(0.000)	0.002(0.000)
california	—	—	—	—	—	—	—	—	—	—	—	—
climate	0.066(0.004)	0.064(0.008)		0.065(0.005) 0.107(0.005)	0.025(0.003)	0.019(0.004)	0.033(0.003)	0.040(0.003)	0.048(0.005)	0.030(0.003)	0.067(0.007)	0.090(0.018)
concrete	—	—	—	—	—	—	—	—	—	—	—	—
slump	—	—	—	—	—	—	—	—	—	—	—	—
sonar	0.023(0.009)	0.015(0.006)		0.016(0.007) 0.011(0.002)	0.016(0.006)	0.002(0.000)	0.017(0.002)	0.017(0.006)	0.011(0.005)	0.006(0.002)	0.014(0.009)	0.008(0.004)
vowel	0.092(0.001)	0.034(0.000)			0.077(0.001) 0.096(0.001) 0.032(0.001)	0.034(0.001)	0.081(0.002)	0.046(0.002)		0.033(0.001) 0.031(0.001) 0.055(0.002)		0.046(0.002)
ecoli	0.070(0.002)	0.039(0.004)		0.061(0.004) 0.057(0.006)	0.039(0.001)	0.028(0.001)	0.067(0.005)	0.048(0.004)	0.037(0.003)	0.034(0.002)	0.048(0.005)	0.041(0.004)
glass	0.098(0.010)	0.069(0.008)		0.080(0.006) 0.086(0.002)	0.063(0.008)	0.055(0.004)	0.088(0.006)	0.072(0.005)		0.062(0.007) 0.053(0.006) 0.069(0.004)		0.064(0.009)
ionosphere	0.074(0.007)	0.045(0.003)		0.051(0.007) 0.035(0.003)	0.036(0.006)	0.022(0.002)	0.048(0.005)	0.037(0.006)	0.026(0.005)	0.028(0.007)	0.033(0.003)	0.029(0.002)
iris		0.027(0.005) 0.001(0.000) 0.007(0.002) 0.002(0.001)			0.002(0.001)	0.002(0.000)		0.001(0.000) 0.002(0.001) 0.001(0.001) 0.002(0.001) 0.001(0.001)				0.003(0.001)
libras	0.000(0.000) 0.000(0.000)			0.007(0.001) 0.009(0.000)	0.008(0.000)	—	0.000(0.000)	0.004(0.000)	0.004(0.000)	0.004(0.001)	0.001(0.000)	0.001(0.000)
parkinsons	0.071(0.004)	0.063(0.002)		0.073(0.005) 0.062(0.003)	0.073(0.004)	0.045(0.004)	0.059(0.003)	0.072(0.005)	0.061(0.003)	0.059(0.002)	0.061(0.004)	0.058(0.004)
planning	0.359(0.017)	0.346(0.021)		0.349(0.018) 0.341(0.022)	0.379(0.017)	0.327(0.010)	0.312(0.004) 0.375(0.011)		0.343(0.017)	0.319(0.018)	0.342(0.028)	0.343(0.013)
qsar	0.080(0.005)	0.057(0.003)		0.057(0.004) 0.052(0.004)	0.059(0.001)	0.047(0.004)	0.055(0.003)	—		0.049(0.002) 0.047(0.002) 0.055(0.004)		0.054(0.004)
seeds	0.017(0.001)	0.004(0.001)		0.007(0.001) 0.004(0.000)	0.005(0.001)	0.002(0.000)	0.007(0.002)		0.006(0.001) 0.002(0.001) 0.002(0.001)		0.003(0.000)	0.005(0.001)
wine	0.001(0.001)	0.000(0.000)		0.000(0.000) 0.001(0.000)	0.000(0.000)	0.000(0.000)	0.001(0.000)			0.000(0.000) 0.000(0.000) 0.000(0.000) 0.000(0.000)		0.000(0.000)
wine (red)	0.209(0.004)	0.171(0.003)		0.199(0.003) 0.182(0.001)	0.177(0.001)	0.179(0.002)	0.207(0.006)	0.170(0.003)		0.169(0.003) 0.161(0.003) 0.178(0.003)		0.175(0.003)
	wine (white) 0.268(0.002) 0.234(0.001) 0.262(0.001) 0.236(0.004)				0.241(0.007)	0.246(0.003)		0.257(0.004) 0.232(0.004) 0.233(0.002) 0.232(0.003) 0.237(0.000)				0.235(0.003)
yacht	—	—	—	—	—	—	—	—	—	—	—	—
yeast	0.152(0.005)	0.104(0.003)				0.146(0.003) 0.160(0.004) 0.092(0.002) 0.093(0.001)	0.153(0.005)	0.107(0.002)	0.111(0.003)	0.099(0.001)	0.153(0.003)	0.114(0.002)

Table 28: Classifier 1-AUROC results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	—	—	—	—	—	—	—	—	—	—	—	—
banknote	0.130(0.006)	0.001(0.001)	0.035(0.004)	0.001(0.000)	0.001(0.001)	0.004(0.001)		0.022(0.005) 0.000(0.000) 0.001(0.000)		0.001(0.000)	0.010(0.002)	0.006(0.001)
blood	0.299(0.009)	0.170(0.010)	0.257(0.005)	0.256(0.005)	0.222(0.006)	0.252(0.019)	0.134(0.014) 0.228(0.008)		0.217(0.004)	0.227(0.019)	0.257(0.027)	0.262(0.033)
cancer	0.007(0.003)	0.002(0.001)	0.004(0.000)	0.002(0.000)		0.002(0.000) 0.000(0.000)	0.003(0.001)	0.003(0.001)	0.000(0.000)	0.000(0.000)	0.002(0.001)	0.001(0.001)
california	—	—	—	—	—	—	—	—	—	—	—	—
climate	0.133(0.019)	0.135(0.018)	0.129(0.019)	0.196(0.009)		0.085(0.025) 0.008(0.005)	0.023(0.006)	0.078(0.022)	0.086(0.041)	0.029(0.014)	0.126(0.027)	0.144(0.047)
concrete	—	—	—	—	—	—	—	—	—	—	—	—
slump	—	—	—	—	—	—	—	—	—	—	—	—
sonar	0.048(0.009)	0.021(0.003)	0.019(0.004)	0.022(0.002)		0.034(0.001) 0.000(0.000)	0.026(0.004)	0.039(0.002)	0.010(0.003)	0.001(0.001)	0.025(0.017)	0.012(0.002)
vowel	0.157(0.003)	0.032(0.002)	0.128(0.002)	0.160(0.002)		0.038(0.003) 0.026(0.002)	0.149(0.002)	0.066(0.003)	0.039(0.003)	0.031(0.003)	0.085(0.003)	0.069(0.003)
ecoli	0.105(0.012)	0.046(0.014)	0.090(0.012)	0.103(0.012)		0.060(0.010) 0.014(0.004)	0.113(0.005)	0.063(0.014)	0.047(0.015)	0.043(0.012)	0.075(0.015)	0.052(0.010)
glass	0.139(0.010)	0.077(0.009)	0.108(0.014)	0.101(0.012)		0.081(0.013) 0.050(0.005)	0.132(0.007)	0.089(0.008)		0.064(0.002) 0.052(0.003) 0.090(0.005)		0.083(0.011)
ionosphere	0.109(0.006)	0.044(0.005)	0.073(0.009)	0.035(0.007)		0.048(0.002) 0.007(0.001)	0.057(0.005)	0.043(0.003)	0.020(0.001)	0.012(0.001)	0.034(0.004)	0.024(0.002)
iris		0.065(0.012) 0.001(0.001) 0.022(0.004)		0.003(0.002)	0.005(0.003)	0.001(0.001)	0.002(0.001)		0.004(0.002) 0.001(0.000) 0.001(0.000)		0.001(0.001)	0.006(0.001)
libras	0.000(0.000)	0.000(0.000)	0.009(0.001)	0.010(0.000)	0.006(0.000)	—	0.000(0.000) 0.002(0.000)		0.000(0.000)		0.001(0.000) 0.000(0.000)	0.000(0.000)
parkinsons	0.093(0.002)	0.051(0.005)	0.089(0.006)	0.051(0.005)		0.084(0.008) 0.013(0.005)	0.034(0.006)	0.084(0.005)	0.032(0.005)	0.023(0.001)	0.057(0.014)	0.051(0.003)
planning	0.343(0.018)	0.339(0.027)		0.258(0.062) 0.169(0.029) 0.359(0.021)		0.199(0.021)	0.235(0.017)	0.355(0.014)	0.252(0.020)	0.213(0.009)	0.303(0.040)	0.261(0.012)
qsar	0.099(0.006)	0.055(0.003)	0.059(0.001)	0.045(0.003)		0.067(0.003) 0.030(0.003)	0.052(0.006)	0.059(nan)		0.038(0.002) 0.031(0.002) 0.075(0.009)		0.059(0.004)
seeds	0.030(0.002)	0.002(0.000)	0.010(0.001)	0.008(0.001)		0.004(0.001) 0.001(0.000)	0.008(0.001)	0.007(0.002)	0.001(0.000)	0.001(0.000)	0.002(0.000)	0.002(0.000)
wine	0.007(0.002)	0.001(0.001)	0.004(0.001)	0.005(0.002)		0.001(0.000) 0.000(0.000)	0.007(0.003)			0.000(0.000) 0.000(0.000) 0.000(0.000) 0.000(0.000)		0.000(0.000)
wine (red)	0.237(0.008)	0.166(0.009)	0.222(0.006)	0.186(0.004)	0.197(0.006)	0.188(0.011)	0.221(0.011)	0.165(0.007)		0.168(0.008) 0.147(0.010) 0.198(0.008)		0.180(0.007)
		wine (white) 0.288(0.004) 0.222(0.012) 0.279(0.005)		0.229(0.006)	0.243(0.004)	0.242(0.006)		0.267(0.012) 0.215(0.012) 0.232(0.005) 0.222(0.006) 0.238(0.004)				0.233(0.002)
yacht	—	—	—	—	—	—	—	—	—	—	—	—
yeast	0.212(0.005)	0.108(0.002)	0.200(0.006)	0.230(0.003)		0.111(0.006) 0.075(0.007)	0.207(0.009)	0.107(0.010)	0.124(0.002)	0.107(0.006)	0.208(0.016)	0.134(0.006)

Table 29: Classifier 1-AUROC results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	—	—	—	—	—	—	—	—	—	—	—	—
banknote	0.209(0.009)	0.003(0.001)		0.055(0.011) 0.000(0.000) 0.004(0.001)		0.009(0.005)	0.060(0.020)	0.021(0.005)	0.002(0.001)	0.003(0.001)	0.042(0.013)	0.015(0.002)
blood		0.338(0.008) 0.152(0.007) 0.258(0.018)		0.264(0.010)	0.298(0.019)	0.186(0.021)	0.168(0.034)	0.268(0.020)	0.227(0.023)	0.246(0.024)	0.291(0.048)	0.231(0.035)
cancer	0.019(0.004)	0.005(0.001)	0.007(0.001)	0.002(0.000)		0.006(0.001) 0.000(0.000)	0.005(0.001)	0.007(0.001)	0.000(0.000)	0.000(0.000)	0.070(0.013)	0.004(0.002)
california	—	—	—	—	—	—	—	—	—	—	—	—
climate	0.254(0.012)	0.207(0.025)	0.242(0.028)	0.301(0.005)		0.257(0.014) 0.004(0.000)	0.040(0.011)	0.236(0.014)	0.159(0.072)	0.045(0.021)	0.303(0.042)	0.204(0.045)
concrete	—	—	—	—	—	—	—	—	—	—	—	—
slump	—	—	—	—	—	—	—	—	—	—	—	—
sonar	0.073(0.005)	0.030(0.004)	0.022(0.003)	0.038(0.005)		0.051(0.004) 0.000(0.000)	0.018(0.003)	0.063(0.013)	0.002(0.000)	0.000(0.000)	0.147(0.014)	0.014(0.009)
vowel	0.232(0.003)	0.031(0.004)	0.182(0.004)	0.199(0.008)		0.062(0.017) 0.016(0.002)	0.222(0.019)	0.081(0.009)	0.055(0.003)	0.043(0.008)	0.147(0.014)	0.097(0.013)
ecoli	0.157(0.022)	0.030(0.008)	0.117(0.018)	0.130(0.021)		0.077(0.012) 0.004(0.002)	0.159(0.015)	0.071(0.008)	0.073(0.021)	0.053(0.013)	0.090(0.017)	0.072(0.016)
glass	0.213(0.009)	0.064(0.009)	0.138(0.013)	0.134(0.013)		0.123(0.008) 0.017(0.003)	0.174(0.006)	0.124(0.015)	0.082(0.004)	0.070(0.007)	0.116(0.004)	0.107(0.007)
ionosphere	0.156(0.019)	0.037(0.008)	0.089(0.020)	0.095(0.017)		0.062(0.012) 0.000(0.000)	0.057(0.002)	0.069(0.010)	0.014(0.009)	0.006(0.003)	0.185(0.020)	0.033(0.008)
iris	0.079(0.012)	0.003(0.002)	0.035(0.013)	0.001(0.001)		0.014(0.005) 0.001(0.000)	0.014(0.005)	0.013(0.006)	0.003(0.001)	0.004(0.001)	0.001(0.001)	0.008(0.002)
libras	0.000(0.000)	0.000(0.000)	0.007(0.001)	0.002(0.000)	0.004(0.000)	—		0.000(0.000) 0.002(0.001)	0.000(0.000)		0.000(0.000) 0.000(0.000)	0.000(0.000)
parkinsons	0.113(0.002)	0.036(0.005)	0.089(0.006)	0.061(0.002)		0.103(0.011) 0.000(0.000)	0.025(0.003)	0.117(0.008)	0.014(0.001)	0.007(0.001)	0.076(0.022)	0.051(0.011)
planning	0.348(0.020)	0.149(0.011)	0.240(0.044)	0.290(0.009)		0.342(0.018) 0.036(0.007)	0.080(0.011)	0.322(0.019)	0.121(0.011)	0.110(0.020)	0.228(0.039)	0.164(0.017)
qsar	0.147(0.005)	0.054(0.005)	0.055(0.008)	0.081(0.006)		0.095(0.012) 0.003(0.002)	0.047(0.005)	0.100(0.008)	0.043(0.007)	0.022(0.005)	0.262(0.014)	0.079(0.010)
seeds	0.070(0.012)	0.001(0.000)	0.027(0.003)	0.018(0.001)		0.011(0.003) 0.000(0.000)	0.030(0.005)	0.013(0.004)	0.002(0.001)	0.001(0.001)	0.004(0.003)	0.003(0.002)
wine	0.055(0.008)	0.004(0.001)	0.029(0.010)	0.035(0.007)		0.016(0.002) 0.000(0.000)	0.043(0.008)	0.008(0.002)	0.000(0.000)	0.000(0.000)	0.004(0.002)	0.002(0.002)
wine (red)		0.285(0.010) 0.151(0.010) 0.265(0.011)		0.188(0.006)	0.245(0.008)	0.205(0.010)	0.230(0.007)	0.206(0.013)		0.157(0.015) 0.141(0.012) 0.263(0.014)		0.191(0.011)
	wine (white) 0.321(0.008)	0.207(0.010)	0.308(0.009)	0.216(0.006)	0.237(0.010)	0.248(0.016)		0.272(0.009) 0.184(0.005) 0.225(0.014)		0.227(0.014)	0.283(0.013)	0.236(0.010)
yacht	—	—	—	—	—	—	—	—	—	—	—	—
yeast	0.270(0.008)	0.104(0.006)	0.245(0.009)	0.293(0.007)		0.197(0.019) 0.049(0.007)	0.252(0.014)	0.170(0.013)	0.159(0.002)	0.108(0.005)	0.301(0.016)	0.166(0.010)

Table 30: Classifier 1-AUROC results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	—	—	—	—	—	—	—	—	—	—	—	—
banknote	0.063(0.029)	0.001(0.000)		0.022(0.009) 0.000(0.000) 0.000(0.000)		0.003(0.001)	0.007(0.001)	0.002(0.002)	0.001(0.000)	0.001(0.000)	0.004(0.003)	0.003(0.001)
blood	0.274(0.012)	0.204(0.009)	0.261(0.008)	0.254(0.007)	0.228(0.008)	0.229(0.005)	0.177(0.026) 0.228(0.006)		0.241(0.002)	0.233(0.006)	0.263(0.014)	0.253(0.011)
cancer	0.004(0.000)	0.003(0.000)	0.003(0.000)	0.002(0.000)		0.003(0.000) 0.001(0.000)	0.003(0.000)	0.002(0.000)	0.002(0.000)	0.001(0.000)	0.002(0.000)	0.002(0.000)
	—	—	—	—	—	—	—	—	—	—	—	—
california		0.058(0.020)	0.060(0.018)	0.097(0.031)		0.026(0.003) 0.013(0.004)		0.037(0.008)	0.051(0.024)	0.031(0.014)	0.071(0.030)	0.078(0.036)
climate	0.061(0.018) —	—	—	—	—	—	0.036(0.009) —	—	—	—	—	—
concrete	—	—	—	—	—	—	—	—	—	—	—	—
slump		0.012(0.005)	0.014(0.003)	0.015(0.004)		0.019(0.005) 0.003(0.002)		0.017(0.007)		0.010(0.004) 0.004(0.002) 0.016(0.006)
sonar	0.019(0.007)			0.085(0.012)		0.033(0.004) 0.029(0.004)	0.016(0.003)	0.043(0.004)		0.032(0.001) 0.030(0.002) 0.050(0.003)		0.004(0.002) 0.044(0.002)
vowel		0.080(0.012) 0.030(0.003) 0.070(0.009) 0.049(0.011)	0.064(0.010)	0.071(0.009)		0.046(0.012) 0.015(0.002)	0.069(0.010)	0.049(0.012)	0.034(0.006)	0.035(0.007)	0.053(0.015)	0.041(0.010)
ecoli	0.076(0.006)						0.064(0.006)
glass		0.085(0.008) 0.054(0.003) 0.076(0.006)		0.078(0.003)		0.062(0.002) 0.055(0.002)	0.085(0.002)			0.069(0.003) 0.057(0.007) 0.055(0.004) 0.067(0.003)		0.067(0.003)
ionosphere	0.066(0.003)	0.054(0.001)	0.060(0.001)	0.046(0.001)		0.039(0.003) 0.018(0.006)	0.036(0.006)	0.038(0.003)		0.028(0.004) 0.023(0.002) 0.030(0.005)		0.032(0.005)
iris		0.019(0.012) 0.000(0.000) 0.003(0.002)		0.001(0.000)	0.000(0.000)	0.001(0.000)	0.000(0.000)	0.001(0.000)	0.000(0.000)	0.002(0.001)	0.001(0.000)	0.002(0.001)
libras	0.000(0.000) 0.000(0.000)		0.008(0.001)	0.009(0.000)	0.007(0.000)	—	0.000(0.000)	0.003(0.000)	0.004(0.000)	0.004(0.000)	0.001(0.000)	0.000(0.000)
parkinsons						0.064(0.005) 0.062(0.007) 0.065(0.008) 0.060(0.006) 0.065(0.008) 0.056(0.009) 0.061(0.010) 0.067(0.008) 0.058(0.006) 0.057(0.007) 0.064(0.007)						0.061(0.006)
planning	0.403(0.028)	0.371(0.019)	0.372(0.023)	0.370(0.022)		0.418(0.014) 0.311(0.028) 0.330(0.027) 0.409(0.027)				0.372(0.036) 0.338(0.029) 0.356(0.032)		0.349(0.025)
qsar	0.072(0.002)	0.060(0.002)	0.056(0.002)	0.054(0.002)		0.059(0.001) 0.050(0.002)	0.060(0.004)	—	0.053(0.001)	0.052(0.001)	0.058(0.002)	0.056(0.002)
seeds	0.009(0.003)	0.003(0.001)	0.006(0.002)	0.004(0.001)		0.004(0.000) 0.002(0.001)	0.005(0.001)		0.004(0.001) 0.002(0.000) 0.002(0.000)		0.004(0.000)	0.004(0.001)
wine	0.002(0.002)	0.002(0.001)	0.002(0.001)	0.001(0.001)		0.000(0.000) 0.000(0.000)	0.000(0.000)	0.000(0.000)		0.000(0.000) 0.000(0.000) 0.000(0.000)		0.000(0.000)
wine (red)	0.200(0.007)	0.180(0.008)	0.197(0.005)	0.186(0.004)	0.179(0.008)	0.180(0.007)				0.190(0.004) 0.174(0.008) 0.172(0.007) 0.166(0.008) 0.184(0.008)		0.178(0.008)
	wine (white) 0.244(0.005)	0.217(0.005)	0.242(0.002)	0.228(0.006)	0.227(0.002)	0.225(0.004)				0.234(0.002) 0.216(0.005) 0.221(0.007) 0.211(0.006) 0.227(0.003)		0.222(0.005)
yacht	—	—	—	—	—	—	—	—	—	—	—	—
yeast	0.146(0.003)	0.112(0.007)	0.143(0.004)	0.154(0.003)		0.098(0.005) 0.093(0.002)	0.142(0.004)	0.117(0.012)	0.117(0.004)	0.103(0.006)	0.155(0.005)	0.121(0.006)

Table 31: Regression CRPS results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		2.021(0.025) 1.370(0.011) 1.795(0.053)		1.269(0.010)	0.723(0.009)	0.967(0.005)		1.355(0.059) 0.630(0.019) 0.654(0.017)		0.628(0.019) 1.174(0.013)		1.064(0.004)
banknote		1.124(0.052) 0.286(0.002) 0.757(0.024)		0.325(0.003)	0.235(0.003)	0.283(0.006)	0.589(0.063)	0.240(0.008)	0.215(0.001) 0.220(0.001)		0.279(0.007)	0.289(0.007)
blood		0.161(0.003) 0.093(0.005) 0.116(0.003)		0.095(0.001)	0.086(0.005)	0.091(0.004)	0.105(0.003)	0.081(0.001)		0.071(0.004) 0.072(0.005) 0.080(0.003)		0.077(0.004)
cancer				1.091(0.022) 0.303(0.012) 0.240(0.015) 0.176(0.010) 0.253(0.008)		0.278(0.012)	0.381(0.014)	0.327(0.010)	0.248(0.012)	0.256(0.013)	0.253(0.007)	0.240(0.006)
california		0.457(0.001) 0.170(0.001) 0.259(0.001)		0.158(0.000)	0.117(0.001)	0.191(0.000)		0.368(0.022) 0.115(0.000) 0.118(0.001)		0.115(0.001) 0.150(0.003)		0.139(0.001)
climate						0.544(0.009) 0.553(0.014) 0.540(0.008) 0.445(0.007) 0.502(0.008) 0.444(0.010) 0.483(0.012)		0.516(0.010)	0.513(0.007)	0.523(0.003)	0.441(0.010)	0.455(0.005)
concrete		4.353(0.042) 3.545(0.021) 3.606(0.328)		1.847(0.021)	1.160(0.027)	1.532(0.036)	3.175(0.212)	1.343(0.030)	1.135(0.021) 1.174(0.013)		1.608(0.027)	1.580(0.038)
slump				1.416(0.108) 1.175(0.020) 1.163(0.053) 0.770(0.034) 1.014(0.040)		1.024(0.051)	1.018(0.073)	1.061(0.073)	1.043(0.064)	1.139(0.010)	0.966(0.069)	1.024(0.080)
sonar				0.846(0.026) 0.541(0.009) 0.474(0.015) 0.403(0.010) 0.439(0.001)		0.458(0.013)	0.772(0.022)	0.522(0.009)	0.559(0.012)	0.485(0.015)	0.450(0.035)	0.459(0.004)
vowel		1.733(0.024) 0.778(0.014) 1.366(0.010)		1.005(0.008)	0.605(0.003)	0.550(0.014)	1.318(0.013)	0.682(0.021)	0.509(0.004)	0.445(0.004) 0.759(0.004)		0.635(0.007)
ecoli		0.935(0.023) 0.538(0.024) 0.723(0.018)		0.492(0.008)	0.460(0.011)	0.488(0.107)	0.800(0.074)	0.547(0.022)	0.418(0.017)	0.429(0.019)	0.406(0.017)	0.395(0.010)
glass				0.824(0.015) 0.487(0.020) 0.557(0.025) 0.379(0.022) 0.426(0.034)		0.405(0.014)	0.587(0.009)	0.466(0.028)	0.405(0.030)		0.391(0.037) 0.379(0.054)	0.382(0.033)
ionosphere		0.729(0.025) 0.469(0.007) 0.502(0.021)		0.436(0.006)		0.388(0.020) 0.346(0.018) 0.525(0.018)		0.446(0.023)	0.390(0.024)	0.384(0.020)	0.364(0.022)	0.394(0.014)
iris		1.105(0.029) 0.278(0.011) 0.468(0.024)		0.272(0.009)		0.304(0.003) 0.236(0.005) 0.339(0.035)		0.295(0.006)	0.299(0.008)	0.301(0.001)	0.245(0.020)	0.271(0.018)
libras				2.350(0.023) 1.350(0.021) 0.292(0.065) 0.061(0.003) 0.146(0.008)		—	0.632(0.031)	0.556(0.018)	0.336(0.018)	0.262(0.007)	0.587(0.017)	0.349(0.002)
parkinsons				0.403(0.022) 0.172(0.012) 0.176(0.009) 0.132(0.004) 0.162(0.011)		0.165(0.010)	0.218(0.004)	0.193(0.010)	0.178(0.011)	0.173(0.013)	0.154(0.007)	0.142(0.013)
planning				0.105(0.005) 0.056(0.003) 0.036(0.002) 0.022(0.000) 0.037(0.003)		0.057(0.005)	0.050(0.003)	0.060(0.004)	0.058(0.006)	0.058(0.005)	0.037(0.003)	0.043(0.002)
qsar		0.705(0.003) 0.343(0.007) 0.343(0.004)		0.288(0.005)	0.218(0.007)	0.306(0.010)	0.411(0.008)	—	0.201(0.007)	0.196(0.003) 0.338(0.013)		0.258(0.005)
seeds				1.152(0.009) 0.272(0.015) 0.307(0.026) 0.188(0.012) 0.238(0.012)		0.256(0.009)	0.324(0.071)	0.294(0.020)	0.266(0.012)	0.268(0.017)	0.218(0.013)	0.243(0.016)
wine		1.148(0.015) 0.504(0.020) 0.708(0.023)		0.495(0.019)		0.495(0.009) 0.452(0.017) 0.630(0.031)		0.540(0.017)	0.548(0.024)	0.508(0.017)	0.469(0.019)	0.474(0.010)
wine (red)		0.148(0.001) 0.091(0.001) 0.124(0.001)		0.094(0.000)	0.086(0.001)	0.086(0.002)		0.122(0.001) 0.079(0.001) 0.081(0.001)		0.080(0.001)	0.086(0.001)	0.083(0.001)
	wine (white) 0.216(0.002) 0.101(0.001) 0.125(0.001)			0.088(0.001)	0.085(0.001)	0.104(0.003)		0.142(0.002) 0.074(0.001) 0.076(0.001)		0.077(0.001)	0.093(0.000)	0.090(0.000)
yacht		2.641(0.101) 0.670(0.033) 1.841(0.074)		1.341(0.060)	0.206(0.019)	1.190(0.027)	1.198(0.024)	1.231(0.065)	0.196(0.003) 0.208(0.005)		0.315(0.029)	0.397(0.028)
yeast		0.648(0.005) 0.505(0.003) 0.594(0.012)		0.519(0.005)		0.498(0.004) 0.434(0.003) 0.610(0.004)		0.452(0.005)	0.445(0.004)	0.454(0.004)	0.502(0.010)	0.441(0.005)

Table 32: Regression CRPS results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil		3.095(0.006) 2.193(0.013) 2.915(0.006)		2.121(0.010)	1.567(0.023)	1.721(0.020)		2.195(0.141) 1.331(0.008) 1.487(0.020)		1.410(0.009)	2.032(0.059)	1.948(0.087)
banknote		1.756(0.015) 0.522(0.018) 1.266(0.017)		0.515(0.005)	0.467(0.016)	0.515(0.009)	0.870(0.012)		0.449(0.010) 0.422(0.008) 0.436(0.009)		0.557(0.019)	0.523(0.010)
blood		0.266(0.005) 0.183(0.004) 0.219(0.004)		0.181(0.002)	0.184(0.006)	0.176(0.003)	0.197(0.003)		0.193(0.002) 0.152(0.005) 0.163(0.003)		0.166(0.005)	0.165(0.001)
cancer				2.063(0.012) 0.572(0.018) 0.535(0.019) 0.388(0.014) 0.510(0.012)		0.615(0.010)	0.790(0.032)	0.573(0.008)	0.583(0.006)	0.571(0.009)	0.722(0.044)	0.558(0.006)
california		0.661(0.001) 0.346(0.001) 0.430(0.000)		0.258(0.001)	0.222(0.002)	0.289(0.007)		0.477(0.030) 0.207(0.002) 0.221(0.002)		0.211(0.001)	0.299(0.013)	0.247(0.002)
climate		0.832(0.013) 0.783(0.015) 0.827(0.015)		0.684(0.009)	0.760(0.004)	0.666(0.009) 0.695(0.006)		0.788(0.020)	0.748(0.016)	0.716(0.010)	0.682(0.018)	0.721(0.026)
concrete		6.917(0.093) 5.508(0.082) 5.860(0.186)			3.730(0.080) 3.044(0.055)	3.487(0.024)	5.346(0.331)	4.546(0.105)	3.162(0.056)	3.219(0.027)	3.732(0.016)	3.513(0.043)
slump				2.611(0.053) 2.138(0.039) 2.293(0.009) 1.861(0.037) 2.071(0.042)		1.842(0.118) 2.040(0.060)		2.081(0.083)	2.189(0.110)	2.309(0.109)	2.032(0.106)	1.942(0.105)
sonar		1.279(0.044) 0.829(0.010) 0.848(0.016)		0.697(0.029)	0.741(0.010)	0.820(0.005)	1.320(0.033)	0.854(0.015)	0.865(0.022)	0.770(0.028)	0.927(0.032)	0.676(0.010)
vowel		2.849(0.039) 1.373(0.019) 2.468(0.031)		1.813(0.014)	1.301(0.008)	1.199(0.007)	2.192(0.079)	1.851(0.012)		1.208(0.029) 1.084(0.006) 1.425(0.005)		1.287(0.007)
ecoli		1.666(0.027) 0.920(0.013) 1.415(0.019)		0.895(0.019)	0.790(0.025)	1.427(0.043)	1.255(0.022)	0.857(0.013)	0.787(0.007)	0.790(0.012)	0.754(0.017)	0.738(0.010)
glass		1.303(0.017) 0.903(0.007) 1.044(0.011)		0.797(0.010)	0.767(0.053)	0.737(0.031)	1.032(0.026)	0.783(0.022)	0.774(0.032)	0.743(0.045)	0.751(0.052)	0.693(0.025)
ionosphere		1.076(0.015) 0.625(0.044) 0.784(0.012)		0.653(0.017)	0.598(0.020)	0.552(0.011) 0.798(0.028)		0.637(0.004)	0.604(0.029)	0.584(0.024)	0.650(0.035)	0.582(0.013)
iris		1.995(0.051) 0.565(0.010) 0.939(0.043)		0.483(0.022)	0.536(0.060)			0.438(0.036) 0.644(0.038) 0.459(0.008) 0.503(0.036)			0.497(0.032) 0.454(0.027)	0.474(0.015)
libras				4.284(0.032) 2.187(0.019) 0.512(0.045) 0.198(0.003) 0.444(0.013)		—	2.405(0.044)	1.166(0.014)	0.931(0.020)	0.769(0.005)	3.374(0.332)	0.893(0.011)
parkinsons				0.663(0.024) 0.336(0.009) 0.352(0.010) 0.271(0.004) 0.331(0.005)		0.355(0.007)	0.399(0.020)	0.350(0.010)	0.360(0.008)	0.359(0.005)	0.372(0.023)	0.300(0.005)
planning				0.132(0.003) 0.080(0.003) 0.061(0.004) 0.044(0.001) 0.062(0.001)		0.090(0.007)	0.077(0.007)	0.075(0.004)	0.086(0.008)	0.091(0.005)	0.068(0.006)	0.077(0.006)
qsar		1.047(0.012) 0.560(0.011) 0.593(0.009)		0.506(0.006)	0.468(0.006)	0.586(0.014)		0.748(0.019) 0.390(nan)	0.456(0.007)	0.422(0.006)	1.056(0.188)	0.498(0.006)
seeds				2.003(0.035) 0.529(0.021) 0.704(0.025) 0.404(0.010) 0.514(0.022)		0.548(0.024)	0.669(0.020)	0.551(0.017)	0.579(0.020)	0.526(0.005)	0.477(0.003)	0.484(0.006)
wine		2.116(0.028) 0.883(0.015) 1.353(0.028)		0.867(0.013)	0.890(0.029)	0.751(0.015) 1.358(0.072)		0.871(0.005)	0.889(0.038)	0.933(0.011)	0.843(0.031)	0.788(0.022)
wine (red)		0.241(0.003) 0.164(0.002) 0.214(0.005)		0.158(0.003)	0.169(0.004)	0.153(0.003)			0.208(0.005) 0.149(0.003) 0.151(0.003) 0.149(0.002) 0.157(0.001)			0.153(0.002)
	wine (white) 0.314(0.002) 0.179(0.001) 0.220(0.002)			0.158(0.000)	0.169(0.003)	0.166(0.004)	0.216(0.015)		0.156(0.001) 0.149(0.000) 0.152(0.000)		0.172(0.001)	0.165(0.001)
yacht		5.540(0.244) 1.573(0.212) 4.019(0.391)			2.579(0.027) 0.471(0.054)	2.648(0.204)	2.208(0.098)	2.339(0.100)	0.714(0.065)	0.747(0.119)	0.710(0.052)	1.013(0.067)
yeast		1.124(0.014) 0.880(0.014) 1.055(0.017)		0.898(0.013)	0.886(0.017)	0.766(0.009) 0.960(0.009)		0.811(0.014)	0.776(0.008)	0.796(0.003)	0.906(0.018)	0.773(0.008)

Table 33: Regression CRPS results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	3.616(0.041)	2.877(0.030)	3.445(0.086)	2.966(0.016)	2.206(0.042)	2.440(0.082)	2.956(0.013)	2.770(0.033)		2.318(0.059) 2.070(0.050) 2.589(0.072)		2.574(0.054)
banknote	1.918(0.010)	0.673(0.017)		1.485(0.030) 0.562(0.019) 0.673(0.027)		0.742(0.013)	1.187(0.101)	0.921(0.017)	0.635(0.031)	0.629(0.036)	0.765(0.053)	0.701(0.032)
blood	0.328(0.003)	0.261(0.002)		0.294(0.003) 0.243(0.004) 0.302(0.023)		0.277(0.017)	0.253(0.008)	0.260(0.008)	0.253(0.002)	0.259(0.003)	0.270(0.019)	0.245(0.006)
cancer	3.003(0.025)	0.953(0.008)		1.091(0.013) 0.741(0.008) 0.861(0.009)		1.162(0.009)	1.168(0.002)	0.957(0.014)	1.188(0.064)	1.057(0.007)	1.734(0.259)	1.049(0.058)
california	0.717(0.002)	0.522(0.004)	0.537(0.003)	0.317(0.001)	0.297(0.002)	0.341(0.008)		0.619(0.054) 0.285(0.003) 0.306(0.001)		0.299(0.003)	0.650(0.047)	0.339(0.003)
climate	1.041(0.012)	0.971(0.007)	1.026(0.015)	0.865(0.016)	0.952(0.022)	0.809(0.017) 0.873(0.019)		0.985(0.022)	0.906(0.019)	0.870(0.010)	1.036(0.031)	0.877(0.006)
concrete	8.173(0.110)	6.472(0.059)		7.296(0.085) 5.175(0.113) 5.502(0.165)		5.392(0.255)	6.098(0.106)			6.106(0.340) 5.274(0.154) 5.130(0.182) 5.187(0.146)		5.177(0.049)
slump	3.381(0.278)	2.911(0.189)	2.897(0.198)	2.814(0.290)	2.845(0.229)	2.926(0.101)	3.061(0.507)	2.962(0.156)		2.796(0.056) 2.696(0.033) 2.721(0.199)		2.716(0.143)
sonar	1.645(0.031)	1.170(0.020)		1.245(0.025) 0.991(0.060) 1.124(0.027)		1.068(0.020)	1.604(0.061)	1.228(0.031)	1.151(0.072)	1.124(0.047)	4.121(0.512)	1.092(0.020)
vowel	3.622(0.011)	2.125(0.024)	3.281(0.015)	2.404(0.022)	1.957(0.071)	1.850(0.057) 2.715(0.077)		2.466(0.038)		1.984(0.043) 1.847(0.040) 2.152(0.043)		1.931(0.029)
ecoli	2.137(0.060)	1.237(0.047)	1.918(0.033)	1.232(0.024)	1.174(0.061)	2.593(0.059)	1.457(0.050)	1.204(0.050)	1.177(0.053)	1.165(0.038)	1.166(0.059)	1.107(0.052)
glass	1.633(0.009)	1.215(0.032)	1.482(0.038)	1.122(0.021)	1.140(0.058)	2.195(0.870)	1.564(0.081)	1.106(0.060)	1.097(0.006)	1.086(0.019)	1.067(0.011)	1.061(0.023)
ionosphere		1.297(0.028) 0.812(0.058) 1.074(0.021)			0.924(0.022) 0.856(0.066)	0.884(0.101)	1.208(0.017)			0.914(0.042) 0.849(0.026) 0.843(0.023) 2.476(0.319)		0.853(0.049)
iris	2.358(0.052)	0.877(0.077)		1.300(0.119) 0.619(0.012) 0.712(0.032)		0.584(0.037) 0.896(0.078)		0.695(0.057)	0.910(0.054)	0.817(0.051)	0.743(0.048)	0.683(0.055)
libras	5.836(0.016)	2.705(0.048)		2.279(0.226) 1.053(0.048) 1.299(0.049)		—	3.934(0.027)	2.166(0.046)	2.542(0.039)	2.503(0.146)	91.685(10.103)	1.717(0.023)
parkinsons	0.892(0.001)	0.529(0.010)		0.544(0.004) 0.437(0.003) 0.533(0.006)		0.600(0.030)	0.736(0.007)	0.544(0.011)	0.517(0.022)	0.564(0.026)	0.737(0.050)	0.466(0.028)
planning	0.149(0.008)		0.125(0.010) 0.102(0.008) 0.139(0.011) 0.106(0.005)			0.143(0.009)	0.184(0.022)	0.125(0.024)	0.121(0.002)	0.120(0.010)	0.135(0.004)	0.130(0.002)
qsar	1.365(0.008)	0.856(0.025)	0.926(0.027)	0.802(0.006)	0.806(0.039)	0.944(0.010)	1.350(0.136)	0.849(0.011)		0.875(0.068) 0.753(0.030) 7.783(1.713)		0.886(0.027)
seeds	2.655(0.071)	0.848(0.026)	1.276(0.047)		0.810(0.033) 0.778(0.022)	0.910(0.041)	1.272(0.111)	0.871(0.044)	0.987(0.094)	0.906(0.064)	0.845(0.031)	0.757(0.027)
wine	2.960(0.036)	1.269(0.018)	2.132(0.065)	1.408(0.051)	1.319(0.077)	0.964(0.020) 1.928(0.075)		1.326(0.066)	1.483(0.136)	1.428(0.105)	1.421(0.038)	1.286(0.072)
wine (red)	0.322(0.005)	0.238(0.009)		0.302(0.004) 0.218(0.006) 0.246(0.010)		0.239(0.009)	0.268(0.006)			0.248(0.007) 0.219(0.004) 0.215(0.006) 0.290(0.013)		0.216(0.006)
	wine (white) 0.357(0.004)	0.255(0.004)		0.297(0.002) 0.221(0.001) 0.231(0.001)		0.242(0.004)	0.277(0.010)		0.223(0.002) 0.220(0.003) 0.225(0.004)		0.314(0.003)	0.241(0.003)
yacht	7.952(0.153)	3.447(0.364)	5.285(0.205)		3.920(0.088) 0.951(0.115)	4.276(0.331)	2.636(0.169)	3.300(0.286)	1.701(0.158)	1.569(0.150)	1.459(0.067)	1.885(0.125)
yeast	1.494(0.011)	1.234(0.002)	1.427(0.011)	1.210(0.009)	1.224(0.026)	1.212(0.045)	1.281(0.018)	1.299(0.017)	1.097(0.010)	1.097(0.011)	1.570(0.072)	1.088(0.007)

Table 34: Regression CRPS results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	2.049(0.009)	1.522(0.010)	1.903(0.035)	1.239(0.042)	0.743(0.025)	0.966(0.057)		1.300(0.079) 0.633(0.046) 0.666(0.048) 0.617(0.043) 1.130(0.024)				1.040(0.046)
banknote	1.190(0.229)	0.331(0.029)	0.791(0.130)	0.356(0.017)	0.253(0.017)	0.351(0.038)	0.504(0.015)			0.458(0.203) 0.231(0.015) 0.238(0.012) 0.324(0.045)		0.296(0.012)
blood	0.161(0.023)	0.102(0.022)	0.124(0.028)		0.097(0.020) 0.096(0.023)	0.100(0.021)	0.109(0.018)			0.111(0.023) 0.078(0.018) 0.085(0.020) 0.087(0.019)		0.089(0.018)
cancer	1.464(0.088)	0.398(0.015)		0.312(0.010) 0.196(0.008) 0.329(0.008)		0.531(0.055)	0.576(0.102)	0.380(0.005)	0.373(0.031)	0.356(0.014)	0.276(0.007)	0.271(0.005)
california	0.417(0.035)	0.182(0.044)	0.220(0.043)		0.144(0.016) 0.103(0.017)	0.155(0.027)		0.312(0.048) 0.105(0.013) 0.112(0.018) 0.098(0.016) 0.157(0.025)				0.130(0.016)
climate	0.492(0.096)	0.500(0.102)		0.490(0.096) 0.404(0.078) 0.459(0.096)				0.395(0.071) 0.428(0.080) 0.462(0.088) 0.455(0.099) 0.456(0.095) 0.397(0.080)				0.408(0.086)
concrete	3.507(1.005)	2.965(0.767)	2.805(0.931)		1.574(0.287) 1.107(0.237)	1.295(0.270)	2.246(0.869)			1.511(0.546) 1.012(0.227) 1.044(0.260) 1.328(0.341)		1.299(0.310)
slump	1.707(0.138)	1.402(0.087)		1.544(0.185) 0.990(0.069) 1.452(0.055)		1.195(0.167)	1.243(0.075)	1.365(0.073)	1.343(0.149)	1.391(0.199)	1.154(0.091)	1.173(0.107)
sonar	0.957(0.078)	0.657(0.071)		0.577(0.056) 0.440(0.043) 0.507(0.034)		0.647(0.071)	0.754(0.059)	0.621(0.068)	0.667(0.069)	0.584(0.073)	0.493(0.074)	0.480(0.073)
vowel	1.650(0.206)	0.794(0.094)	1.327(0.174)	0.992(0.129)	0.725(0.076)	0.634(0.054)	1.204(0.165)			0.826(0.058) 0.554(0.066) 0.505(0.061) 0.768(0.100)		0.640(0.090)
ecoli	0.850(0.078)	0.468(0.047)	0.675(0.039)	0.486(0.020)	0.423(0.031)	1.665(0.614)	0.561(0.057)			0.471(0.030) 0.397(0.046) 0.387(0.048) 0.383(0.033)		0.375(0.035)
glass	0.780(0.108)	0.485(0.036)		0.545(0.081) 0.381(0.040) 0.438(0.032)		0.449(0.024)	0.527(0.084)	0.491(0.042)	0.425(0.037)		0.443(0.036) 0.362(0.035)	0.397(0.037)
ionosphere	0.695(0.040)	0.440(0.035)	0.473(0.022)	0.425(0.021)	0.361(0.024)	0.335(0.025)	0.468(0.025)	0.416(0.026)	0.358(0.030)		0.343(0.032) 0.317(0.025)	0.356(0.024)
iris	1.248(0.052)	0.262(0.040)	0.493(0.097)	0.273(0.026)	0.271(0.044)		0.231(0.023) 0.242(0.029) 0.275(0.041)		0.272(0.024)		0.281(0.031) 0.234(0.033)	0.252(0.031)
libras	2.157(0.103)	1.376(0.075)		0.172(0.031) 0.060(0.005) 0.148(0.009)		—	0.688(0.028)	0.550(0.006)	0.288(0.023)	0.264(0.008)	0.458(0.073)	0.336(0.014)
parkinsons	0.533(0.074)	0.254(0.034)		0.238(0.036) 0.157(0.010) 0.221(0.025)		0.327(0.047)	0.316(0.058)	0.251(0.032)	0.272(0.030)	0.274(0.038)	0.220(0.039)	0.187(0.023)
planning	0.114(0.002)	0.056(0.003)		0.040(0.001) 0.025(0.003) 0.037(0.002)		0.065(0.007)	0.051(0.004)	0.063(0.002)	0.060(0.004)	0.065(0.005)	0.036(0.002)	0.045(0.003)
qsar	0.743(0.056)	0.410(0.044)	0.397(0.043)		0.312(0.041) 0.270(0.027)	0.403(0.040)	0.508(0.058)	—		0.254(0.039) 0.252(0.027) 0.323(0.008)		0.288(0.033)
seeds	1.227(0.058)	0.306(0.015)		0.394(0.026) 0.216(0.012) 0.283(0.023)		0.295(0.027)	0.292(0.026)	0.314(0.017)	0.299(0.032)	0.309(0.018)	0.247(0.024)	0.256(0.018)
wine	1.171(0.061)	0.484(0.047)		0.707(0.087) 0.463(0.037) 0.489(0.031)		0.441(0.040)	0.637(0.036)	0.513(0.047)	0.514(0.029)		0.492(0.026) 0.434(0.032)	0.439(0.024)
wine (red)		0.117(0.019) 0.077(0.011) 0.098(0.016)			0.081(0.014) 0.077(0.010)	0.072(0.011)		0.092(0.014) 0.070(0.011) 0.068(0.010) 0.068(0.010) 0.073(0.010)				0.070(0.009)
	wine (white) 0.232(0.011)	0.114(0.007)	0.133(0.014)	0.092(0.011)	0.093(0.010)	0.105(0.012)		0.148(0.019) 0.089(0.009) 0.083(0.008) 0.084(0.008) 0.101(0.010)				0.097(0.009)
yacht	4.184(0.095)	0.912(0.005)	2.782(0.124)		1.992(0.097) 0.277(0.009)	1.731(0.039)	1.734(0.250)	1.782(0.101)	0.329(0.038)	0.321(0.037)	0.424(0.038)	0.616(0.016)
yeast	0.524(0.049)	0.432(0.034)	0.491(0.046)	0.431(0.039)	0.433(0.030)	0.602(0.048)	0.443(0.042)			0.417(0.026) 0.375(0.034) 0.385(0.030) 0.432(0.051)		0.375(0.034)

Table 35: Regression parameter percent bias results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	21.045(1.359)	10.460(0.292)	17.333(0.746)	13.165(2.723)	7.589(1.477)	10.515(0.793)	14.425(1.985)	6.175(1.091)	3.425(0.282)	2.805(0.250)	16.482(1.733)	11.222(0.739)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	77.777(13.927)	38.181(2.644)	32.442(5.355)	9.444(0.715)	46.917(17.458)	51.159(8.679)	85.863(1.048)	23.385(2.098)	63.626(24.175)	16.535(1.369)	254.163(184.353)	55.665(15.974)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	302.035(20.400)	302.235(19.217)		294.243(24.996) 136.610(35.935) 110.254(27.506) 305.614(11.213)			413.372(46.552)	369.057(135.365)	165.339(48.861)	204.463(38.303)	268.286(17.456)	310.936(26.129)
slump		78.495(20.796) 110.543(14.815)	125.968(52.009)		854.288(336.502) 72.560(10.998)			62.165(21.648) 90.105(42.687) 73.605(29.381) 108.091(48.617)		96.965(27.279)	181.395(34.612)	75.143(12.226)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	172.779(25.802)	68.338(14.578)	101.911(27.655)	48.407(4.136)	92.803(36.892)	78.343(17.559)	201.439(50.128)	142.143(21.999)	63.660(1.413)	59.686(10.312)	45.167(5.614)	42.894(14.592)
	wine (white) 224.253(2.751)	54.640(4.806)	100.118(15.993)	78.161(2.669)	109.051(9.369)	147.694(7.847)	207.932(3.897)	160.160(93.121)	56.121(8.199)		37.912(8.597) 114.708(11.422)	88.611(11.489)
yacht		191.655(19.215) 214.740(124.081) 281.762(104.642) 1770.158(372.069)						340.446(141.402) 91.521(20.292) 391.877(120.655) 308.173(185.478) 865.391(130.019) 823.299(425.832) 484.072(172.121)				210.212(117.268)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 36: Regression parameter percent bias results: mean and standard error. MCAR 50% missingness.

		OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
	Mean
airfoil	30.164(1.345)	25.026(2.215)	28.638(1.167)	31.068(1.060)	17.061(1.933)	17.780(4.203)	25.868(7.516)	10.711(1.417)	6.408(0.863)	6.662(2.472)	31.016(2.336)	27.933(4.398)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	195.899(78.059)	230.868(123.919)		196.887(109.694) 121.330(61.414) 261.737(119.741) 135.197(51.557) 515.142(43.557)				266.139(170.538)	245.931(69.111)	194.144(71.127)	248.495(44.091)	137.803(36.286)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete								490.087(162.544) 519.608(185.116) 531.738(191.664) 608.442(280.425) 474.851(311.127) 532.359(188.055) 910.557(605.482) 679.202(318.341) 557.863(241.775) 618.978(335.091) 538.531(260.705) 543.975(278.703)
slump		147.970(79.960) 190.120(86.435)		217.407(99.146) 214.062(134.258) 172.069(31.218)		209.055(87.663) 264.016(78.852)		228.191(120.423)		232.867(106.208) 188.507(74.634) 221.421(100.098) 231.405(119.389)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons
planning	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —	— —
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine wine (red)		154.261(11.164)	190.113(34.844)	164.927(16.038)			290.472(94.224)	577.749(315.259)	153.290(41.436)	194.434(18.321)	176.538(28.078)	182.659(4.271)
	298.755(35.916)				109.440(25.823)	184.634(51.887)
wine (white)	302.386(9.308)	98.927(16.342)	151.003(16.046)	116.727(26.766)	145.214(51.431)	196.399(34.081)	387.879(108.687)	784.462(510.438)	136.034(16.359)	151.505(22.129)	173.901(14.611)	156.762(25.278)
yacht	134.562(56.651)	208.142(66.425)	191.385(65.517)	1102.681(662.672)	538.860(445.865)	89.544(17.955) 458.770(363.241)		142.544(21.163)	194.995(96.794)	212.337(75.022)	182.392(77.395)	242.751(106.013)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 37: Regression parameter percent bias results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	30.916(2.873)	37.036(4.657)	28.294(4.907)	49.193(0.312)	16.371(1.987)	30.887(8.592)	40.296(13.014)	31.175(6.761)	23.317(5.226)	25.268(7.701)	40.258(5.365)	38.396(3.663)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california		674.354(254.270) 994.691(379.388)	659.889(30.226)		650.030(173.367) 564.243(176.231)	381.406(75.526)		5371.358(3421.220) 5999.860(4875.670) 626.232(306.192) 423.213(108.353) 101.168(21.362) 335.730(79.650)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete slump	139.754(23.056)	199.362(33.550)		390.200(86.410) 76.532(13.378) 350.394(37.257)		275.736(64.425) 356.269(96.549) 266.267(35.671) 296.744(84.523) 409.639(234.340) 328.830(78.522) 602.158(325.394) 142.198(11.597)	216.516(53.702)	403.673(150.452) 318.459(82.535)	235.437(73.063)	232.057(73.652)	219.188(79.416)	449.981(117.986) 381.937(90.245) 262.211(76.465) 298.210(89.966) 226.209(52.651)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	248.858(32.138)	298.209(27.160)	203.154(51.330)	124.053(43.398)	131.231(24.669)	176.373(47.994)	479.186(93.116)	167.319(49.848)	266.174(88.039)			248.902(84.530) 104.401(13.114) 217.148(49.364)
	wine (white) 581.875(69.429)	419.985(57.398)	518.416(20.856)	380.030(63.350)	407.376(90.624)	744.367(167.116)		2034.211(342.315) 5433.388(2353.917) 456.634(95.353)				471.801(92.505) 155.494(3.133) 396.736(122.326)
yacht	259.578(50.103)	234.774(48.184)		274.075(80.997) 106.727(32.515) 213.112(121.591)		186.218(11.032)	195.323(78.984)	316.952(62.806)	321.317(101.557) 302.058(88.222)		249.788(93.300)	325.439(72.934)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 38: Regression parameter percent bias results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	17.910(2.865)	22.072(5.118)	16.229(2.977)	13.842(0.571)	9.019(1.987)	13.938(4.971)	13.386(0.795)	7.606(1.445)	4.756(0.824)	6.571(2.166)	16.045(4.591)	13.329(6.179)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	116.667(23.119)	115.693(25.595)	92.953(27.443)	133.244(43.606)	91.862(27.450)		58.699(3.113) 153.980(45.762)	97.171(5.573)		100.526(26.860) 59.861(16.501) 125.827(22.206)		97.815(57.493)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	184.238(95.158)	268.534(31.339)	150.671(74.187)		323.966(156.688) 56.144(20.017) 196.802(27.871)			236.334(69.737) 232.987(173.016)	183.770(13.659)	174.546(66.237)	134.795(48.271)	194.912(60.502)
slump	97.591(24.462)	144.515(25.307)	101.595(23.030)			688.735(78.899) 91.907(6.560) 97.697(20.210) 179.398(56.994)		113.122(18.663)	155.591(30.783)	108.455(18.741)	166.790(40.481)	142.635(31.505)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	156.258(34.179)	83.134(32.005)		94.996(41.069) 159.112(52.801)		137.773(35.157) 77.645(39.814) 258.936(52.106)			128.033(15.995) 97.160(26.664)	94.162(29.954)	92.379(34.330)	91.136(31.600)
	wine (white) 239.966(46.588)	131.554(24.993)	176.835(32.540)	174.912(43.386)	140.419(17.636)	192.892(43.826)	239.809(94.380)		222.686(88.846) 103.041(12.901) 109.775(13.764) 158.546(26.907)			116.690(8.476)
yacht		332.889(86.576) 312.161(125.931) 1022.673(379.932) 588.291(145.693) 717.056(170.545) 260.489(97.738) 927.175(316.770) 509.025(193.582) 1004.272(524.084) 726.963(278.646) 850.603(248.990)										691.960(212.749)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 39: Regression parameter 1-CR results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	0.722(0.056)	0.244(0.040)	0.533(0.033)	0.567(0.102)	0.322(0.097)	0.367(0.038)	0.456(0.073)			0.222(0.056) 0.000(0.000) 0.000(0.000) 0.611(0.044)		0.333(0.058)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.704(0.037)	0.607(0.096)	0.667(0.064)	0.119(0.030)	0.578(0.089)	0.726(0.020)	0.859(0.032)	0.593(0.027)		0.696(0.085) 0.074(0.030) 0.800(0.013)		0.689(0.046)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	0.875(0.000)	0.875(0.000)	0.833(0.022)		0.317(0.094) 0.067(0.044)	0.758(0.008)	0.800(0.014)	0.408(0.210)	0.333(0.220)	0.458(0.191)	0.750(0.000)	0.750(0.000)
slump			0.000(0.000) 0.042(0.042) 0.000(0.000) 0.533(0.101) 0.000(0.000)			0.008(0.008)		0.050(0.050) 0.000(0.000) 0.058(0.058)		0.025(0.025)	0.133(0.109)	0.008(0.008)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.212(0.030)		0.097(0.097) 0.012(0.012) 0.048(0.022)		0.121(0.074)	0.012(0.012) 0.339(0.058)		0.242(0.088)	0.048(0.048)	0.055(0.038)	0.030(0.006)	0.018(0.010)
	wine (white) 0.556(0.056)		0.450(0.079) 0.244(0.128) 0.261(0.102) 0.433(0.059)			0.622(0.053)	0.422(0.011)	0.500(0.101)		0.300(0.108) 0.178(0.106) 0.506(0.024)		0.456(0.015)
yacht	0.000(0.000) 0.010(0.010)		0.143(0.059)		0.314(0.087) 0.000(0.000)	0.010(0.010)	0.143(0.029)	0.010(0.010)	0.048(0.034)	0.162(0.162)	0.048(0.034)	0.029(0.029)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 40: Regression parameter 1-CR results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	0.667(0.096)	0.667(0.000)	0.633(0.088)	0.689(0.011)	0.644(0.091)	0.633(0.184)	0.611(0.029)		0.378(0.113) 0.122(0.062) 0.111(0.095) 0.700(0.051)			0.689(0.022)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.815(0.037)	0.881(0.007)		0.844(0.034) 0.526(0.007) 0.800(0.000)		0.911(0.013)	0.919(0.027)	0.822(0.071)	0.815(0.037)	0.756(0.046)	0.867(0.013)	0.793(0.049)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	0.875(0.000)	0.917(0.042)		0.925(0.025) 0.650(0.115) 0.458(0.217)		0.950(0.038)	0.825(0.063)	0.825(0.050)	0.792(0.042)	0.717(0.046)	0.825(0.038)	0.783(0.017)
slump	0.000(0.000) 0.117(0.104)		0.125(0.066)	0.283(0.159)	0.075(0.038)	0.158(0.123)	0.500(0.095)	0.117(0.073)	0.300(0.123)	0.308(0.085)	0.283(0.142)	0.325(0.115)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.273(0.052)	0.224(0.106)	0.115(0.074)		0.261(0.022) 0.055(0.036)	0.218(0.038)	0.552(0.107)	0.424(0.176)	0.218(0.046)	0.236(0.055)	0.345(0.018)	0.345(0.046)
	wine (white) 0.583(0.000)			0.556(0.036) 0.394(0.061) 0.383(0.076) 0.528(0.102)		0.656(0.039)	0.728(0.011)	0.806(0.096)	0.561(0.024)	0.561(0.020)	0.600(0.017)	0.594(0.020)
yacht	0.095(0.095) 0.324(0.085)		0.362(0.124)			0.467(0.234) 0.086(0.049) 0.105(0.042) 0.333(0.076)			0.219(0.042) 0.105(0.042) 0.200(0.125)		0.200(0.033)	0.276(0.062)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 41: Regression parameter 1-CR results: mean and standard error. MCAR 75% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil			0.500(0.000) 0.689(0.056) 0.478(0.154) 0.822(0.011)		0.633(0.051)			0.567(0.120) 0.722(0.080) 0.500(0.107) 0.567(0.088) 0.589(0.211) 0.722(0.029)				0.844(0.048)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.778(0.064) 0.904(0.015)		0.926(0.037)	0.852(0.032)	0.941(0.020)	0.926(0.037)	0.926(0.020)	0.970(0.007)	0.904(0.039)	0.844(0.034)	0.859(0.020)	0.881(0.037)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	0.833(0.042)	0.792(0.042)	0.683(0.088)		0.792(0.055) 0.650(0.058)	0.917(0.017)		0.750(0.126) 0.600(0.076) 0.833(0.058)		0.783(0.102)	0.758(0.036)	0.792(0.030)
slump	0.042(0.042) 0.400(0.063)		0.358(0.079)	0.325(0.101)	0.375(0.080)	0.275(0.104)	0.567(0.073)	0.358(0.058)	0.425(0.170)	0.392(0.071)	0.425(0.138)	0.475(0.088)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)			0.303(0.030) 0.673(0.028) 0.255(0.056) 0.455(0.073)		0.430(0.026)	0.467(0.070)		0.764(0.031) 0.255(0.092) 0.624(0.024)		0.552(0.054)	0.800(0.028)	0.485(0.053)
	wine (white) 0.528(0.028) 0.683(0.051)		0.561(0.096)	0.717(0.042)	0.617(0.054)	0.850(0.029)	0.783(0.029)	0.928(0.031)	0.722(0.031)	0.789(0.091)	0.833(0.000)	0.783(0.092)
yacht	0.095(0.095) 0.571(0.099)		0.552(0.050)		0.333(0.126) 0.152(0.101)	0.467(0.117)	0.543(0.016)	0.390(0.159)	0.486(0.119)	0.429(0.218)	0.486(0.115)	0.610(0.156)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 42: Regression parameter 1-CR results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)		CSDI (short) CFMI (ours, short)
airfoil	0.556(0.056)	0.722(0.147)	0.533(0.033)	0.389(0.022)	0.300(0.153)	0.433(0.150)	0.656(0.040)		0.256(0.087) 0.022(0.011) 0.111(0.111)		0.444(0.099)	0.433(0.145)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.778(0.064)	0.859(0.071)	0.844(0.034)	0.711(0.056)	0.859(0.041)	0.748(0.020)	0.896(0.027)	0.867(0.044)		0.756(0.077) 0.504(0.112) 0.852(0.049)		0.733(0.046)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	0.542(0.273)	0.867(0.008)	0.417(0.216)		0.492(0.202) 0.092(0.092)	0.658(0.036)	0.608(0.130)	0.458(0.234)	0.425(0.088)	0.317(0.219)	0.308(0.183)	0.583(0.129)
slump	0.042(0.042) 0.225(0.052)		0.067(0.036)		0.458(0.134) 0.050(0.038)			0.033(0.017) 0.392(0.101) 0.042(0.022) 0.158(0.088)		0.067(0.067)	0.175(0.139)	0.125(0.072)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.303(0.080)		0.073(0.056) 0.030(0.030) 0.139(0.024)		0.170(0.119)	0.048(0.022) 0.442(0.044)		0.109(0.052)	0.085(0.016)	0.091(0.046)	0.073(0.018)	0.085(0.040)
	wine (white) 0.667(0.048)	0.622(0.059)	0.528(0.164)	0.572(0.106)	0.567(0.135)	0.611(0.036)	0.606(0.087)		0.617(0.134) 0.339(0.143) 0.339(0.140) 0.572(0.063)			0.533(0.025)
yacht	0.048(0.048) 0.124(0.019)		0.476(0.125)			0.114(0.033) 0.029(0.029) 0.029(0.016) 0.371(0.092)		0.086(0.044)	0.162(0.162)	0.200(0.186)	0.171(0.115)	0.152(0.138)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 43: Regression parameter AW results: mean and standard error. MCAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.733(0.005)	0.675(0.006)		0.769(0.008) 0.635(0.005) 0.645(0.001)		0.691(0.006)	0.715(0.012)	0.649(0.002)	0.656(0.003)	0.655(0.003)	0.727(0.028)	0.686(0.029)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.035(0.000)	0.039(0.001)	0.048(0.001)	0.040(0.001)	0.041(0.001)	0.029(0.000)	0.036(0.001)	0.038(0.001)	0.034(0.001)		0.040(0.001) 0.026(0.001)	0.038(0.000)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	2.299(0.023)	2.100(0.017) 2.534(0.145)		3.536(0.123)	3.233(0.036)	2.531(0.024)	2.490(0.009)	4.734(0.117)	3.059(0.055)	3.092(0.049)	2.534(0.015)	2.612(0.031)
slump		4.311(0.043) 4.321(0.147) 5.501(0.384)		14.119(3.073)	5.100(0.170)	4.465(0.237)	5.053(0.006)	4.779(0.150)	4.928(0.260)	4.621(0.112)	5.199(0.238)	4.739(0.193)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.089(0.000)	0.095(0.001)	0.099(0.000)	0.085(0.001)	0.096(0.001)	0.087(0.000)	0.092(0.001)	0.095(0.001)	0.089(0.001)		0.089(0.001) 0.084(0.000)	0.084(0.001)
	wine (white) 0.059(0.000) 0.071(0.001)		0.081(0.003)	0.085(0.006)		0.082(0.005) 0.060(0.001) 0.067(0.001)		0.099(0.012)	0.073(0.001)	0.073(0.002)	0.064(0.001)	0.063(0.000)
yacht	3.192(0.093)	2.832(0.035) 3.445(0.128)		6.897(0.354)	5.974(0.600)	3.367(0.035)	3.216(0.081)	3.811(0.024)	5.889(0.529)	5.677(0.524)	3.828(0.165)	3.777(0.148)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 44: Regression parameter AW results: mean and standard error. MCAR 50% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.898(0.006)	0.729(0.016)		0.930(0.005) 0.613(0.001) 0.690(0.016)		0.764(0.009)	0.874(0.024)	0.665(0.020)	0.714(0.015)	0.674(0.008)	0.793(0.035)	0.729(0.042)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.050(0.008)	0.050(0.006)	0.067(0.008)	0.045(0.004)	0.046(0.003)	0.034(0.004)	0.048(0.008)	0.051(0.005)	0.035(0.005)		0.044(0.004) 0.026(0.001)	0.038(0.003)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	2.667(0.021)	2.175(0.032)	2.973(0.165)	2.698(0.194)	3.068(0.017)	2.375(0.007)	2.900(0.070)	2.827(0.029)	2.548(0.037)		2.556(0.086) 2.121(0.017)	2.183(0.022)
slump	4.968(0.147)	4.328(0.418)		5.720(0.189) 3.395(0.214) 5.528(0.302)		4.159(0.268)	4.448(0.433)	5.291(0.290)	4.204(0.304)	3.996(0.390)	4.273(0.428)	3.997(0.340)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.105(0.001)	0.111(0.001)	0.122(0.001)	0.085(0.001)	0.115(0.003)	0.093(0.003)	0.107(0.000)	0.128(0.010)	0.092(0.002)		0.092(0.003) 0.079(0.002)	0.082(0.002)
	wine (white) 0.069(0.000)	0.077(0.001)	0.093(0.002)	0.084(0.002)	0.100(0.002)	0.064(0.001)	0.075(0.001)	0.143(0.005)	0.064(0.001)		0.062(0.001) 0.054(0.001)	0.056(0.001)
yacht		3.985(0.061) 2.762(0.011) 3.584(0.135)		11.053(5.210)	5.013(0.608)	2.910(0.124)	3.405(0.283)	3.859(0.006)	2.980(0.088)	2.845(0.084)	2.788(0.070)	2.814(0.028)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 45: Regression parameter AW results: mean and standard error. MCAR 75% missingness.

		OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
	Mean
airfoil	1.292(0.012)	0.914(0.036)		1.351(0.031) 0.626(0.004) 0.880(0.027)		1.048(0.007)	1.525(0.076)	1.343(0.023)	0.989(0.009)	0.853(0.023)	0.935(0.035)	0.861(0.067)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.098(0.002)	0.084(0.001)	0.113(0.003)	0.054(0.002)	0.055(0.001)	0.049(0.001)	0.069(0.003)	0.068(0.004)	0.044(0.005)		0.051(0.001) 0.012(0.001)	0.040(0.000)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	3.792(0.032)	2.809(0.037)		4.133(0.098) 1.961(0.071) 3.785(0.276)		2.743(0.139)	3.990(0.358)	4.194(0.239)	2.519(0.032)	2.391(0.069)	2.269(0.055)	2.098(0.086)
slump	7.047(0.181)	4.827(0.480)		6.954(0.367) 3.643(0.146) 9.358(0.687)		5.166(0.342)	4.203(0.257)	8.630(0.714)	5.261(0.215)	4.929(0.487)	4.955(0.545)	4.521(0.398)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.148(0.002)	0.141(0.003)	0.179(0.003)	0.080(0.001)	0.153(0.005)	0.112(0.005)	0.158(0.004)	0.159(0.007)	0.091(0.003)		0.091(0.003) 0.036(0.002)	0.078(0.002)
	wine (white) 0.095(0.001)	0.095(0.001)	0.122(0.003)	0.063(0.001)	0.120(0.003)	0.074(0.002)	0.107(0.001)	0.171(0.017)	0.059(0.001)		0.058(0.001) 0.023(0.000)	0.052(0.001)
yacht	6.241(0.048)	3.521(0.103)		4.446(0.275) 2.243(0.109) 4.114(0.234)		3.314(0.149)	3.363(0.122)	5.366(0.361)	3.183(0.059)	3.160(0.186)	3.025(0.171)	2.813(0.181)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

Table 46: Regression parameter AW results: mean and standard error. MAR 25% missingness.

	Mean	OT imputer	SoftImpute	MICE	MissForest	ForestDiffusion	GAIN	HyperImpute	CSDI	CFMI (ours)	CSDI (short)	CFMI (ours, short)
airfoil	0.762(0.007)	0.675(0.019)		0.776(0.013) 0.636(0.006) 0.672(0.019)		0.705(0.007)	0.744(0.023)	0.662(0.017)	0.669(0.004)	0.658(0.002)	0.730(0.020)	0.691(0.038)
banknote	—	—	—	—	—	—	—	—	—	—	—	—
blood	—	—	—	—	—	—	—	—	—	—	—	—
cancer	—	—	—	—	—	—	—	—	—	—	—	—
california	0.042(0.004)	0.045(0.003)	0.052(0.002)	0.045(0.003)	0.042(0.003)	0.035(0.003)	0.040(0.004)	0.041(0.002)	0.037(0.003)		0.043(0.003) 0.030(0.002)	0.040(0.002)
climate	—	—	—	—	—	—	—	—	—	—	—	—
concrete	2.322(0.073)	2.158(0.016) 2.708(0.050)		3.688(0.173)	3.190(0.133)	2.542(0.072)	2.662(0.072)	3.903(0.320)	3.109(0.076)	3.077(0.085)	2.566(0.025)	2.667(0.039)
slump	4.423(0.037)	3.934(0.084) 4.688(0.218)		12.416(1.408)	5.001(0.326)	4.443(0.118)	4.335(0.388)	4.735(0.203)	4.746(0.141)	4.473(0.152)	5.255(0.275)	4.446(0.173)
sonar	—	—	—	—	—	—	—	—	—	—	—	—
vowel	—	—	—	—	—	—	—	—	—	—	—	—
ecoli	—	—	—	—	—	—	—	—	—	—	—	—
glass	—	—	—	—	—	—	—	—	—	—	—	—
ionosphere	—	—	—	—	—	—	—	—	—	—	—	—
iris	—	—	—	—	—	—	—	—	—	—	—	—
libras	—	—	—	—	—	—	—	—	—	—	—	—
parkinsons	—	—	—	—	—	—	—	—	—	—	—	—
planning	—	—	—	—	—	—	—	—	—	—	—	—
qsar	—	—	—	—	—	—	—	—	—	—	—	—
seeds	—	—	—	—	—	—	—	—	—	—	—	—
wine	—	—	—	—	—	—	—	—	—	—	—	—
wine (red)	0.091(0.001)	0.098(0.000)	0.103(0.002)	0.090(0.002)	0.096(0.001)	0.088(0.001)	0.096(0.002)	0.095(0.000)	0.090(0.000)	0.090(0.000)	0.087(0.000)	0.086(0.000)
	wine (white) 0.060(0.000) 0.073(0.001)		0.086(0.003)	0.092(0.002)	0.095(0.003)	0.064(0.001)	0.071(0.001)	0.111(0.013)	0.075(0.001)	0.074(0.000)	0.062(0.001)	0.063(0.000)
yacht	3.671(0.011)	2.916(0.093) 4.615(0.374)		7.842(0.284)	6.365(0.551)	3.230(0.069)	3.661(0.116)	4.164(0.061)	5.543(0.274)	4.599(0.289)	4.298(0.320)	3.808(0.180)
yeast	—	—	—	—	—	—	—	—	—	—	—	—

B.2.2 Raw results box plots

Figure 7: Wasserstein-2 results: box-plot of 3 runs. MCAR 25% missingness.

Figure 8: Wasserstein-2 results: box-plot of 3 runs. MCAR 50% missingness.

Figure 9: Wasserstein-2 results: box-plot of 3 runs. MCAR 75% missingness.

Figure 10: Wasserstein-2 results: box-plot of 3 runs. MAR 25% missingness.

Figure 11: Average RMSE results: box-plot of 3 runs. MCAR 25% missingness.

Figure 12: Average RMSE results: box-plot of 3 runs. MCAR 50% missingness.

Figure 13: Average RMSE results: box-plot of 3 runs. MCAR 75% missingness.

Figure 14: Average RMSE results: box-plot of 3 runs. MAR 25% missingness.

Figure 15: CRPS results: box-plot of 3 runs. MCAR 25% missingness.

Figure 16: CRPS results: box-plot of 3 runs. MCAR 50% missingness.

Figure 17: CRPS results: box-plot of 3 runs. MCAR 75% missingness.

Figure 18: CRPS results: box-plot of 3 runs. MAR 25% missingness.

Figure 19: Energy MMD results: box-plot of 3 runs. MCAR 25% missingness.

Figure 20: Energy MMD results: box-plot of 3 runs. MCAR 50% missingness.

Figure 21: Energy MMD results: box-plot of 3 runs. MCAR 75% missingness.

Figure 22: Energy MMD results: box-plot of 3 runs. MAR 25% missingness.

Figure 23: Gaussian MMD results: box-plot of 3 runs. MCAR 25% missingness.

Figure 24: Gaussian MMD results: box-plot of 3 runs. MCAR 50% missingness.

Figure 25: Gaussian MMD results: box-plot of 3 runs. MCAR 75% missingness.

Figure 26: Gaussian MMD results: box-plot of 3 runs. MAR 25% missingness.

Figure 27: Laplacian MMD results: box-plot of 3 runs. MCAR 25% missingness.

Figure 28: Laplacian MMD results: box-plot of 3 runs. MCAR 50% missingness.

Figure 29: Laplacian MMD results: box-plot of 3 runs. MCAR 75% missingness.

Figure 30: Laplacian MMD results: box-plot of 3 runs. MAR 25% missingness.

Figure 31: Classifier 1-AUROC results: box-plot of 3 runs. MCAR 25% missingness.

Figure 32: Classifier 1-AUROC results: box-plot of 3 runs. MCAR 50% missingness.

Figure 33: Classifier 1-AUROC results: box-plot of 3 runs. MCAR 75% missingness.

Figure 34: Classifier 1-AUROC results: box-plot of 3 runs. MAR 25% missingness.

Figure 35: Regression CRPS results: box-plot of 3 runs. MCAR 25% missingness.

Figure 36: Regression CRPS results: box-plot of 3 runs. MCAR 50% missingness.

Figure 37: Regression CRPS results: box-plot of 3 runs. MCAR 75% missingness.

Figure 38: Regression CRPS results: box-plot of 3 runs. MAR 25% missingness.

Figure 39: Regression parameter percent bias results: box-plot of 3 runs. MCAR 25% missingness.

Figure 40: Regression parameter percent bias results: box-plot of 3 runs. MCAR 50% missingness.

Figure 41: Regression parameter percent bias results: box-plot of 3 runs. MCAR 75% missingness.

Figure 42: Regression parameter percent bias results: box-plot of 3 runs. MAR 25% missingness.

Figure 43: Regression parameter 1-CR results: box-plot of 3 runs. MCAR 25% missingness.

Figure 44: Regression parameter 1-CR results: box-plot of 3 runs. MCAR 50% missingness.

Figure 45: Regression parameter 1-CR results: box-plot of 3 runs. MCAR 75% missingness.

Figure 46: Regression parameter 1-CR results: box-plot of 3 runs. MAR 25% missingness.

Figure 47: Regression parameter AW results: box-plot of 3 runs. MCAR 25% missingness.

Figure 48: Regression parameter AW results: box-plot of 3 runs. MCAR 50% missingness.

Figure 49: Regression parameter AW results: box-plot of 3 runs. MCAR 75% missingness.

Figure 50: Regression parameter AW results: box-plot of 3 runs. MAR 25% missingness.

B.2.3 Average relative performance matrices

CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion GAIN HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −0.8 −0.6 −0.4 −0.2 0.0

Figure 51: Wasserstein-2 results: average relative distance (blue means row method outperforms the

Figure 52: Wasserstein-2 results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 54: Wasserstein-2 results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

Figure 55: Average RMSE results: average relative distance (blue means row method outperforms the

Figure 56: Average RMSE results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 57: Average RMSE results: average relative distance (blue means row method outperforms the

Figure 58: Average RMSE results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

GAIN HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −0.3 −0.2 −0.1

Figure 59: CRPS results: average relative distance (blue means row method outperforms the column

Figure 60: CRPS results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 61: CRPS results: average relative distance (blue means row method outperforms the column

Figure 62: CRPS results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −1.5 −1.0 −0.5 1.0

Figure 63: Energy MMD results: average relative distance (blue means row method outperforms the

Figure 64: Energy MMD results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 65: Energy MMD results: average relative distance (blue means row method outperforms the

Figure 66: Energy MMD results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −1.5 −1.0 −0.5 0.5 1.0

Figure 67: Gaussian MMD results: average relative distance (blue means row method outperforms the

Figure 68: Gaussian MMD results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 69: Gaussian MMD results: average relative distance (blue means row method outperforms the

Figure 70: Gaussian MMD results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −1.5 −1.0 −0.5 1.0

Figure 71: Laplacian MMD results: average relative distance (blue means row method outperforms the

Figure 72: Laplacian MMD results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 73: Laplacian MMD results: average relative distance (blue means row method outperforms the

Figure 74: Laplacian MMD results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

Figure 75: Classifier 1-AUROC results: average relative distance (blue means row method outperforms

Figure 76: Classifier 1-AUROC results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 77: Classifier 1-AUROC results: average relative distance (blue means row method outperforms

Figure 78: Classifier 1-AUROC results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −0.4 −0.3 −0.2 −0.1 0.2

Figure 79: Regression CRPS results: average relative distance (blue means row method outperforms

Figure 80: Regression CRPS results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 81: Regression CRPS results: average relative distance (blue means row method outperforms

Figure 82: Regression CRPS results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

MICE MissForest ForestDiffusion GAIN HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) ⁻0.⁶ −0.4 −0.2 0.2 0.4 0.6

Figure 83: Regression parameter percent bias results: average relative distance (blue means row method outperforms the column method). MCAR

Figure 84: Regression parameter percent bias results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 85: Regression parameter percent bias results: average relative distance (blue means row method outperforms the column method). MCAR

Figure 86: Regression parameter percent bias results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −0.6 −0.4 −0.2

Figure 87: Regression parameter 1-CR results: average relative distance (blue means row method outperforms the column method). MCAR 25% missing-

Figure 88: Regression parameter 1-CR results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 89: Regression parameter 1-CR results: average relative distance (blue means row method outperforms the column method). MCAR 75% missing-

Figure 90: Regression parameter 1-CR results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

CFMI (ours) CSDI (short) CFMI (ours, short) OT imputer SoftImpute MICE MissForest ForestDiffusion HyperImpute CSDI CFMI (ours) CSDI (short) CFMI (ours, short) −0.4 −0.3 −0.2 −0.1 0.2 0.3 0.4

Figure 91: Regression parameter AW results: average relative distance (blue means row method outperforms the column method). MCAR 25% miss-

Figure 92: Regression parameter AW results: average relative distance (blue means row method outperforms the column method). MCAR 50% missingness.

Figure 93: Regression parameter AW results: average relative distance (blue means row method outperforms the column method). MCAR 75% miss-

Figure 94: Regression parameter AW results: average relative distance (blue means row method outperforms the column method). MAR 25% missingness.

−0.3 −0.2 −0.1

0.1

B.2.4 Average rank tables

Table 47: Average rank of each method for all metrics, mean and standard error. Lowest mean value are highlighted in bold, as well as values that fall within lowest mean + standard error range. MCAR 25% missingness.

Method	W2	RMSE	CRPS	Energy MMD	Gaussian MMD	Laplacian MMD	1-AUROC	Pred. CRPS	Reg. PB	Reg. 1-CR	Reg. AW
Mean	9.67(0.51)	8.96(0.58)	10.75(0.11)	10.54(0.12)	9.46(0.26)	10.67(0.12)	9.53(0.61)	10.83(0.08)	7.43(1.46)	6.86(1.81)	3.00(1.41)
OT imputer	5.00(0.43)	4.04(0.46)	7.33(0.37)	6.92(0.30)	6.79(0.43)	6.92(0.30)	4.47(0.58)	7.25(0.34)	4.57(0.90)	5.71(1.15)	3.86(1.35)
SoftImpute	5.17(0.64)	3.33(0.53)	8.00(0.57)	8.12(0.39)	5.88(0.78)	8.21(0.38)	8.63(0.29)	8.12(0.55)	6.29(1.02)	4.57(1.46)	8.43(1.21)
MICE	7.21(0.63)	8.00(0.60)	4.38(0.70)	4.46(0.61)	4.08(0.48)	4.46(0.61)	7.32(0.64)	4.04(0.69)	5.00(1.76)	5.57(1.74)	7.43(1.72)
MissForest		1.08(0.26) 0.75(0.20)	3.62(0.40)	3.42(0.51)	3.04(0.49)	3.42(0.51)	5.37(0.78)	3.25(0.37)	4.00(1.02)	3.14(1.03)	8.14(1.22)
ForestDiffusion	7.04(0.51)	8.00(0.41)	4.58(0.57)	5.38(0.48)	7.38(0.53)	5.25(0.47)	2.74(0.79)	4.33(0.58)	4.71(1.29)	5.86(1.32)	2.86(0.77)
GAIN	9.04(0.25)	8.50(0.32)	8.58(0.39)	9.75(0.23)	10.33(0.19)	9.67(0.21)	6.68(0.89)	8.54(0.40)	8.86(0.86)	8.29(0.92)	4.86(0.94)
HyperImpute	4.25(0.64)	3.29(0.67)	6.04(0.65)	7.00(0.50)	5.42(0.66)	7.04(0.46)	5.95(0.76)	6.00(0.67)	5.57(1.39)	4.71(1.15)	7.00(1.23)
CSDI	3.25(0.42)	3.96(0.40)	3.38(0.57)	3.04(0.52)	3.04(0.40)	3.00(0.52)	2.68(0.34)	3.58(0.60)	4.86(1.32)	4.57(1.17)	5.57(0.90)
CFMI (ours)	2.17(0.43)	3.04(0.41)	3.12(0.61)	2.25(0.43)	1.92(0.45)	2.38(0.42)	2.11(0.46)	3.46(0.58)		3.14(1.26) 3.86(1.42)	6.00(0.93)
CSDI (short)	6.62(0.63)	7.50(0.50)	3.38(0.44)	2.83(0.38)	4.92(0.56)	2.71(0.40)	5.47(0.50)	3.29(0.45)	7.14(1.34)	7.71(1.06)	4.86(1.39)
CFMI (ours, short)	5.50(0.44)	6.62(0.39)	2.83(0.28)	2.29(0.32)	3.75(0.46)	2.29(0.32)	5.05(0.54)	3.29(0.34)	4.43(1.17)	5.14(0.59)	4.00(0.85)

Table 48: Average rank of each method for all metrics, mean and standard error. Lowest mean value are highlighted in bold, as well as values that fall within lowest mean + standard error range. MCAR 50% missingness.

Method	W2	RMSE	CRPS	Energy MMD	Gaussian MMD	Laplacian MMD	1-AUROC	Pred. CRPS	Reg. PB	Reg. 1-CR	Reg. AW
Mean	9.00(0.50)	7.17(0.63)	10.79(0.10)	10.29(0.16)	8.25(0.33)	10.42(0.17)	10.00(0.44)	10.79(0.10)	4.86(1.65)	4.86(1.18)	7.57(0.72)
OT imputer	4.46(0.46)	2.75(0.40)	6.33(0.37)	6.29(0.37)	6.00(0.57)	6.46(0.34)	4.05(0.57)	6.29(0.32)	3.43(0.87)	6.14(1.14)	5.00(1.23)
SoftImpute	5.21(0.68)	2.08(0.42)	8.04(0.67)	8.17(0.39)	5.00(0.85)	8.25(0.40)	8.42(0.43)	8.21(0.58)	5.29(0.68)	5.29(1.44)	9.86(0.55)
MICE	6.75(0.63)	8.25(0.55)	4.00(0.68)	3.04(0.58)	2.67(0.43)	3.12(0.60)	6.53(0.75)	3.83(0.57)	5.57(1.72)	4.71(1.69)	4.71(1.60)
MissForest	2.42(0.38)	2.42(0.39)	3.75(0.51)	3.50(0.49)	4.33(0.49)	3.58(0.50)	6.37(0.53)	3.67(0.48)	3.71(1.57)	1.71(0.84)	8.43(1.09)
ForestDiffusion	6.71(0.54)	7.79(0.43)	5.00(0.57)	6.12(0.41)	8.00(0.35)	5.83(0.44)	2.16(0.78)	4.21(0.72)	3.86(1.03)	6.29(1.41)	3.86(0.70)
GAIN	9.38(0.36)	8.29(0.45)	8.21(0.39)	10.29(0.15)	10.79(0.08)	10.08(0.16)	6.53(0.84)	8.38(0.39)	9.57(0.69)	8.57(1.23)	7.29(0.47)
HyperImpute	3.88(0.63)	2.58(0.67)	5.67(0.54)	6.71(0.36)	4.58(0.64)	6.75(0.37)	6.21(0.79)	5.38(0.63)	7.71(1.52)	6.29(1.25)	8.29(1.30)
CSDI	2.92(0.48)	4.38(0.41)	3.46(0.59)	3.33(0.47)	3.42(0.49)	3.29(0.47)	2.63(0.32)	3.96(0.60)	4.57(1.39)	3.86(0.80)	3.71(0.36)
CFMI (ours)	2.33(0.41)	3.50(0.39)	3.50(0.66)	3.25(0.52)	3.58(0.63)	3.21(0.49)	1.84(0.33)	3.29(0.59)	5.00(1.18)	3.71(1.15)	2.86(0.51)
CSDI (short)	7.58(0.71)	9.00(0.39)	4.54(0.64)	3.29(0.56)	5.08(0.68)	3.25(0.57)	5.95(0.52)	5.17(0.63)	6.29(0.92)	7.71(0.68)	2.00(1.21)
CFMI (ours, short)	5.38(0.48)	7.79(0.29)	2.71(0.29)	1.71(0.29)	4.29(0.54)	1.75(0.28)	5.32(0.53)	2.83(0.38)	6.14(0.86)	6.86(1.22)	2.43(0.65)

Table 49: Average rank of each method for all metrics, mean and standard error. Lowest mean value are highlighted in bold, as well as values that fall within lowest mean + standard error range. MCAR 75% missingness.

Method	W2	RMSE	CRPS	Energy MMD	Gaussian MMD	Laplacian MMD	1-AUROC	Pred. CRPS	Reg. PB	Reg. 1-CR	Reg. AW
Mean	8.12(0.41)	4.58(0.60)	10.29(0.16)	9.88(0.14)	6.21(0.53)	10.00(0.15)	10.11(0.30)	10.58(0.12)	5.29(1.06)	1.71(1.25)	8.43(0.65)
OT imputer	3.42(0.49)	1.79(0.32)	5.46(0.51)	6.04(0.42)	5.33(0.53)	6.33(0.42)	2.74(0.42)	5.33(0.46)	6.14(1.01)	6.71(0.89)	6.00(0.76)
SoftImpute	4.42(0.58)	1.46(0.44)	7.71(0.54)	8.50(0.20)	3.58(0.69)	8.75(0.16)	8.00(0.37)	8.12(0.49)	5.86(1.22)	3.14(1.34)	9.86(0.40)
MICE	6.54(0.56)	8.33(0.43)	3.38(0.68)	1.12(0.36)	1.50(0.35)	1.33(0.42)	6.42(0.71)	3.08(0.62)	3.14(1.53)	4.43(1.15)	1.57(0.81)
MissForest	4.12(0.54)	5.33(0.45)	3.96(0.58)	3.21(0.39)	5.67(0.60)	3.21(0.40)	7.16(0.41)	3.42(0.48)	4.43(1.39)	4.00(1.23)	7.43(0.95)
ForestDiffusion	6.71(0.59)	7.42(0.48)	6.21(0.63)	6.17(0.45)	8.79(0.38)	5.92(0.41)	1.58(0.72)	5.79(0.72)	3.86(1.01)	6.29(1.34)	5.14(0.46)
GAIN	9.92(0.17)	9.04(0.35)	7.17(0.54)	10.46(0.16)	10.67(0.12)	10.29(0.18)	6.11(0.85)	7.96(0.39)	8.29(1.39)	7.86(1.01)	7.43(1.21)
HyperImpute	3.25(0.49)	2.71(0.37)	5.12(0.48)	4.83(0.41)	3.25(0.52)	5.08(0.40)	6.68(0.56)	5.29(0.48)	8.14(1.08)	4.57(1.73)	9.71(0.52)
CSDI	3.29(0.61)	4.79(0.51)	3.50(0.55)	4.00(0.41)	4.25(0.55)	4.08(0.41)	2.74(0.23)	3.96(0.49)	6.86(1.26)	6.57(0.87)	4.29(0.64)
CFMI (ours)	2.00(0.43)	3.17(0.39)	3.08(0.57)	4.08(0.53)	3.79(0.41)	4.17(0.52)	2.11(0.31)	2.54(0.47)	5.86(0.91)	5.14(0.86)	2.86(0.40)
CSDI (short)	8.75(0.64)	9.46(0.45)	6.83(0.82)	5.54(0.74)	7.46(0.65)	4.88(0.74)	7.63(0.73)	6.96(0.72)	2.71(1.38)	7.43(1.11)	2.00(0.85)
CFMI (ours, short)	5.46(0.51)	7.92(0.38)	3.29(0.49)	2.17(0.48)	5.50(0.55)	1.96(0.46)	4.74(0.43)	2.96(0.53)	5.43(1.31)	8.14(1.01)	1.29(0.18)

Table 50: Average rank of each method for all metrics, mean and standard error. Lowest mean value are highlighted in bold, as well as values that fall within lowest mean + standard error range. MAR 25% missingness.

Method	W2	RMSE	CRPS	Energy MMD	Gaussian MMD	Laplacian MMD	1-AUROC	Pred. CRPS	Reg. PB	Reg. 1-CR	Reg. AW
Mean	9.58(0.49)	8.79(0.55)	10.79(0.08)	10.75(0.09)	9.79(0.42)	10.75(0.09)	9.74(0.61)	10.79(0.08)	6.57(1.49)	6.43(1.48)	4.14(1.37)
OT imputer	5.21(0.39)	3.88(0.44)	7.04(0.38)	6.75(0.30)	6.33(0.42)	6.96(0.27)	4.47(0.69)	7.04(0.37)	5.71(1.55)	8.14(1.28)	4.14(1.62)
SoftImpute	5.42(0.72)	3.29(0.58)	8.17(0.62)	7.88(0.48)	5.25(0.69)	7.88(0.48)	8.53(0.39)	8.17(0.56)	5.71(1.29)	4.86(1.42)	8.71(0.87)
MICE	6.38(0.68)	7.83(0.65)	4.29(0.71)	3.42(0.57)	2.83(0.51)	3.50(0.59)	7.16(0.68)	3.92(0.66)	8.29(1.08)	5.57(1.21)	8.00(1.54)
MissForest		1.50(0.24) 1.08(0.26)	3.88(0.44)	3.25(0.43)	3.33(0.52)	3.42(0.43)	5.26(0.74)	3.75(0.43)	3.29(1.13)	3.86(1.34)	8.14(0.86)
ForestDiffusion	7.67(0.41)	8.33(0.33)	6.00(0.60)	6.62(0.51)	7.92(0.55)	6.08(0.55)	1.95(0.73)	5.67(0.69)	3.43(1.41)	4.00(1.45)	2.71(0.78)
GAIN	9.79(0.20)	9.04(0.35)	7.62(0.46)	9.67(0.18)	10.25(0.17)	9.67(0.18)	6.21(0.76)	7.75(0.46)	9.29(0.78)	9.71(0.52)	4.57(1.11)
HyperImpute	4.58(0.47)	2.62(0.55)	6.00(0.52)	6.71(0.34)	5.62(0.52)	7.00(0.35)	6.00(0.66)	6.29(0.51)	5.43(1.00)	5.43(1.25)	6.86(1.24)
CSDI	2.88(0.49)	4.00(0.41)	3.42(0.64)	3.21(0.54)	3.46(0.50)	3.17(0.50)	3.21(0.42)	3.46(0.58)	4.86(1.44)	3.71(1.08)	5.86(1.06)
CFMI (ours)	2.79(0.55)	3.62(0.44)	3.54(0.60)	3.33(0.54)	3.88(0.62)	3.25(0.50)	2.21(0.50)	3.50(0.66)		3.00(0.85) 3.43(1.25)	5.29(0.94)
CSDI (short)	5.21(0.69)	7.00(0.51)	2.75(0.47)	2.42(0.35)	3.92(0.51)	2.29(0.36)	6.42(0.54)	3.00(0.47)	6.14(1.20)	5.71(1.02)	4.14(1.47)
CFMI (ours, short)	5.00(0.46)	6.50(0.46)	2.50(0.26)	2.00(0.31)	3.42(0.42)	2.04(0.32)	4.84(0.50)	2.67(0.30)	4.29(0.64)	5.14(0.77)	3.43(0.75)

B.2.5 Average rank vs fraction of missingness

Figure 95: Average rank on Wasserstein-2 over all data sets versus missingness fraction. MCAR missingness.

Figure 96: Average rank on average RMSE over all data sets versus missingness fraction. MCAR missingness.

Figure 97: Average rank on CRPS over all data sets versus missingness fraction. MCAR missingness.

Figure 98: Average rank on energy MMD over all data sets versus missingness fraction. MCAR missingness.

Figure 99: Average rank on Gaussian MMD over all data sets versus missingness fraction. MCAR missingness.

Figure 100: Average rank on Laplacian MMD over all data sets versus missingness fraction. MCAR missingness.

Figure 101: Average rank on classifier 1-AUROC over all data sets versus missingness fraction. MCAR missingness.

Figure 102: Average rank on regression CRPS over all data sets versus missingness fraction. MCAR missingness.

Figure 103: Average rank on regression parameter PB over all data sets versus missingness fraction. MCAR missingness.

Figure 104: Average rank on regression parameter 1-CR over all data sets versus missingness fraction. MCAR missingness.

Figure 105: Average rank on regression parameter AW over all data sets versus missingness fraction. MCAR missingness.

B.2.6 Imputation metrics vs number of training steps

Figure 106: Wasserstein-2 metric vs number of training steps. MCAR 25% missingness.

Figure 107: Wasserstein-2 metric vs number of training steps. MAR 25% missingness.

Figure 108: Average RMSE metric vs number of training steps. MCAR 25% missingness.

Figure 109: Average RMSE metric vs number of training steps. MAR 25% missingness.

Figure 110: CRPS metric vs number of training steps. MCAR 25% missingness.

Figure 111: CRPS metric vs number of training steps. MAR 25% missingness.

Figure 112: Energy MMD metric vs number of training steps. MCAR 25% missingness.

Figure 113: Energy MMC metric vs number of training steps. MAR 25% missingness.

Figure 114: Gaussian MMD metric vs number of training steps. MCAR 25% missingness.

Figure 115: Gaussian MMC metrics vs number of training steps. MAR 25% missingness.

Figure 116: Laplacian MMD metric vs number of training steps. MCAR 25% missingness.

Figure 117: Laplacian MMC metric vs number of training steps. MAR 25% missingness.

Figure 118: Classifier 1-AUROC metric vs number of training steps. MCAR 25% missingness.

Figure 119: Classifier 1-AUROC metric vs number of training steps. MAR 25% missingness.

Figure 120: Regression CRPS metric vs number of training steps. MCAR 25% missingness.

Figure 121: Regression CRPS metric vs number of training steps. MAR 25% missingness.

Figure 122: Regression parameter PB metric vs number of training steps. MCAR 25% missingness.

Figure 123: Regression parameter PB metric vs number of training steps. MAR 25% missingness.

Figure 124: Regression parameter 1-CR metric vs number of training steps. MCAR 25% missingness.

Figure 125: Regression parameter 1-CR metric vs number of training steps. MAR 25% missingness.

Figure 126: Regression parameter AW metric vs number of training steps. MCAR 25% missingness.

Figure 127: Regression parameter AW metric vs number of training steps. MAR 25% missingness.

B.3 Time-series data sets

B.3.1 Raw results tables

In this appendix section, we compare the performance of the proposed method, CFMI, with both our implementation of CSDI and the results reported by Tashiro et al. (2021). Overall, our CSDI implementation achieves results comparable to the original, with some notable differences. Specifically, our implementation demonstrates improved performance on the PM2.5 data set but slightly underperforms on the PhysioNet data set. This discrepancy is likely attributable to implementation differences, as detailed in appendix A.1.

Despite these variations, CFMI consistently matches the performance of CSDI while demonstrating better efficiency, as discussed in section 4.3.

Table 51: CRPS on the time-series data. We report the mean and standard error for five runs.

		PhysioNet		PM2.5
	10% missing	50% missing	90% missing
CSDI CFMI (ours)	0.219(0.002) 0.220(0.004)	0.294(0.005) 0.282(0.010)	0.446(0.008) 0.410(0.014)	6.767(0.018) 6.726(0.036)

Table 52: Magnitude-normalised CRPS on the time-series data. We report the mean and standard error for five runs. Asterisks mean the results were copied from (Tashiro et al., 2021).

		PhysioNet		PM2.5
	10% missing	50% missing	90% missing
Multitask GP (Bonilla et al., 2007) (*)	0.489(0.005)	0.581(0.003)	0.942(0.010)	0.301(0.003)
GP-VAE (Fortuin et al., 2020) (*)	0.574(0.003)	0.774(0.004)	0.998(0.001)	0.397(0.009)
V-RIN (Mulyadi et al., 2020) (*)	0.808(0.008)	0.831(0.005)	0.922(0.003)	0.526(0.025)
CSDI (Tashiro et al., 2021) (*)	0.238(0.001)	0.330(0.002)	0.522(0.002)	0.108(0.001)
CSDI (Tashiro et al., 2021)	0.240(0.003)	0.333(0.003)	0.550(0.006)	0.099(0.000)
CFMI (ours)	0.250(0.004)	0.338(0.007)	0.538(0.019)	0.098(0.001)

Table 53: MAE on the time-series data. We report the mean and standard error for five runs. Asterisks mean the results were copied from (Tashiro et al., 2021).

		PhysioNet		PM2.5
	10% missing	50% missing	90% missing
V-RIN (Mulyadi et al., 2020) (*)	0.271(0.001)	0.365(0.002)	0.606(0.006)	25.4(0.62)
BRITS (Cao et al., 2018) (*)	0.284(0.001)	0.368(0.002)	0.517(0.002)	14.11(0.26)
GLIMA (Suo et al., 2020) (*)	0.265	−	−	10.54
RDIS (Choi et al., 2023) (*)	0.319(0.002)	0.419(0.002)	0.631(0.002)	22.11(0.35)
CSDI (Tashiro et al., 2021) (*)	0.217(0.001)	0.301(0.002)	0.481(0.003)	9.60(0.04)
CSDI (Tashiro et al., 2021)	0.228(0.002)	0.319(0.002)	0.530(0.005)	9.430(0.021)
CFMI (ours)	0.234(0.002)	0.319(0.005)	0.504(0.009)	9.404(0.050)

Table 54: RMSE on the time-series data. We report the mean and standard error for five runs. Asterisks mean the results were copied from (Tashiro et al., 2021).

		PhysioNet		PM2.5
	10% missing	50% missing	90% missing
V-RIN (Mulyadi et al., 2020) (*)	0.628(0.025)	0.693(0.022)	0.928(0.013)	40.11(1.14)
BRITS (Cao et al., 2018) (*)	0.619(0.022)	0.693(0.023)	0.836(0.015)	24.47(0.73)
RDIS (Choi et al., 2023) (*)	0.633(0.021)	0.741(0.018)	0.934(0.013)	37.49(0.28)
SSGAN (Miao et al., 2021) (*)	0.598	0.762	0.818	−
CSDI (Tashiro et al., 2021) (*)	0.498(0.020)	0.614(0.017)	0.803(0.012)	19.30(0.13)
CSDI (Tashiro et al., 2021)	0.582(0.055)	0.678(0.044)	0.850(0.013)	17.978(0.089)
CFMI (ours)	0.538(0.052)	0.624(0.019)	0.822(0.019)	17.701(0.106)

B.3.2 Imputation metrics vs number of training epochs

Figure 128: CRPS versus number of training epochs.

Figure 129: Magnitude-normalised CRPS versus number of training epochs.

Figure 130: MAE versus number of training epochs.

Figure 131: RMSE versus number of training epochs.