Hybrid Methods for Robust Tabular Data Imputation

Jinwei Li; Michelle Bruch; Daniel Tenbrinck

cs.LGSep 30, 2026

Hybrid Methods for Robust Tabular Data Imputation

Authors: Jinwei Li, Michelle Bruch, Daniel Tenbrinck

Organizations: Department of Data Science Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen, Germany

Abstract

Missing data are a fundamental challenge in statistical analysis and machine learning, as the choice of imputation method substantially impacts downstream inference. In this work, we propose two hybrid imputation methods called NuclearForest and SoftForest, which combine nuclear-norm-based low-rank initialization using Singular Value Thresholding (SVT) and SoftImpute, respectively, with a non-iterative Random Forest refinement. For the SVT-based component, we further introduce an adaptive step-size rule, prove adaptive step-size bounds, and establish convergence for the corresponding zero-initialized iteration. The low-rank initialization provides a structured warm start that captures the global covariance patterns in the data, while the subsequent Random Forest step recovers residual nonlinear signals encoding local dependencies. We conduct an extensive benchmark on diverse datasets from different application domains, comparing the proposed methods with seven established imputation methods under the Missing Completely at Random (MCAR), Missing at Random (MAR), and Missing Not at Random (MNAR) mechanisms across varying missingness rates. Our results demonstrate that NuclearForest and SoftForest match or exceed the imputation fidelity of state-of-the-art iterative methods such as MissForest, while significantly reducing computational cost. In particular, they achieve speedups of approximately 5.81 times and 9.52 times over MissForest by replacing iterative cycles with a single refinement step. Our approach effectively exploits the low-rank structure of real-world tabular data and accommodates mixed-type variables, providing an efficient and robust solution for data imputation in bioinformatics, economics, and beyond.

Figures & tables

Panel of Figure 2 — Figure 2: Comparison of imputation methods on the metabolomics dataset under MCAR/MAR.

Mechanism	MissForest	SoftForest	NuclearForest
MCAR/MAR	77.787 ± 21.426	8.171 ± 1.931	13.389 ± 2.365
MNAR	89.438 ± 22.845	4.344 ± 2.137	9.797 ± 2.381

Table 1: Runtime comparison for metabolomics dataset, reported in seconds as mean ± std.

Panel of Figure 3 — Figure 3: Comparison of the imputation methods using metabolomics dataset under MNAR

Panel of Figure 4 — Figure 4: Comparison of imputation methods under MCAR (a–f) and MAR (g–l).

Mechanism	MissForest	SoftForest	NuclearForest
MCAR	5.934 ± 0.187	0.664 ± 0.045	0.659 ± 0.047
MAR	5.973 ± 0.198	0.661 ± 0.039	0.661 ± 0.034

Table 2: Runtime comparison on the housing dataset, reported in seconds as mean ± std.

Appendix figures & tables45 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Configuration / hyperparameters
Mean	Column-wise mean imputation.
Median	Column-wise median imputation.
Half-min	Missing entries are replaced by one half of the observed column minimum when the minimum is positive; otherwise a floor value of $10^{-6}$ is used.
kNN	$k=10$ nearest neighbors with distance-based weighting.
MissForest	Iterative Random Forest imputation using 100 trees, maximum iterations $=10$ , median initialization, and n_jobs=-1 .
SVT_OGpaper	The original SVT method with a fixed step size and $\delta=\min(1.2/p,2)$ , zero initialization, skip-ahead dual initialization, maximum iterations $=1000$ , and tolerance $=10^{-5}$ .

Appendix

Table 3: Hyperparameter settings used in the additional experiments. SVT_OGpaper follows the fixed step size configuration of Cai et al. (2010) . SoftImpute follows the soft-thresholded SVD formulation of Mazumder et al. (2010) .

Setting	Value
Number of repeated runs	10
Random seeds	$42,43,\ldots,51$ , assigned sequentially to the 10 repeated runs
Missingness rates	$10\%,20\%,30\%,40\%,50\%,60\%,70\%,80\%,90\%$

Appendix

Table 4: Global experimental settings

Experiment	Total runtime (min)	Total runtime (h)
Metabolomics dataset under MCAR/MAR missingness	192.5	3.21
Metabolomics dataset under MNAR missingness	192.0	3.20
Housing dataset under MCAR missingness	12.9	0.22
Housing dataset under MAR missingness	13.0	0.22

Appendix

Table 5: Total wall-clock runtime.

Panel of Figure 5 — Figure 5: Comparison of SVT_OGpaper and NF_OGpaper with the adapted SVT and NuclearForest methods on the metabolomics dataset under MCAR/MAR (subfigures a–d) and MNAR (subfigures e–h).

Panel of Figure 6 — Figure 6: Comparison of SVT_OGpaper and NF_OGpaper with the adapted SVT and NuclearForest methods on the housing dataset under MCAR (subfigures a–f) and MAR (subfigures g–l).

Panel of Figure 7 — Figure 7: Comparison of imputation methods on the housing dataset under MCAR (subfigure a) and MAR (subfigure b).

Rate	SVT_OGpaper	SVT_partial	SVT
10%	0.806 ± 0.011	0.804 ± 0.010	0.740 ± 0.050
20%	0.920 ± 0.029	0.917 ± 0.028	0.814 ± 0.032
30%	0.999 ± 0.012	0.995 ± 0.014	0.874 ± 0.015
40%	1.104 ± 0.027	1.096 ± 0.028	0.938 ± 0.028
50%	1.646 ± 0.067	1.646 ± 0.067	1.023 ± 0.023
60%	1.579 ± 0.014	1.579 ± 0.014	1.140 ± 0.013

Appendix

Table 6: NRMSE

Rate	SVT_OGpaper	SVT_partial	SVT
10%	0.0028 ± 0.0006	0.0028 ± 0.0007	0.0019 ± 0.0007
20%	0.0095 ± 0.0026	0.0096 ± 0.0027	0.0059 ± 0.0011
30%	0.0216 ± 0.0029	0.0215 ± 0.0030	0.0119 ± 0.0014
40%	0.0393 ± 0.0045	0.0385 ± 0.0042	0.0226 ± 0.0048
50%	0.1020 ± 0.0178	0.1020 ± 0.0178	0.0488 ± 0.0123
60%	0.1600 ± 0.0309	0.1600 ± 0.0309	0.0952 ± 0.0312

Appendix

Table 7: PCA Procrustes distance

Rate	SVT_OGpaper	SVT_partial	SVT
10%	0.982 ± 0.004	0.982 ± 0.004	0.995 ± 0.003
20%	0.956 ± 0.012	0.956 ± 0.012	0.991 ± 0.002
30%	0.916 ± 0.020	0.915 ± 0.020	0.980 ± 0.003
40%	0.858 ± 0.015	0.858 ± 0.015	0.960 ± 0.006
50%	0.770 ± 0.025	0.770 ± 0.025	0.938 ± 0.009
60%	0.695 ± 0.039	0.695 ± 0.039	0.889 ± 0.018

Appendix

Table 8: Pearson correlation of

-\log_{10}p

Rate	SVT_OGpaper	SVT_partial	SVT
10%	0.0045 ± 0.0004	0.0045 ± 0.0005	0.0041 ± 0.0005
20%	0.0124 ± 0.0004	0.0125 ± 0.0005	0.0094 ± 0.0012
30%	0.0272 ± 0.0017	0.0271 ± 0.0015	0.0203 ± 0.0018
40%	0.0562 ± 0.0051	0.0560 ± 0.0049	0.0403 ± 0.0022
50%	0.1630 ± 0.0243	0.1630 ± 0.0243	0.0740 ± 0.0089
60%	0.2020 ± 0.0048	0.2020 ± 0.0048	0.1272 ± 0.0116

Appendix

Table 9: PLS Procrustes distance

Method	Mean runtime (s)	Std runtime (s)	Min runtime (s)	Max runtime (s)
SVT_OGpaper	5.465 ± 1.817	1.817	3.933	14.748
SVT_partial	15.810 ± 5.069	5.069	8.206	29.109
SVT	5.292 ± 1.267	1.267	3.896	9.304

Appendix

Table 10: Runtime comparison between partial SVD and full SVD in SVT.

Figure 8: Ablation study on the metabolomics dataset under MCAR/MAR missingness.

Rate	Zero · fixed	Zero · adaptive	Warm · fixed	Warm · adaptive
10%	0.811 ± 0.027	0.811 ± 0.027	0.734 ± 0.042	0.736 ± 0.043
20%	0.922 ± 0.024	0.923 ± 0.024	0.809 ± 0.026	0.810 ± 0.025
30%	1.008 ± 0.020	1.005 ± 0.020	0.877 ± 0.017	0.874 ± 0.016
40%	1.113 ± 0.025	1.104 ± 0.023	0.949 ± 0.027	0.938 ± 0.026
50%	1.656 ± 0.058	1.237 ± 0.011	1.052 ± 0.020	1.027 ± 0.017
60%	1.620 ± 0.077	1.392 ± 0.024	1.168 ± 0.025	1.127 ± 0.022

Appendix

Table 11: NRMSE across missing rates

Rate	Zero · fixed	Zero · adaptive	Warm · fixed	Warm · adaptive
10%	0.003 ± 0.001	0.003 ± 0.001	0.002 ± 0.001	0.002 ± 0.001
20%	0.010 ± 0.002	0.010 ± 0.002	0.006 ± 0.001	0.006 ± 0.001
30%	0.021 ± 0.004	0.021 ± 0.004	0.012 ± 0.003	0.012 ± 0.003
40%	0.037 ± 0.005	0.037 ± 0.004	0.022 ± 0.003	0.022 ± 0.004
50%	0.105 ± 0.017	0.081 ± 0.018	0.049 ± 0.010	0.048 ± 0.010
60%	0.175 ± 0.027	0.167 ± 0.021	0.095 ± 0.023	0.095 ± 0.025

Appendix

Table 12: PCA Procrustes distance across missing rates

Rate	Zero · fixed	Zero · adaptive	Warm · fixed	Warm · adaptive
10%	0.005 ± 0.000	0.005 ± 0.000	0.004 ± 0.000	0.004 ± 0.000
20%	0.012 ± 0.001	0.012 ± 0.001	0.009 ± 0.001	0.009 ± 0.001
30%	0.027 ± 0.003	0.027 ± 0.003	0.021 ± 0.003	0.021 ± 0.003
40%	0.052 ± 0.006	0.052 ± 0.006	0.040 ± 0.002	0.040 ± 0.002
50%	0.163 ± 0.022	0.105 ± 0.009	0.072 ± 0.009	0.071 ± 0.008
60%	0.220 ± 0.023	0.194 ± 0.018	0.126 ± 0.017	0.125 ± 0.015

Appendix

Table 13: PLS Procrustes distance across missing rates

Rate	Zero · fixed	Zero · adaptive	Warm · fixed	Warm · adaptive
10%	0.982 ± 0.006	0.982 ± 0.006	0.996 ± 0.002	0.996 ± 0.002
20%	0.951 ± 0.016	0.951 ± 0.016	0.990 ± 0.003	0.990 ± 0.002
30%	0.906 ± 0.020	0.906 ± 0.020	0.979 ± 0.006	0.978 ± 0.005
40%	0.856 ± 0.020	0.857 ± 0.021	0.962 ± 0.006	0.962 ± 0.006
50%	0.765 ± 0.024	0.781 ± 0.025	0.936 ± 0.010	0.935 ± 0.009
60%	0.682 ± 0.034	0.693 ± 0.047	0.894 ± 0.012	0.893 ± 0.014

Appendix

Table 14: Pearson log-

p

correlation across missing rates

Method	Mean ± Std.	Min.	Max.
Zero init + fixed step size	6.440 ± 1.616	4.385	12.716
Zero init + adaptive step size	6.147 ± 1.152	4.377	11.303
Warm start + fixed step size	6.378 ± 1.355	4.421	11.093
Warm start + adaptive step size	6.459 ± 1.669	4.229	12.460

Appendix

Table 15: Runtime comparison of SVT ablation variants. Runtime is reported in seconds as mean ± standard deviation

Missing	Pre. (ms)	SVD (ms)	SVT (s)	RF (s)	NF (s)	MF (s)	SI (s)
20%	0.18 ms	4.09 ms	8.97 s	10.31 s	19.28 s	105.50 s	0.48 s
50%	0.22 ms	10.18 ms	10.28 s	7.75 s	18.03 s	76.64 s	0.80 s
80%	0.19 ms	12.81 ms	8.22 s	5.08 s	13.30 s	50.84 s	0.46 s

Appendix

Table 16: Stage-decomposed runtime on the metabolomics dataset (

198\times 130

, MCAR). Times presented are means over three measured runs. Pre denotes Preprocess, SVD denotes one SVD. NF denotes NuclearForest, MF denotes MissForest, and SI denotes SoftImpute.

Matrix size	Entries	SVT stage	RF stage	Total	RF share
$(1000\times 130)$	130,000	19.58 s	46.44 s	66.02 s	70.3%
$(2000\times 130)$	260,000	25.84 s	104.19 s	130.02 s	80.1%
$(1000\times 500)$	500,000	169.73 s	653.89 s	823.62 s	79.4%
$(5000\times 200)$	1,000,000	106.67 s	727.89 s	834.56 s	87.2%
$(2000\times 1000)$	2,000,000	893.83 s	5991.90 s	6885.72 s	87.0%

Appendix

Table 17: Runtime scaling study on rank-10 Gaussian matrices with 5% noise and 30% MCAR masking.

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.618 ± 0.020	0.888 ± 0.012	1.006 ± 0.001	1.029 ± 0.002	2.389 ± 0.063	0.811 ± 0.027
20%	0.646 ± 0.021	0.901 ± 0.009	1.005 ± 0.001	1.029 ± 0.002	2.353 ± 0.043	0.922 ± 0.024
30%	0.680 ± 0.007	0.922 ± 0.005	1.006 ± 0.001	1.031 ± 0.003	2.340 ± 0.049	1.008 ± 0.020
40%	0.719 ± 0.009	0.949 ± 0.005	1.007 ± 0.001	1.030 ± 0.002	2.323 ± 0.027	1.113 ± 0.025
50%	0.760 ± 0.008	0.970 ± 0.008	1.007 ± 0.001	1.031 ± 0.003	2.311 ± 0.030	1.656 ± 0.058
60%	0.812 ± 0.009	1.001 ± 0.010	1.008 ± 0.001	1.031 ± 0.002	2.314 ± 0.016	1.620 ± 0.077

Appendix

Table 18: NRMSE across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.002 ± 0.001	0.005 ± 0.001	0.007 ± 0.001	0.007 ± 0.001	0.038 ± 0.007	0.003 ± 0.001
20%	0.005 ± 0.001	0.016 ± 0.003	0.018 ± 0.003	0.018 ± 0.004	0.081 ± 0.013	0.010 ± 0.002
30%	0.010 ± 0.002	0.037 ± 0.007	0.037 ± 0.006	0.039 ± 0.007	0.153 ± 0.043	0.021 ± 0.004
40%	0.020 ± 0.003	0.062 ± 0.020	0.062 ± 0.016	0.064 ± 0.015	0.243 ± 0.041	0.037 ± 0.005
50%	0.030 ± 0.004	0.101 ± 0.028	0.099 ± 0.035	0.101 ± 0.034	0.400 ± 0.077	0.105 ± 0.017
60%	0.053 ± 0.010	0.164 ± 0.049	0.215 ± 0.066	0.221 ± 0.073	0.625 ± 0.072	0.175 ± 0.027

Appendix

Table 19: PCA Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.003 ± 0.001	0.009 ± 0.001	0.009 ± 0.001	0.010 ± 0.001	0.057 ± 0.008	0.005 ± 0.000
20%	0.008 ± 0.001	0.021 ± 0.001	0.021 ± 0.001	0.022 ± 0.001	0.116 ± 0.015	0.012 ± 0.001
30%	0.016 ± 0.002	0.043 ± 0.005	0.040 ± 0.006	0.042 ± 0.006	0.193 ± 0.034	0.027 ± 0.003
40%	0.027 ± 0.003	0.069 ± 0.009	0.065 ± 0.009	0.067 ± 0.009	0.274 ± 0.023	0.052 ± 0.006
50%	0.043 ± 0.002	0.101 ± 0.009	0.097 ± 0.011	0.098 ± 0.012	0.371 ± 0.037	0.163 ± 0.022
60%	0.073 ± 0.010	0.166 ± 0.023	0.177 ± 0.014	0.181 ± 0.017	0.537 ± 0.036	0.220 ± 0.023

Appendix

Table 20: PLS Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.995 ± 0.003	0.993 ± 0.003	0.992 ± 0.003	0.991 ± 0.003	0.892 ± 0.018	0.982 ± 0.006
20%	0.991 ± 0.002	0.985 ± 0.002	0.983 ± 0.003	0.981 ± 0.004	0.814 ± 0.037	0.951 ± 0.016
30%	0.984 ± 0.004	0.975 ± 0.004	0.970 ± 0.006	0.966 ± 0.007	0.719 ± 0.047	0.906 ± 0.020
40%	0.972 ± 0.007	0.958 ± 0.010	0.957 ± 0.008	0.952 ± 0.008	0.666 ± 0.080	0.856 ± 0.020
50%	0.956 ± 0.008	0.942 ± 0.008	0.940 ± 0.010	0.931 ± 0.012	0.582 ± 0.072	0.765 ± 0.024
60%	0.938 ± 0.010	0.916 ± 0.013	0.909 ± 0.017	0.900 ± 0.022	0.511 ± 0.088	0.682 ± 0.034

Appendix

Table 21: Pearson log-

p

correlation across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	81.200 ± 7.815	111.400 ± 7.575	126.900 ± 5.990	97.600 ± 7.619	83.900 ± 12.991	49.900 ± 9.231
20%	158.300 ± 9.031	216.800 ± 7.495	253.000 ± 13.241	182.700 ± 14.244	153.300 ± 18.252	119.900 ± 14.873
30%	245.000 ± 19.067	328.300 ± 12.650	380.200 ± 11.013	271.700 ± 16.207	222.600 ± 27.240	162.300 ± 15.805
40%	327.600 ± 17.494	434.600 ± 12.011	509.400 ± 13.032	369.000 ± 22.784	279.400 ± 19.873	215.300 ± 16.687
50%	395.600 ± 27.293	538.500 ± 13.938	624.700 ± 10.605	441.900 ± 19.451	356.300 ± 29.208	306.200 ± 17.415
60%	480.600 ± 23.114	654.100 ± 23.473	740.500 ± 16.939	518.100 ± 16.462	398.600 ± 23.177	345.900 ± 28.136

Appendix

Table 22: SOR across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.001 ± 0.001	0.004 ± 0.001	0.005 ± 0.002	0.004 ± 0.002	0.001 ± 0.000	0.001 ± 0.001
20%	0.003 ± 0.001	0.010 ± 0.005	0.012 ± 0.005	0.009 ± 0.005	0.002 ± 0.001	0.003 ± 0.002
30%	0.011 ± 0.003	0.038 ± 0.018	0.037 ± 0.017	0.029 ± 0.012	0.004 ± 0.001	0.005 ± 0.002
40%	0.021 ± 0.011	0.056 ± 0.021	0.066 ± 0.035	0.047 ± 0.020	0.006 ± 0.001	0.006 ± 0.002
50%	0.032 ± 0.008	0.108 ± 0.036	0.122 ± 0.033	0.088 ± 0.023	0.009 ± 0.001	0.010 ± 0.003
60%	0.050 ± 0.013	0.170 ± 0.061	0.191 ± 0.040	0.129 ± 0.027	0.010 ± 0.002	0.014 ± 0.003

Appendix

Table 23: PCA Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.002 ± 0.001	0.005 ± 0.003	0.006 ± 0.003	0.005 ± 0.002	0.001 ± 0.000	0.002 ± 0.001
20%	0.006 ± 0.002	0.013 ± 0.004	0.013 ± 0.004	0.009 ± 0.003	0.002 ± 0.000	0.004 ± 0.001
30%	0.015 ± 0.005	0.041 ± 0.021	0.040 ± 0.016	0.031 ± 0.012	0.005 ± 0.001	0.009 ± 0.004
40%	0.024 ± 0.008	0.061 ± 0.029	0.067 ± 0.025	0.051 ± 0.019	0.007 ± 0.002	0.009 ± 0.002
50%	0.039 ± 0.006	0.090 ± 0.028	0.107 ± 0.020	0.080 ± 0.015	0.010 ± 0.001	0.018 ± 0.003
60%	0.057 ± 0.015	0.128 ± 0.052	0.155 ± 0.040	0.114 ± 0.029	0.012 ± 0.003	0.024 ± 0.005

Appendix

Table 24: PLS Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.991 ± 0.007	0.962 ± 0.024	0.942 ± 0.036	0.956 ± 0.027	0.995 ± 0.003	0.997 ± 0.001
20%	0.983 ± 0.006	0.948 ± 0.022	0.924 ± 0.030	0.946 ± 0.024	0.994 ± 0.002	0.992 ± 0.004
30%	0.964 ± 0.015	0.890 ± 0.048	0.849 ± 0.042	0.888 ± 0.032	0.989 ± 0.004	0.988 ± 0.006
40%	0.950 ± 0.012	0.842 ± 0.046	0.787 ± 0.037	0.838 ± 0.031	0.987 ± 0.003	0.986 ± 0.003
50%	0.948 ± 0.013	0.847 ± 0.035	0.788 ± 0.028	0.842 ± 0.023	0.986 ± 0.004	0.975 ± 0.006
60%	0.921 ± 0.016	0.798 ± 0.079	0.734 ± 0.061	0.803 ± 0.042	0.982 ± 0.005	0.967 ± 0.009

Appendix

Table 25: Pearson log-

p

correlation across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.842 ± 0.032	1.049 ± 0.029	1.003 ± 0.001	1.147 ± 0.012	1.829 ± 0.057	1.451 ± 0.037
20%	0.892 ± 0.020	1.072 ± 0.026	1.003 ± 0.001	1.141 ± 0.011	1.814 ± 0.032	1.460 ± 0.038
30%	0.950 ± 0.015	1.060 ± 0.024	1.002 ± 0.001	1.134 ± 0.005	1.818 ± 0.020	1.520 ± 0.021
40%	1.020 ± 0.023	1.011 ± 0.010	1.003 ± 0.001	1.138 ± 0.005	1.807 ± 0.056	1.401 ± 0.016
50%	1.072 ± 0.021	1.006 ± 0.006	1.002 ± 0.001	1.137 ± 0.006	1.799 ± 0.039	1.689 ± 0.036
60%	1.124 ± 0.031	1.018 ± 0.005	1.003 ± 0.001	1.140 ± 0.004	1.759 ± 0.047	1.610 ± 0.031

Appendix

Table 26: NRMSE across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.145 ± 0.017	0.271 ± 0.027	0.253 ± 0.014	0.253 ± 0.014	0.338 ± 0.021	0.271 ± 0.016
20%	0.173 ± 0.007	0.254 ± 0.004	0.253 ± 0.012	0.253 ± 0.012	0.333 ± 0.010	0.274 ± 0.010
30%	0.194 ± 0.010	0.243 ± 0.018	0.247 ± 0.007	0.247 ± 0.007	0.331 ± 0.010	0.281 ± 0.006
40%	0.227 ± 0.010	0.236 ± 0.008	0.247 ± 0.006	0.247 ± 0.006	0.332 ± 0.009	0.245 ± 0.018
50%	0.252 ± 0.014	0.243 ± 0.005	0.250 ± 0.006	0.250 ± 0.006	0.333 ± 0.009	0.295 ± 0.013
60%	0.270 ± 0.017	0.254 ± 0.005	0.253 ± 0.004	0.253 ± 0.004	0.333 ± 0.006	0.289 ± 0.007

Appendix

Table 27: PFC across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.130 ± 0.008	0.228 ± 0.019	0.215 ± 0.009	0.212 ± 0.009	0.342 ± 0.018	0.265 ± 0.009
20%	0.147 ± 0.005	0.216 ± 0.008	0.215 ± 0.008	0.209 ± 0.009	0.337 ± 0.008	0.269 ± 0.012
30%	0.161 ± 0.006	0.208 ± 0.012	0.212 ± 0.006	0.205 ± 0.006	0.333 ± 0.008	0.280 ± 0.007
40%	0.185 ± 0.007	0.202 ± 0.005	0.211 ± 0.004	0.205 ± 0.005	0.335 ± 0.008	0.246 ± 0.012
50%	0.204 ± 0.008	0.207 ± 0.003	0.214 ± 0.005	0.208 ± 0.005	0.335 ± 0.008	0.304 ± 0.012
60%	0.221 ± 0.010	0.216 ± 0.004	0.217 ± 0.003	0.211 ± 0.003	0.333 ± 0.006	0.297 ± 0.006

Appendix

Table 28: Gower’s distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.023 ± 0.008	0.038 ± 0.017	0.100 ± 0.021	0.113 ± 0.020	0.330 ± 0.044	0.339 ± 0.045
20%	0.034 ± 0.012	0.114 ± 0.029	0.192 ± 0.026	0.217 ± 0.030	0.479 ± 0.026	0.449 ± 0.034
30%	0.050 ± 0.023	0.173 ± 0.033	0.276 ± 0.033	0.324 ± 0.041	0.571 ± 0.056	0.499 ± 0.056
40%	0.065 ± 0.017	0.217 ± 0.020	0.356 ± 0.040	0.414 ± 0.052	0.630 ± 0.041	0.760 ± 0.096
50%	0.059 ± 0.032	0.325 ± 0.048	0.448 ± 0.035	0.559 ± 0.038	0.672 ± 0.034	1.180 ± 0.118
60%	0.060 ± 0.034	0.480 ± 0.054	0.562 ± 0.066	0.697 ± 0.109	0.689 ± 0.055	1.152 ± 0.096

Appendix

Table 29: Downstream predictive

R^{2}

degradation across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.024 ± 0.004	0.045 ± 0.006	0.043 ± 0.003	0.060 ± 0.005	0.122 ± 0.007	0.112 ± 0.011
20%	0.067 ± 0.008	0.096 ± 0.007	0.091 ± 0.006	0.121 ± 0.008	0.227 ± 0.020	0.225 ± 0.020
30%	0.124 ± 0.010	0.154 ± 0.020	0.142 ± 0.005	0.185 ± 0.009	0.335 ± 0.015	0.361 ± 0.022
40%	0.208 ± 0.011	0.198 ± 0.011	0.198 ± 0.007	0.257 ± 0.008	0.436 ± 0.017	0.367 ± 0.019
50%	0.319 ± 0.021	0.284 ± 0.011	0.282 ± 0.014	0.354 ± 0.016	0.559 ± 0.018	0.492 ± 0.025
60%	0.441 ± 0.015	0.393 ± 0.018	0.378 ± 0.011	0.455 ± 0.013	0.657 ± 0.024	0.584 ± 0.020

Appendix

Table 30: PCA Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.027 ± 0.004	0.047 ± 0.007	0.046 ± 0.004	0.062 ± 0.006	0.140 ± 0.011	0.134 ± 0.014
20%	0.069 ± 0.007	0.100 ± 0.006	0.096 ± 0.006	0.125 ± 0.009	0.254 ± 0.022	0.257 ± 0.024
30%	0.124 ± 0.007	0.157 ± 0.017	0.148 ± 0.007	0.187 ± 0.010	0.364 ± 0.020	0.393 ± 0.026
40%	0.207 ± 0.012	0.203 ± 0.013	0.206 ± 0.009	0.260 ± 0.010	0.466 ± 0.020	0.403 ± 0.020
50%	0.314 ± 0.020	0.290 ± 0.012	0.292 ± 0.014	0.358 ± 0.019	0.589 ± 0.020	0.526 ± 0.024
60%	0.439 ± 0.021	0.396 ± 0.020	0.385 ± 0.016	0.455 ± 0.016	0.680 ± 0.025	0.610 ± 0.023

Appendix

Table 31: PLS Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.993 ± 0.004	0.993 ± 0.004	0.983 ± 0.008	0.995 ± 0.002	0.892 ± 0.031	0.761 ± 0.033
20%	0.970 ± 0.008	0.971 ± 0.015	0.952 ± 0.016	0.984 ± 0.008	0.854 ± 0.035	0.631 ± 0.045
30%	0.926 ± 0.015	0.966 ± 0.015	0.927 ± 0.024	0.973 ± 0.014	0.763 ± 0.040	0.486 ± 0.047
40%	0.885 ± 0.032	0.959 ± 0.018	0.920 ± 0.023	0.970 ± 0.015	0.661 ± 0.056	0.294 ± 0.053
50%	0.846 ± 0.023	0.927 ± 0.021	0.890 ± 0.029	0.951 ± 0.015	0.675 ± 0.082	0.522 ± 0.113
60%	0.794 ± 0.032	0.919 ± 0.029	0.875 ± 0.036	0.941 ± 0.029	0.600 ± 0.064	0.325 ± 0.119

Appendix

Table 32: Pearson log-

p

correlation across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.852 ± 0.033	1.063 ± 0.061	1.009 ± 0.005	1.119 ± 0.012	1.808 ± 0.066	1.415 ± 0.046
20%	0.920 ± 0.029	1.060 ± 0.021	1.012 ± 0.003	1.132 ± 0.005	1.779 ± 0.050	1.458 ± 0.044
30%	0.979 ± 0.020	1.068 ± 0.019	1.018 ± 0.003	1.124 ± 0.008	1.774 ± 0.039	1.534 ± 0.029
40%	1.032 ± 0.027	1.055 ± 0.010	1.023 ± 0.003	1.124 ± 0.007	1.733 ± 0.040	1.371 ± 0.030
50%	1.084 ± 0.019	1.068 ± 0.004	1.033 ± 0.004	1.127 ± 0.002	1.702 ± 0.031	2.007 ± 0.065
60%	1.107 ± 0.026	1.099 ± 0.008	1.046 ± 0.008	1.132 ± 0.006	1.687 ± 0.020	1.965 ± 0.056

Appendix

Table 33: NRMSE across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.134 ± 0.019	0.256 ± 0.030	0.229 ± 0.022	0.229 ± 0.022	0.301 ± 0.018	0.243 ± 0.017
20%	0.178 ± 0.012	0.244 ± 0.024	0.247 ± 0.013	0.247 ± 0.013	0.318 ± 0.010	0.276 ± 0.014
30%	0.204 ± 0.016	0.236 ± 0.011	0.235 ± 0.013	0.235 ± 0.013	0.310 ± 0.016	0.270 ± 0.015
40%	0.237 ± 0.016	0.242 ± 0.006	0.240 ± 0.006	0.240 ± 0.006	0.310 ± 0.007	0.241 ± 0.011
50%	0.258 ± 0.012	0.247 ± 0.005	0.244 ± 0.002	0.244 ± 0.002	0.315 ± 0.003	0.291 ± 0.009
60%	0.266 ± 0.010	0.259 ± 0.005	0.242 ± 0.005	0.242 ± 0.005	0.313 ± 0.006	0.293 ± 0.008

Appendix

Table 34: PFC across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.128 ± 0.012	0.216 ± 0.019	0.203 ± 0.017	0.194 ± 0.018	0.323 ± 0.017	0.250 ± 0.012
20%	0.151 ± 0.008	0.214 ± 0.012	0.220 ± 0.011	0.212 ± 0.010	0.341 ± 0.009	0.278 ± 0.012
30%	0.168 ± 0.012	0.214 ± 0.010	0.211 ± 0.010	0.202 ± 0.010	0.334 ± 0.010	0.280 ± 0.012
40%	0.188 ± 0.008	0.223 ± 0.003	0.219 ± 0.004	0.213 ± 0.004	0.337 ± 0.005	0.248 ± 0.009
50%	0.205 ± 0.007	0.226 ± 0.005	0.222 ± 0.003	0.217 ± 0.004	0.338 ± 0.005	0.356 ± 0.005
60%	0.215 ± 0.009	0.238 ± 0.003	0.225 ± 0.004	0.219 ± 0.004	0.338 ± 0.005	0.347 ± 0.009

Appendix

Table 35: Gower’s distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.021 ± 0.008	0.026 ± 0.024	0.068 ± 0.022	0.057 ± 0.023	0.206 ± 0.040	0.215 ± 0.042
20%	0.031 ± 0.015	0.066 ± 0.019	0.143 ± 0.019	0.129 ± 0.022	0.318 ± 0.050	0.320 ± 0.058
30%	0.028 ± 0.019	0.144 ± 0.034	0.225 ± 0.030	0.210 ± 0.039	0.374 ± 0.055	0.353 ± 0.062
40%	0.060 ± 0.010	0.228 ± 0.068	0.275 ± 0.051	0.264 ± 0.052	0.385 ± 0.047	0.513 ± 0.081
50%	0.070 ± 0.031	0.323 ± 0.050	0.341 ± 0.036	0.354 ± 0.056	0.394 ± 0.044	1.264 ± 0.138
60%	0.057 ± 0.027	0.453 ± 0.075	0.421 ± 0.045	0.469 ± 0.049	0.418 ± 0.059	1.339 ± 0.156

Appendix

Table 36: Downstream predictive

R^{2}

degradation across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.022 ± 0.004	0.038 ± 0.004	0.038 ± 0.003	0.049 ± 0.005	0.091 ± 0.006	0.081 ± 0.005
20%	0.070 ± 0.009	0.085 ± 0.006	0.086 ± 0.004	0.109 ± 0.004	0.179 ± 0.010	0.186 ± 0.010
30%	0.131 ± 0.008	0.139 ± 0.005	0.141 ± 0.005	0.162 ± 0.006	0.243 ± 0.011	0.284 ± 0.018
40%	0.213 ± 0.029	0.224 ± 0.015	0.214 ± 0.010	0.233 ± 0.011	0.310 ± 0.009	0.289 ± 0.012
50%	0.312 ± 0.021	0.345 ± 0.012	0.307 ± 0.011	0.320 ± 0.010	0.365 ± 0.015	0.458 ± 0.021
60%	0.404 ± 0.027	0.516 ± 0.021	0.421 ± 0.018	0.400 ± 0.010	0.410 ± 0.009	0.494 ± 0.024

Appendix

Table 37: PCA Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.023 ± 0.004	0.039 ± 0.003	0.039 ± 0.003	0.049 ± 0.005	0.101 ± 0.006	0.095 ± 0.006
20%	0.071 ± 0.009	0.089 ± 0.007	0.091 ± 0.004	0.113 ± 0.005	0.195 ± 0.011	0.208 ± 0.012
30%	0.130 ± 0.008	0.145 ± 0.009	0.147 ± 0.007	0.166 ± 0.009	0.263 ± 0.018	0.307 ± 0.023
40%	0.206 ± 0.025	0.235 ± 0.018	0.224 ± 0.013	0.239 ± 0.016	0.332 ± 0.012	0.318 ± 0.012
50%	0.313 ± 0.021	0.358 ± 0.016	0.319 ± 0.013	0.327 ± 0.011	0.385 ± 0.017	0.465 ± 0.021
60%	0.398 ± 0.033	0.537 ± 0.024	0.435 ± 0.021	0.409 ± 0.015	0.430 ± 0.010	0.499 ± 0.023

Appendix

Table 38: PLS Procrustes distance across missing rates

Rate	MissForest	kNN	Mean	Median	Half-min	SVT_OGpaper
10%	0.994 ± 0.003	0.995 ± 0.004	0.968 ± 0.010	0.987 ± 0.006	0.886 ± 0.021	0.808 ± 0.026
20%	0.972 ± 0.011	0.988 ± 0.005	0.918 ± 0.025	0.952 ± 0.017	0.808 ± 0.034	0.682 ± 0.047
30%	0.940 ± 0.023	0.970 ± 0.015	0.889 ± 0.023	0.921 ± 0.022	0.698 ± 0.028	0.551 ± 0.042
40%	0.879 ± 0.021	0.931 ± 0.020	0.832 ± 0.027	0.854 ± 0.029	0.638 ± 0.042	0.488 ± 0.045
50%	0.810 ± 0.021	0.827 ± 0.037	0.783 ± 0.042	0.767 ± 0.041	0.533 ± 0.041	0.629 ± 0.045
60%	0.751 ± 0.040	0.707 ± 0.063	0.739 ± 0.036	0.674 ± 0.046	0.473 ± 0.056	0.541 ± 0.064

Appendix

Table 39: Pearson log-

p

correlation across missing rates

Method	Mean (s)	Std (s)	Min (s)	Max (s)
MissForest	77.787	21.426	49.290	116.269
kNN	0.040	0.009	0.021	0.063
Mean	0.001	0.001	0.000	0.004
Median	0.002	0.001	0.001	0.006
Half-min	0.000	0.000	0.000	0.002
SVT_OGpaper	5.889	1.261	4.180	12.519

Appendix

Table 40: Runtime comparison of imputation methods on the metabolomics dataset under MCAR/MAR missingness. Runtime is reported in seconds.

Method	Mean (s)	Std (s)	Min (s)	Max (s)
MissForest	89.438	22.845	11.206	111.187
kNN	0.025	0.012	0.006	0.060
Mean	0.001	0.001	0.000	0.005
Median	0.002	0.001	0.001	0.005
Half-min	0.000	0.000	0.000	0.001
SVT_OGpaper	5.611	0.769	4.911	9.745

Appendix

Table 41: Runtime comparison of imputation methods on the metabolomics dataset under MNAR missingness. Runtime is reported in seconds.

Method	Mean (s)	Std (s)	Min (s)	Max (s)
MissForest	5.934	0.187	5.516	6.454
kNN	0.034	0.024	0.010	0.144
Mean	0.001	0.000	0.000	0.004
Median	0.001	0.000	0.001	0.003
Half-min	0.000	0.000	0.000	0.003
SVT_OGpaper	0.119	0.096	0.002	0.285

Appendix

Table 42: Runtime comparison of imputation methods on the housing dataset under MCAR missingness. Runtime is reported in seconds.

Method	Mean (s)	Std (s)	Min (s)	Max (s)
MissForest	5.973	0.198	5.719	6.754
kNN	0.030	0.018	0.008	0.140
Mean	0.001	0.001	0.000	0.007
Median	0.001	0.000	0.001	0.003
Half-min	0.000	0.000	0.000	0.001
SVT_OGpaper	0.120	0.096	0.002	0.305

Appendix

Table 43: Runtime comparison of imputation methods on the housing dataset under MAR missingness. Runtime is reported in seconds.

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