Comparison of Methods for Quantifying Dataset Similarity

The following interactive table contains the results of the article "Comparison of Methods for Quantifying Dataset Similarity" (DOI: ...). In this, 114 methods were compared with respect to the criteria specified in the columns here. For details on the comparisons and additional information, please refer to the article. If you have additions or corrections to the criteria for a method, please add an issue in the corresponding GitHub repository. We will be happy to incorporate them here.

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✔ Fulfilled (✔) Fulfilled for certain parameter choices ✖ Unfulfilled -- Inapplicable

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Method/ Article	Also target?	Numeric?	Categorical?	$n_{i} \neq n_{j}$ ?	$p > n_{i}$ ?	$k > 2$ ?	Without training?	No assumptions?	No parameters?	Implemented?	Complexity?	Interpretable units?	Lower bound?	Upper bound?	Rotation invariant?	Location change invariant?	Homogeneous scale invariant?	Positive definite?	Symmetric?	Triangle inequality?	No. fulfilled criteria	No. cond. fulfilled criteria	No. unfulfilled criteria	No. NAs	Class	Subclass	Link
Energy statistic (Zech and Aslan, 2003)	✖	✔	✖	✔	✔	✔	✔	✖	✔	✔	$O (N^{2})$	✖	✔	NA	✔	✔	✖	✔	✔	✔	13	0	5	1	Comparison based on inter-point distances	Comparison based on inter-point distances	Energy statistic (Zech and Aslan, 2003)
Generalized energy statistic (Sejdinovic et al., 2013)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	✔	✔	NA	(✔)	✔	(✔)	9	2	5	3	Comparison based on inter-point distances	Comparison based on inter-point distances	Generalized energy statistic (Sejdinovic et al., 2013)
DISCO (Rizzo and Székely, 2010)	✖	✔	✖	✔	✔	✔	✔	✖	✖	✔	$O (N^{2})$	✖	✔	NA	✔	✔	✖	NA	✔	NA	10	0	6	3	Comparison based on inter-point distances	Comparison based on inter-point distances	DISCO (Rizzo and Székely, 2010)
Huang and Huo (2017)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	$O (m N \log (N))$	✖	✔	NA	NA	NA	NA	✖	✔	NA	6	0	7	6	Comparison based on inter-point distances	Comparison based on inter-point distances	Huang and Huo (2017)
Deb and Sen (2021)	✖	✔	✖	✔	✔	✔	✔	✖	✔	✔	$O (N^{3}) + O (n_{1} n_{2} p)$	✖	✔	NA	NA	✔	✔	✔	✔	NA	12	0	4	3	Comparison based on inter-point distances	Comparison based on inter-point distances	Deb and Sen (2021)
Al-Labadi et al. (2022)	✖	✔	✖	✔	✔	✔	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	NA	✔	NA	7	0	5	7	Comparison based on inter-point distances	Comparison based on inter-point distances	Al-Labadi et al. (2022)
Baringhaus and Franz (2010)	✖	✔	✖	✔	✔	✔	✔	✖	✖	✔	NA	✖	✔	NA	✔	✔	NA	✔	✔	NA	11	0	5	3	Comparison based on inter-point distances	Comparison based on inter-point distances	Baringhaus and Franz (2010)
Liu and Modarres (2011)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	$Independent of p$	✖	NA	NA	✔	✔	✔	NA	(✔)	NA	8	1	5	5	Comparison based on inter-point distances	Comparison based on inter-point distances	Liu and Modarres (2011)
Biswas and Ghosh (2014)	✖	✔	✖	✔	✔	✔	✔	✖	✔	NA	NA	✖	✔	NA	✔	✔	✔	NA	✔	NA	11	0	4	4	Comparison based on inter-point distances	Comparison based on inter-point distances	Biswas and Ghosh (2014)
Sarkar and Ghosh (2018)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	✔	✔	NA	NA	✔	NA	8	0	6	5	Comparison based on inter-point distances	Comparison based on inter-point distances	Sarkar and Ghosh (2018)
Montero-Manso and Vilar (2019)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	$O (N^{2} \log (N))$	✖	NA	NA	✔	✔	NA	NA	✔	NA	8	0	5	6	Comparison based on inter-point distances	Comparison based on inter-point distances	Montero-Manso and Vilar (2019)
Tsukada (2019)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	NA	✖	✔	NA	✔	✔	NA	NA	✔	NA	9	0	5	5	Comparison based on inter-point distances	Comparison based on inter-point distances	Tsukada (2019)
DataSpheres (Johnson and Dasu, 1998)	✖	✔	✔	✔	✖	✖	✔	✖	✔	NA	NA	✖	NA	NA	NA	NA	NA	NA	✔	NA	6	0	5	8	Comparison based on summary statistics	Comparison based on summary statistics	DataSpheres (Johnson and Dasu, 1998)
Constrained minimum distance (Tatti, 2007)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	$O (N^{3})$	✖	✔	NA	NA	NA	NA	✔	✔	✖	8	0	6	5	Comparison based on summary statistics	Comparison based on summary statistics
Bickel (1969)	✖	✔	✖	✔	✖	✖	✔	✖	✔	NA	$O (c o n s t^{N})$	✖	✔	NA	✖	✖	✖	✔	✔	NA	7	0	9	3	Comparison of CDFs, density or characteristic functions	Comparison of CDFs	Bickel (1969)
Biau and Gyorfi (2005)	✖	✔	✖	✖	✔	✖	✔	✔	✖	NA	NA	✖	✔	NA	✖	✖	✖	✔	✔	NA	7	0	9	3	Comparison of CDFs, density or characteristic functions	Comparison of CDFs	Biau and Gyorfi (2005)
Ntoutsi et al. (2008)	✔	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	✔	✖	✔	✔	✖	✔	NA	12	0	5	2	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Ntoutsi et al. (2008)
Ganti et al. (1999)	✔	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✔	✔	✔	✖	✔	✔	✖	✖	NA	12	0	5	2	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Ganti et al. (1999)
Roederer et al. (2001)	✖	✔	✖	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	NA	✖	✔	✖	✖	✖	NA	7	0	9	3	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Roederer et al. (2001)
Wang and Pei (2005)	✔	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	✔	✖	✖	✖	✖	✔	NA	10	0	7	2	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Wang and Pei (2005)
Ahmad and Cerrito (1993)	✖	✔	✖	✔	NA	✖	✔	✖	✖	NA	NA	✖	✔	✔	✖	✔	✖	✔	✔	NA	8	0	8	3	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Ahmad and Cerrito (1993)
Anderson et al. (1994)	✖	✔	✖	✔	NA	✖	✔	✖	✖	NA	NA	✖	✔	NA	✖	✔	✖	✔	✔	✔	8	0	8	3	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Anderson et al. (1994)
Cao and van Keilegom (2006)	✖	✔	✖	✔	NA	✔	✔	✖	✖	NA	NA	✖	NA	NA	✖	✖	✖	✖	✖	✔	5	0	10	4	Comparison of CDFs, density or characteristic functions	Comparison of density functions	Cao and van Keilegom (2006)
Alba-Fernández et al. (2004)	✖	✔	NA	✔	NA	✖	✔	✖	✖	NA	NA	✖	✔	NA	✖	✖	✖	✔	✔	NA	6	0	8	5	Comparison of CDFs, density or characteristic functions	Comparison of characteristic functions	Alba-Fernández et al. (2004)
Alba Fernández et al. (2008)	✖	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	NA	✖	✖	✖	(✔)	✔	NA	8	1	7	3	Comparison of CDFs, density or characteristic functions	Comparison of characteristic functions	Alba Fernández et al. (2008)
Liu et al. (2015)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	✖	✖	✖	NA	✔	NA	6	0	8	5	Comparison of CDFs, density or characteristic functions	Comparison of characteristic functions	Liu et al. (2015)
Li et al. (2022)	✖	✔	✔	✔	✔	✖	✔	✔	✔	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	NA	10	0	3	6	Comparison of CDFs, density or characteristic functions	Comparison of characteristic functions	Li et al. (2022)
Boeckel et al. (2018)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	NA	✔	NA	6	0	6	7	Comparison of CDFs, density or characteristic functions	Comparison of CDFs	Boeckel et al. (2018)
Engineer metric (Rachev, 1991)	✖	✔	✖	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	✖	✔	✔	✔	✖	✔	✔	11	0	7	1	Discrepancy measure for distributions	Probability metric
Zolotarev`s semimetric (Rachev, 1991)	✖	✔	✔	NA	NA	✖	NA	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	(✔)	✔	✔	5	1	5	8	Discrepancy measure for distributions	Probability metric
Ky Fan metric (Rachev, 1991)	✖	✔	✔	NA	NA	✖	NA	✔	✖	NA	NA	✖	✔	NA	✔	✔	NA	✔	✔	✔	9	0	4	6	Discrepancy measure for distributions	Probability metric
Prokhorov metric (Rachev, 1991)	✖	✔	✖	NA	NA	✖	NA	✔	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✔	6	0	5	8	Discrepancy measure for distributions	Probability metric
Dudley metric	✖	✔	✔	NA	NA	✖	NA	✔	✔	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✔	8	0	3	8	Discrepancy measure for distributions	Probability metric
Total variation metric (Zolotarev, 1984)	✖	✔	✔	NA	NA	✖	NA	✔	✔	NA	NA	✖	✔	✔	NA	NA	NA	✔	✔	✔	9	0	3	7	Discrepancy measure for distributions	Probability metric
Kantorovich-Rubinstein metric (Zolotarev, 1984, Dudley, 1989)	✖	✔	✔	NA	NA	✖	NA	✔	✔	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✔	8	0	3	8	Discrepancy measure for distributions	Probability metric	Kantorovich-Rubinstein metric (Zolotarev, 1984, Dudley, 1989)
$L_{q}$ metric	✖	✔	✔	NA	NA	✖	NA	✔	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✔	7	0	4	8	Discrepancy measure for distributions	Probability metric
(Squared) Hellinger distance	✖	✔	✔	NA	NA	✖	NA	✔	✔	✔	NA	✖	✔	✔	NA	NA	NA	✔	✔	(✔)	9	1	3	6	Discrepancy measure for distributions	Divergence
Vincze Le Cam distance (Vincze, 1981, Le Cam, 1986)	✖	✔	✔	NA	NA	✖	NA	✔	✔	NA	NA	✖	✔	✔	NA	NA	NA	✔	✔	(✔)	8	1	3	7	Discrepancy measure for distributions	Divergence
KL divergence (Kullback and Leibler, 1951)	✖	✔	✔	✔	NA	✖	✔	✔	✔	✔	NA	✖	✔	✖	NA	NA	✔	✔	✖	✖	10	0	6	3	Discrepancy measure for distributions	Divergence	KL divergence (Kullback and Leibler, 1951)
Jeffrey`s divergence	✖	✔	✔	✔	NA	✖	✔	✔	✔	✔	NA	✖	✔	✖	NA	NA	✔	✔	✔	✖	11	0	5	3	Discrepancy measure for distributions	Divergence
Extended $ϕ_{α}$ divergence	✖	✔	✔	NA	NA	✖	✔	✔	✖	NA	NA	✖	✔	✔	NA	NA	NA	✔	✔	(✔)	8	1	4	6	Discrepancy measure for distributions	Divergence
Jensen Shannon divergence	✖	✔	✔	NA	NA	✖	NA	✔	✔	✔	NA	✖	✔	✔	NA	NA	✔	✔	✔	(✔)	10	1	3	5	Discrepancy measure for distributions	Divergence
Pearson divergence (Pearson, 1900)	✖	✔	✔	✔	NA	✖	✔	✔	✔	NA	NA	✖	✔	✖	NA	NA	✔	✔	✖	✖	9	0	6	4	Discrepancy measure for distributions	Divergence	Pearson divergence (Pearson, 1900)
Relative Pearson divergence (Yamada et al., 2013)	✖	✔	✔	✔	NA	✖	✔	✔	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✖	✖	7	0	6	6	Discrepancy measure for distributions	Divergence	Relative Pearson divergence (Yamada et al., 2013)
Rényi divergence (Rényi, 1961)	✖	✔	✔	NA	NA	✖	NA	✔	✖	NA	NA	✖	✔	✖	NA	NA	NA	(✔)	✖	✖	4	1	7	7	Discrepancy measure for distributions	Divergence	Rényi divergence (Rényi, 1961)
Relative information of type s (Taneja and Kumar, 2004)	✖	✔	✔	NA	NA	✖	NA	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✖	NA	4	0	6	9	Discrepancy measure for distributions	Divergence	Relative information of type s (Taneja and Kumar, 2004)
H-divergence (Zhao et al., 2021)	✖	✔	✔	✖	✔	✖	✔	✔	✖	NA	NA	✖	✔	NA	NA	NA	NA	(✔)	(✔)	NA	6	2	5	6	Discrepancy measure for distributions	Divergence	H-divergence (Zhao et al., 2021)
Muñoz et al. (2012)	✖	✔	✖	✔	NA	✖	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✖	6	0	7	6	Discrepancy measure for distributions	Divergence	Muñoz et al. (2012)
Muñoz et al. (2013)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✖	7	0	7	5	Discrepancy measure for distributions	Divergence	Muñoz et al. (2013)
f-dissimilarity (Györfi and Nemetz, 1975)	✖	✔	✔	✔	NA	✔	NA	✔	✖	NA	NA	✖	✔	✔	NA	NA	NA	✔	NA	NA	7	1	2	9	Discrepancy measure for distributions	Divergence	f-Dissimilarity (Györfi and Nemetz, 1975, García-García & Williamson, 2012)
Feurer et al. (2015)	✔	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✖	✔	(✔)	NA	NA	NA	NA	✔	NA	9	1	3	6	Distance/ similarity measure for datasets	Distance/ similarity measure for datasets	Feurer et al. (2015)
Gromov-Hausdoff distance (Mémoli, 2017)	✖	✔	(✔)	NA	NA	✖	NA	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✔	✔	✔	5	1	5	8	Distance/ similarity measure for datasets	Distance/ similarity measure for datasets	Gromov-Hausdoff distance (Mémoli, 2017)
Leite et al. (2012), Leite and Brazdil (2021)	✔	✔	✔	✔	(✔)	✔	✔	✔	✖	NA	NA	(✔)	✔	✔	(✔)	(✔)	(✔)	✖	✔	✖	10	5	3	1	Distance/ similarity measure for datasets	Distance/ similarity measure for datasets	Leite et al. (2012), Leite and Brazdil (2021)
DeDiMs (Calderon Ramirez et al., 2022)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✖	✖	✖	6	0	8	5	Distance/ similarity measure for datasets	Distance/ similarity measure for datasets	DeDiMs (Calderon Ramirez et al., 2022)
Alvarez-Melis and Fusi (2020)	✔	✔	✔	✔	✔	✖	✔	✖	✖	✔	$?$	✖	✔	NA	NA	NA	NA	✔	✔	✔	11	0	4	4	Distance/ similarity measure for datasets	Distance/ similarity measure for datasets	Alvarez-Melis and Fusi (2020)
Friedman and Rafsky (1979)	✖	✔	✖	✔	✔	✖	✔	✖	✔	✔	NA	✔	✔	✔	✔	✔	✔	✖	✔	NA	13	0	5	1	Graph-based	Graph-based	Friedman and Rafsky (1979)
Rosenbaum (2005)	✖	✔	✖	✔	✔	✖	✔	✖	✔	✔	$O (N^{3})$	✔	✔	✔	✔	✔	✔	✖	✔	NA	13	0	5	1	Graph-based	Graph-based	Rosenbaum (2005)
Chen and Zhang (2013)	✖	✖	✔	✔	✔	✖	✔	✖	✖	✔	$O (K^{2}), O (M)$	✔	NA	NA	--	--	--	✖	✔	NA	7	0	6	3	Graph-based	Graph-based	Chen and Zhang (2013)
Biswas et al. (2014)	✖	✔	✖	✔	✔	✖	✔	✔	✔	NA	$O (N^{2} \log (N))$	✔	✔	✔	✔	✔	✔	✖	✔	NA	13	0	4	2	Graph-based	Graph-based	Biswas et al. (2014)
Petrie (2016)	✖	✔	✖	✔	✔	✔	✔	✖	✖	✔	$O (N^{2} \log (N)), O (N^{3}), O (N \log (N))$	✔	NA	NA	✔	✔	✔	NA	✔	NA	11	0	4	4	Graph-based	Graph-based	Petrie (2016)
Chen and Friedman (2017)	✖	✔	✔	✔	✔	✖	✔	✖	✖	✔	NA	✖	NA	NA	✔	✔	✔	NA	✔	NA	10	0	5	4	Graph-based	Graph-based	Chen and Friedman (2017)
Chen et al. (2018)	✖	✔	✔	✔	✔	✖	✔	✖	✖	✔	NA	✖	NA	NA	✔	✔	✔	NA	✔	NA	10	0	5	4	Graph-based	Graph-based	Chen et al. (2018)
Sarkar et al. (2020)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	✔	✔	NA	NA	✔	NA	7	0	6	6	Graph-based	Graph-based	Sarkar et al. (2020)
Mukhopadhyay and Wang (2020)	✖	✔	✔	✔	✔	✔	✔	✖	✖	✔	NA	✖	NA	NA	NA	NA	NA	NA	✔	NA	8	0	4	7	Graph-based	Graph-based	Mukhopadhyay and Wang (2020)
Mukherjee et al. (2022)	✖	✔	✔	✔	✔	✔	✔	✔	✔	✔	NA	✖	✔	NA	✔	✔	✔	NA	✔	NA	14	0	2	3	Graph-based	Graph-based	Mukherjee et al. (2022)
Weiss (1960)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	NA	✔	✔	NA	✔	✔	✔	NA	✖	NA	10	0	5	4	Graph-based	Graph-based (NN)	Weiss (1960)
Friedman and Steppel (1973)	✖	✔	(✔)	✔	✔	✖	✔	✔	✖	NA	$O (2^{p} N \log_{2} (N)), O (p N^{2})$	✖	NA	NA	✔	✔	✔	NA	✖	NA	8	1	5	5	Graph-based	Graph-based (NN)
Schilling (1986), Henze (1988)	✖	✔	✖	✔	✔	(✔)	✔	✖	✖	NA	NA	✔	✔	✔	✔	✔	✔	✖	✔	NA	11	1	5	2	Graph-based	Graph-based (NN)	Schilling (1986), Henze (1988)
Barakat et al. (1996)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	NA	✖	NA	✔	✔	✔	✔	✖	✔	NA	10	0	6	3	Graph-based	Graph-based (NN)	Barakat et al. (1996)
Nettleton and Banerjee (2001)	✖	✖	✔	✔	✔	✔	✔	✔	✖	NA	NA	✔	✔	✔	--	--	--	NA	✔	NA	10	0	3	3	Graph-based	Graph-based (NN)	Nettleton and Banerjee (2001)
Hall and Tajvidi (2002)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	✔	✔	✔	NA	✔	NA	8	0	6	5	Graph-based	Graph-based (NN)	Hall and Tajvidi (2002)
Chen et al. (2013)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	✔	✔	✔	✔	NA	✔	NA	10	0	6	3	Graph-based	Graph-based (NN)	Chen et al. (2013)
Mondal et al. (2015)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	✔	✔	✔	NA	✔	NA	8	0	6	5	Graph-based	Graph-based (NN)	Mondal et al. (2015)
Zhang and Chen (2019) (weighted)	✖	✖	✔	✔	✔	✖	✔	✔	✖	✔	NA	✖	NA	NA	--	--	--	✖	✔	NA	7	0	6	3	Graph-based	Graph-based	Zhang and Chen (2019) (weighted)
Zhang and Chen (2019) (generalized)	✖	✖	✔	✔	✔	✖	✔	✔	✖	✔	NA	✖	NA	NA	--	--	--	✖	✔	NA	7	0	6	3	Graph-based	Graph-based	Zhang and Chen (2019) (generalized)
Zhang and Chen (2019) (max-type)	✖	✖	✔	✔	✔	✖	✔	✔	✖	✔	NA	✖	NA	NA	--	--	--	✖	✔	NA	7	0	6	3	Graph-based	Graph-based	Zhang and Chen (2019) (max-type)
Song and Chen (2022)	✖	✔	✖	✔	✔	✔	✔	✖	✖	✔	NA	✖	NA	NA	✔	✔	✔	NA	✔	NA	10	0	5	4	Graph-based	Graph-based	Song and Chen (2022)
MMD (Gretton et al., 2009)	✖	✔	✔	(✔)	✔	✖	✔	✖	✖	✔	$O (N^{2} p), O (N p)$	✖	✔	NA	(✔)	(✔)	(✔)	✔	✔	✔	9	4	5	1	Kernel-based	Kernel-based (MMD)	(Linear) MMD^2 (Muandet et al., 2017, Gretton et al., 2009, , Gretton et al., 2012)
Block MMD (Zaremba et al., 2013)	✖	✔	✔	✖	✔	✖	✔	✖	✖	✔	$O (N^{1 + γ} p), γ \in (0,1)$	✖	✔	NA	(✔)	(✔)	(✔)	✔	✔	✔	9	3	6	1	Kernel-based	Kernel-based (MMD)	Block MMD (Zaremba et al., 2013)
fastMMD (Zhao and Meng, 2015)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	$O (L N p), O (L N \log p)$	✖	✔	NA	(✔)	✔	(✔)	✔	✔	✔	10	2	5	2	Kernel-based	Kernel-based (MMD)	fastMMD (Zhao and Meng, 2015)
ME (Chwialkowski et al., 2015, Jitkrittum et al., 2016)	✖	✔	NA	✖	✔	✖	✖	✖	✖	✔	$O (N^{3})$	✖	✔	NA	(✔)	(✔)	(✔)	✔	✔	NA	6	3	7	3	Kernel-based	Kernel-based (MMD)	ME (Chwialkowski et al., 2015, Jitkrittum et al., 2016)
SCF (Chwialkowski et al., 2015, Jitkrittum et al., 2016)	✖	✔	NA	✖	✔	✖	✖	✖	✖	✔	$O (N)$	✖	✔	NA	(✔)	(✔)	(✔)	NA	✔	NA	5	3	7	4	Kernel-based	Kernel-based (MMD)	SCF (Chwialkowski et al., 2015, Jitkrittum et al., 2016)
regularized MMD (Danafar et al., 2014)	✖	✔	✔	✔	✔	✔	✔	✖	✖	NA	$O (N)$	✖	NA	NA	(✔)	(✔)	(✔)	NA	✔	NA	7	3	4	5	Kernel-based	Kernel-based (MMD)	regularized MMD (Danafar et al., 2014)
Anisotropic kernel MMD (Cheng et al., 2020)	✖	✔	✖	✔	✖	✖	(✔)	✖	✖	NA	$O (N^{2})$	✖	✔	NA	NA	NA	NA	✖	✔	NA	4	1	8	6	Kernel-based	Kernel-based (MMD)	Anisotropic kernel MMD (Cheng et al., 2020)
DMMD/ DFDA (Kirchler et al., 2020)	✖	✔	✔	✔	✔	✖	✖	✖	✖	✔	$O (N N_{R})$	✖	✔	NA	(✔)	(✔)	(✔)	NA	✔	NA	7	3	6	3	Kernel-based	Kernel-based (MMD)	DMMD/ DFDA (Kirchler et al., 2020)
GPK (Song and Chen, 2021)	✖	✔	✔	✔	✔	✖	✔	✖	✖	✔	NA	✖	✔	NA	(✔)	(✔)	(✔)	NA	✔	NA	8	3	5	3	Kernel-based	Kernel-based (MMD)	GPK (Song and Chen, 2023)
Kernel FDA (Eric et al., 2007)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	NA	NA	NA	NA	✔	NA	6	0	5	8	Kernel-based	Kernel-based	Kernel FDA (Eric et al., 2007)
Fromont et al. (2012)	✖	✔	✔	✔	✔	✖	✔	✖	✖	NA	NA	✖	NA	NA	NA	NA	NA	NA	✔	NA	6	0	5	8	Kernel-based	Kernel-based	Fromont et al. (2012)
Scetbon and Varoquaux (2019)	✖	✔	✖	✔	✔	✖	(✔)	✖	✖	NA	NA	✖	✔	NA	(✔)	(✔)	(✔)	✖	✔	NA	5	4	7	3	Kernel-based	Kernel-based	Scetbon and Varoquaux (2019)
Kernel-based quadratic distance (Chen and Markatou, 2020)	✖	✔	✖	✔	✔	✔	✔	✖	✖	NA	NA	✖	✔	NA	(✔)	(✔)	(✔)	NA	✔	NA	7	3	5	4	Kernel-based	Kernel-based	Kernel-based quadratic distance (Chen and Markatou, 2020)
Bayesian kernel test (Zhang et al., 2022)	✖	✔	✔	✖	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	✔	✔	✖	NA	✔	NA	8	0	7	4	Kernel-based	Kernel-based	Bayesian kernel test (Zhang et al., 2022)
KMD (Huang and Sen, 2023)	✖	✔	✔	✔	✔	✔	✔	✔	✖	✔	$O (K N \log N)$	✖	✔	✔	✔	✔	✔	✔	✔	NA	15	0	3	1	Kernel-based	Kernel-based	KMD (Huang & Sen, 2023)
Friedman (2004)	✖	✔	(✔)	✔	(✔)	✖	(✔)	✖	✖	NA	NA	✖	NA	NA	(✔)	(✔)	(✔)	NA	✔	NA	3	6	5	5	Method based on binary classification	Method based on binary classification
C2ST (Lopez-Paz and Oquab, 2017)	✖	✔	(✔)	✔	(✔)	✖	✖	✖	✖	✔	NA	✔	✔	✔	(✔)	(✔)	(✔)	NA	✔	NA	7	5	5	2	Method based on binary classification	Method based on binary classification	C2ST (Lopez-Paz and Oquab, 2017)
Regression based test (Kim et al., 2019)	✖	✔	(✔)	✔	(✔)	✖	✖	✖	✖	NA	NA	✔	✔	✔	(✔)	(✔)	(✔)	NA	✖	NA	5	5	6	3	Method based on binary classification	Method based on binary classification	Regression based test (Kim et al., 2019)
Cheng and Cloninger (2022)	✖	✔	✔	✔	✔	✖	✖	✖	✖	NA	$O (N), O (p)$	✖	✔	✔	(✔)	(✔)	(✔)	NA	✔	NA	7	3	6	3	Method based on binary classification	Method based on binary classification	Cheng and Cloninger (2022)
Yu et al. (2007)	✖	✔	✔	✔	✔	✖	✖	✔	✖	NA	NA	✔	✔	✔	✖	✔	✔	NA	✔	NA	11	0	5	3	Method based on binary classification	Method based on binary classification	Yu et al. (2007)
DiProPerm test (Wei et al., 2016)	✖	✔	(✔)	✔	✔	✖	✔	✖	✖	✔	NA	✖	NA	NA	(✔)	(✔)	(✔)	NA	✖	NA	5	4	6	4	Method based on binary classification	Method based on binary classification	DiProPerm test (Wei et al., 2016)
Classifier Probability Test (Cai et al., 2020)	✖	✔	(✔)	✔	(✔)	✖	✖	✖	✖	NA	NA	✖	NA	NA	(✔)	(✔)	(✔)	NA	✖	NA	2	5	7	5	Method based on binary classification	Method based on binary classification	Classifier Probability Test (Cai et al., 2020)
Kim et al. (2021)	✖	✔	(✔)	✔	(✔)	✖	✖	✖	✖	NA	NA	✔	✔	✔	(✔)	(✔)	(✔)	NA	✔	NA	6	5	5	3	Method based on binary classification	Method based on binary classification	Kim et al. (2021)
Hediger et al. (2021)	✖	✔	✔	(✔)	✔	✖	(✔)	✔	✖	✔	NA	✔	✔	✔	✖	✔	✔	NA	✔	NA	11	2	4	2	Method based on binary classification	Method based on binary classification	Hediger et al. (2021)
Ghosh and Biswas (2016)	✖	✔	✖	✔	✔	✖	✖	✔	✖	NA	NA	✖	✔	NA	NA	✔	NA	✖	✔	NA	7	0	7	5	Rank-based	Rank-based	Ghosh and Biswas (2016)
Ghosal and Sen (2021)	✖	✔	✖	✔	NA	✖	✔	✖	✔	NA	NA	✖	✔	NA	NA	NA	NA	NA	✔	NA	6	0	5	8	Rank-based	Rank-based	Ghosal and Sen (2021)
Deb et al. (2021)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	$O (N^{3})$	✖	✔	NA	NA	NA	NA	NA	✔	NA	6	0	6	7	Rank-based	Rank-based	Deb et al. (2021)
Romano (1989)	✖	✔	✔	✔	✔	✔	✔	✔	✖	NA	NA	✖	✔	NA	NA	(✔)	NA	NA	✔	NA	9	1	3	6	Testing approach	Testing approach	Romano (1989)
Burke (2000)	✖	✔	✔	✔	✔	✖	✔	✔	✖	NA	NA	✖	NA	NA	NA	NA	NA	✖	✖	NA	6	0	6	7	Testing approach	Testing approach	Burke (2000)
Ping (2000)	✖	✔	✖	✔	✔	✔	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	NA	✔	NA	7	0	5	7	Testing approach	Testing approach	Ping (2000)
Chen and Hanson (2014)	✖	✔	✖	✔	NA	✔	✔	✖	✖	NA	$O (p N^{2})$	✖	✔	NA	NA	NA	NA	✖	✔	✖	6	0	7	6	Testing approach	Testing approach	Chen and Hanson (2014)
Zhou et al. (2017)	✖	✔	✖	✔	✔	✖	✔	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	✖	✖	NA	5	0	8	6	Testing approach	Testing approach	Zhou et al. (2017)
Pan et al. (2018)	✖	✔	✖	✔	✔	✔	✔	✔	✖	✔	NA	✖	✔	NA	NA	NA	NA	✔	✔	✖	10	0	5	4	Testing approach	Testing approach	Pan et al. (2018)
Wan et al. (2018)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	NA	✖	NA	NA	NA	NA	NA	NA	✔	NA	6	0	5	8	Testing approach	Testing approach	Wan et al. (2018)
Kim et al. (2020)	✖	✔	✖	✔	✔	✖	✔	✖	✔	NA	$O (N^{3})$	✖	✔	NA	✔	NA	NA	✔	✔	NA	9	0	5	5	Testing approach	Testing approach	Kim et al. (2020)
Li and Zhang (2020)	✖	✔	✖	✔	✔	✖	✔	✔	✔	NA	$O (N^{2})$	✖	✔	NA	NA	NA	NA	✔	✔	✔	10	0	4	5	Testing approach	Testing approach	Li and Zhang (2020)
Liu et al. (2022)	✖	✔	✖	✔	✔	✔	✖	✖	✖	NA	NA	✖	✔	NA	NA	NA	NA	NA	✔	NA	6	0	6	7	Testing approach	Testing approach	Liu et al. (2022)