Hypothesis testing using pairwise distances and associated kernels

By Dino Sejdinovic, Arthur Gretton, Bharath K. Sriperumbudur, Kenji Fukumizu. 2012

42 citations recorded by the source corpus.

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. The equivalence holds when energy distances are computed with semimetrics of negative type, in which case a kernel may be defined such that the RKHS distance between distributions corresponds exactly to the energy distance. We determine the class of probability distributions for which kernels induced by semimetrics are characteristic (that is, for which embeddings of the distributions to an RKHS are injective). Finally, we investigate the performance of this family of kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.

Subfields: Control and Systems Engineering, Statistics and Probability, Statistical Learning Theory

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Control and Systems Engineering

In Statistics and Probability

In Statistical Learning Theory

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “1 Sejdinovic et al. (2012) shows that the distance covariance, originally defined with Euclidean distance, can be generalized with a semi-metric of negative type.”
NoPeek: Information leakage reduction to share activations in distributed deep learning (2020, methodology): “Distance correlation has been shown to be a simpler special case of other recent popular measures of dependence such as Hilbert-Schmidt Independence Criterion (HSIC), Maximum Mean Discrepancy (MMD) and Kernelized Mutual Information (KMI) that have been extensively studied and used recently in the ma”
Prototypal Analysis and Prototypal Regression (2017, background): “2 Prototypal Learning on Distributions through Kernel Embedding Probability distributions or samples thereof can also be mapped to a reproducing kernel Hilbert space via kernel embedding (see Berlinet and Thomas-Agnan, 2004; Gretton et al., 2006; Smola et al., 2007; Sriperumbudur et al., 2010; Sejdi”
A Kernel Test for Three-Variable Interactions (2013, methodology): “When a bounded characteristic kernel is used, the above tests are consistent against all alternatives, and their alternative interpretation is as a generalization [29, 3] of energy distance [30, 31] and distance covariance [2, 32].”
B-tests: Low Variance Kernel Two-Sample Tests (2013, methodology): “Kernel test statistics include the maximum mean discrepancy [10, 6] (of which the energy distance is an example [18, 2, 22]), which is the distance between expected features of P and Q in the RKHS; the kernel Fisher discriminant [12], which is the distance between expected feature maps normalized by”
Energy distance (2016, methodology): “Measures of the energy distance type have also been studied in the machine learning literature.(21) TESTING FOR EQUAL DISTRIBUTIONS”
B-test: A Non-parametric, Low Variance Kernel Two-sample Test (2013, background): “These include the maximum mean discrepancy [9, 6] (of which the energy distance is an example [15, 2, 19]), which is the distance between expected features of P and Q in the RHKS; the kernel Fisher discriminant [11], which is the distance between expected feature maps normalized by the feature space”
Kernel-Based Methods for Hypothesis Testing: A Unified View (2013, unknown): “Several interpretations of the MMD test statistic are discussed and reviewed in [38], [15], and [47], respectively.”
Markov Decision Processes (2004, unknown): “In particular, we adopt MMD (Nguyen-Tang et al., 2021; Sejdinovic et al., 2012) as the distribution metric, i.”