A Kernel Two-Sample Test

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD). We present two distributionfree tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests. ∗. Also at Gatsby Computational Neuroscience Unit, CSML, 17 Queen Square, London WC1N 3AR, UK. †. This work was carried out while K.M.B. was with the Ludwig-Maximilians-Universität München. ‡. This work was carried out while M.J.R. was with the Graz University of Technology. §. Also at The Australian National University, Canberra, ACT 0200, Australia. c ©2012 Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf and Alexander Smola. GRETTON, BORGWARDT, RASCH, SCHÖLKOPF AND SMOLA