Maximum Mean Discrepancy Meets Neural Networks: The Radon-Kolmogorov-Smirnov Test

Oregon Convention Center 

Maximum mean discrepancy (MMD) refers to a class of nonparametric two-sample tests based on maximizing the mean difference between samples from distribution P versus Q, over all data transformations f in a function space F. Inspired by recent work connecting the functions of Radon bounded variation (RBV) and neural networks (NN), we study the MMD taking F as the unit ball in the RBV space of a given smoothness degree k ≥ 0. This test, named the Radon-Kolmogorov-Smirnov (RKS) test, can be viewed as a generalization of the well-known and classical Kolmogorov-Smirnov (KS) test to multiple dimensions and higher orders of smoothness. It is also intimately connected to NN: we prove the RKS test's witness – the function f achieving the MMD – is always a ridge spline of degree k, i.e., a single neuron in NN. We can thus leverage the modern NN optimization toolkits to (approximately) maximize the criterion that underlies the RKS test. We prove the RKS test has asymptotically full power at distinguishing any P ≠ Q, derive its asymptotic null distribution, and carry out experiments to elucidate the strengths and weaknesses of the RKS test versus the more traditional kernel MMD test.

nonparametric two-sample testing

maximum mean discrepancy (integral probability metric)

neural network based test

ridge spline

Kolmogorov-Smirnov test 

Section on Nonparametric Statistics