Remember the Curse of Dimensionality: The Case of Goodness-of-Fit Testing in Arbitrary Dimension

TL;DR

This paper investigates the fundamental difficulty of goodness-of-fit testing in high dimensions, revealing a curse of dimensionality and proposing adaptive chi-squared tests that perform well even with low intrinsic dimensions.

Contribution

It derives the minimax rate for goodness-of-fit testing in arbitrary dimensions, demonstrating the curse of dimensionality, and extends chi-squared tests to adapt to unknown smoothness and intrinsic dimension.

Findings

Minimax rate exhibits curse of dimensionality in arbitrary dimensions.

Chi-squared test achieves the minimax rate.

Multiscale chi-squared test adapts to unknown smoothness and intrinsic dimension.

Abstract

Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions spanning decades, there is no mention there of any curse of dimensionality. Only more recently Ramdas et al. (2015) have discussed this issue in the context of kernel methods by showing that their performance degrades with the dimension even when the underlying distributions are isotropic Gaussians. We take a minimax perspective and follow in the footsteps of Ingster (1987) to derive the minimax rate in arbitrary dimension when the discrepancy is measured in the L2 metric. That rate is revealed to be nonparametric and exhibit a prototypical curse of dimensionality. We further extend Ingster's work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test can be made to work when the distributions have support of low intrinsic dimension. Finally, inspired by Ingster (2000), we consider a multiscale version of the chi-square test which can adapt to unknown smoothness and/or unknown intrinsic dimensionality without much loss in power.