Distribution-free two-sample testing with blurred total variation distance

TL;DR

This paper introduces the blurred total variation distance, a relaxed measure enabling distribution-free two-sample testing without assumptions, with theoretical guarantees and high-dimensional analysis.

Contribution

It proposes the blurred TV distance for distribution-free inference and provides theoretical bounds and properties in high-dimensional settings.

Findings

The blurred TV distance allows distribution-free bounds on distribution similarity.

Theoretical guarantees are established for the blurred TV distance.

Properties of the blurred TV distance are analyzed in high-dimensional spaces.

Abstract

Two-sample testing, where we aim to determine whether two distributions are equal or not equal based on samples from each one, is challenging if we cannot place assumptions on the properties of the two distributions. In particular, certifying equality of distributions, or even providing a tight upper bound on the total variation (TV) distance between the distributions, is impossible to achieve in a distribution-free regime. In this work, we examine the blurred TV distance, a relaxation of TV distance that enables us to perform inference without assumptions on the distributions. We provide theoretical guarantees for distribution-free upper and lower bounds on the blurred TV distance, and examine its properties in high dimensions.