Testing Product Distributions: A Closer Look

TL;DR

This paper explores the sample complexity of identity and closeness testing for high-dimensional product distributions, extending prior binary alphabet results to tolerant testing over arbitrary alphabets and measures.

Contribution

It provides a detailed analysis of how sample complexity varies with different distance measures and extends bounds to bounded-degree Bayes nets.

Findings

Sample complexity varies with distance measures in tolerant testing.

Extended bounds to bounded-degree Bayes nets.

Provided a comprehensive map of testing complexities for product distributions.

Abstract

We study the problems of identity and closeness testing of $n$-dimensional product distributions. Prior works by Canonne, Diakonikolas, Kane and Stewart (COLT 2017) and Daskalakis and Pan (COLT 2017) have established tight sample complexity bounds for non-tolerant testing over a binary alphabet: given two product distributions $P$ and $Q$ over a binary alphabet, distinguish between the cases $P = Q$ and $d_{\mathrm{TV}}(P, Q) > \epsilon$. We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating tolerant testing with respect to several natural distance measures and over an arbitrary alphabet. Our study gives a fine-grained understanding of how the sample complexity of tolerant testing varies with the distance measures for product distributions. In addition, we also extend one of our upper bounds on product distributions to bounded-degree Bayes nets.