Testing Identity of Distributions under Kolmogorov Distance in Polylogarithmic Space

TL;DR

This paper presents a space-efficient streaming algorithm for identity testing of continuous distributions under the Kolmogorov distance, significantly reducing space complexity while maintaining optimal sample complexity.

Contribution

It introduces a novel algorithm that tests distribution identity with logarithmic space in the streaming model, improving over previous linear space methods.

Findings

Uses $O( ext{log}^4 rac{1}{ ext{epsilon}})$ space for testing

Achieves asymptotically optimal sample complexity

Contrasts with discrete case where space reduction is impossible

Abstract

Suppose we have a sample from a distribution $D$ and we want to test whether $D = D^*$ for a fixed distribution $D^*$. Specifically, we want to reject with constant probability, if the distance of $D$ from $D^*$ is $\geq \varepsilon$ in a given metric. In the case of continuous distributions, this has been studied thoroughly in the statistics literature. Namely, for the well-studied Kolmogorov metric a test is known that uses the optimal $O(1/\varepsilon^2)$ samples. However, this test naively uses also space $O(1/\varepsilon^2)$, and previous work improved this to $O(1/\varepsilon)$. In this paper, we show that much less space suffices -- we give an algorithm that uses space $O(\log^4 \varepsilon^{-1})$ in the streaming setting while also using an asymptotically optimal number of samples. This is in contrast with the standard total variation distance on discrete distributions for which such space reduction is known to be impossible. Finally, we state 9 related open problems that we hope will spark interest in this and related problems.