New Lower Bounds for Testing Monotonicity and Log Concavity of Distributions

TL;DR

This paper introduces a novel technique for establishing lower bounds in distribution testing, specifically for properties like monotonicity and log-concavity, by constructing specialized distribution pairs.

Contribution

The authors develop a new method for proving lower bounds in distribution property testing, achieving tight bounds for log-concavity and improved bounds for monotonicity testing.

Findings

New lower bounds for monotonicity testing over discrete cubes

Tight lower bounds for log-concavity testing

A general technique for distribution property lower bounds

Abstract

We develop a new technique for proving distribution testing lower bounds for properties defined by inequalities involving the bin probabilities of the distribution in question. Using this technique we obtain new lower bounds for monotonicity testing over discrete cubes and tight lower bounds for log-concavity testing. Our basic technique involves constructing a pair of moment-matching families of distributions by tweaking the probabilities of pairs of bins so that one family maintains the defining inequalities while the other violates them.