$L_p$-Testers for Bounded Derivative Properties on Product Distributions

TL;DR

This paper develops $L_p$-testers for bounded derivative properties on hypergrids under product distributions, achieving optimal sample complexity and extending previous uniform distribution results to more general distributions.

Contribution

It introduces $L_p$-testers that work for arbitrary product distributions with optimal bounds, extending known results from uniform to general distributions.

Findings

Testers match bounds for uniform distributions on arbitrary product distributions.

Testers are optimal for a broad class of bounded derivative properties.

Time complexity remains consistent across different distributions.

Abstract

We consider the problem of $L_p$-testing of class of bounded derivative properties over hypergrid domain with points distributed according to some product distribution. This class includes monotonicity, the Lipschitz property, $(\alpha,\beta)$-generalized Lipschitz and many more properties. Previous results for $L_p$ testing on $[n]^d$ for this class were known for monotonicity and $c$-Lipschitz properties over uniformly distributed domains. \medskip Our results imply testers that give the same upper bound for arbitrary product distributions as the hitherto known testers, which use uniformly randomly chosen samples from $[n]^d$, for monotonicity and Lipschitz testing. Also, our testers are \emph{optimal} for a large class of bounded derivative properties, that includes $(\alpha, \beta)$-generalized Lipschitz property, over uniform distributions. Infact, each edge in $[n]^d$ is allowed to have it's own left and right Lipschitz constants. The time complexity is \emph{same} for arbitrary product distributions.