Distribution-Free Testing of Linear Functions on R^n

TL;DR

This paper develops distribution-free property testing methods to determine whether a function on R^n is linear, achieving query complexities independent of dimension when sampling is available, but requiring linear samples otherwise.

Contribution

It introduces a new distribution-free linearity testing algorithm with query complexity independent of dimension, and establishes lower bounds for sampling-only testing.

Findings

Query complexity is O((1/eps)log(1/eps)) with sampling access.

Testing without sampling requires Omega(n) samples.

Extension to continuous functions enables distribution-free linearity testing.

Abstract

We study the problem of testing whether a function f:R^n->R is linear (i.e., both additive and homogeneous) in the distribution-free property testing model, where the distance between functions is measured with respect to an unknown probability distribution over R. We show that, given query access to f, sampling access to the unknown distribution as well as the standard Gaussian, and eps>0, we can distinguish additive functions from functions that are eps-far from additive functions with O((1/eps)log(1/eps)) queries, independent of n. Furthermore, under the assumption that f is a continuous function, the additivity tester can be extended to a distribution-free tester for linearity using the same number of queries. On the other hand, we show that if we are only allowed to get values of f on sampled points, then any distribution-free tester requires Omega(n) samples, even if the underlying distribution is the standard Gaussian.