Testing Junta Truncation

TL;DR

This paper investigates the statistical challenge of detecting whether a distribution over the Boolean hypercube is uniform or truncated by a junta, providing bounds on the sample complexity needed for this task.

Contribution

It establishes upper and lower bounds on the sample complexity for testing junta truncation, highlighting the necessity of learning relevant variables.

Findings

Sample complexity depends on junta size and combinatorial factors.

Testing junta truncation requires identifying relevant variables.

Sample bounds are tight up to constant factors.

Abstract

We consider the basic statistical problem of detecting truncation of the uniform distribution on the Boolean hypercube by juntas. More concretely, we give upper and lower bounds on the problem of distinguishing between i.i.d. sample access to either (a) the uniform distribution over $\{0,1\}^n$, or (b) the uniform distribution over $\{0,1\}^n$ conditioned on the satisfying assignments of a $k$-junta $f: \{0,1\}^n\to\{0,1\}$. We show that (up to constant factors) $\min\{2^k + \log{n\choose k}, {2^{k/2}\log^{1/2}{n\choose k}}\}$ samples suffice for this task and also show that a $\log{n\choose k}$ dependence on sample complexity is unavoidable. Our results suggest that testing junta truncation requires learning the set of relevant variables of the junta.