On combinatorial testing problems

TL;DR

This paper investigates hypothesis testing problems involving Gaussian vectors, focusing on the detectability of contaminated subsets within a class of sets, and establishes conditions for when testing is feasible or impossible.

Contribution

It provides general conditions that determine the possibility of successful testing based on the combinatorial and geometric structure of the set class.

Findings

Testing is possible under certain structural conditions.

Testing becomes impossible with small risk under other conditions.

Theoretical bounds are illustrated with various examples.

Abstract

We study a class of hypothesis testing problems in which, upon observing the realization of an $n$-dimensional Gaussian vector, one has to decide whether the vector was drawn from a standard normal distribution or, alternatively, whether there is a subset of the components belonging to a certain given class of sets whose elements have been ``contaminated,'' that is, have a mean different from zero. We establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk. The combinatorial and geometric structure of the class of sets is shown to play a crucial role. The bounds are illustrated on various examples.