Detecting positive correlations in a multivariate sample

TL;DR

This paper investigates statistical tests for detecting positive correlations in high-dimensional multivariate normal data, proposing computationally feasible methods and establishing theoretical bounds.

Contribution

It introduces near-optimal, computationally efficient tests for identifying sparse positive correlations and applies these results to bounds in random geometric graphs.

Findings

Derived a general lower bound for testing correlation matrices.

Developed near-optimal, computationally feasible tests.

Applied results to establish new bounds on clique numbers.

Abstract

We consider the problem of testing whether a correlation matrix of a multivariate normal population is the identity matrix. We focus on sparse classes of alternatives where only a few entries are nonzero and, in fact, positive. We derive a general lower bound applicable to various classes and study the performance of some near-optimal tests. We pay special attention to computational feasibility and construct near-optimal tests that can be computed efficiently. Finally, we apply our results to prove new lower bounds for the clique number of high-dimensional random geometric graphs.