Hypothesis testing by convex optimization

TL;DR

This paper introduces a convex optimization-based method for multiple hypotheses testing where each hypothesis involves a convex set of parameters, providing a computationally efficient and nearly optimal testing procedure applicable to various classical detection problems.

Contribution

The paper develops a convex programming approach for pairwise hypothesis testing with convex parameter sets, offering a nearly optimal and computationally efficient solution.

Findings

The proposed test is nearly optimal under certain assumptions.

The method is applicable to classical detection problems like signal change detection.

The approach is computationally efficient due to convex programming.

Abstract

We discuss a general approach to handling "multiple hypotheses" testing in the case when a particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. With our approach, this problem reduces to testing the hypotheses pairwise. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, so that our construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection.