Compressed Hypothesis Testing: To Mix or Not to Mix?

TL;DR

This paper explores how mixed observations can improve the efficiency of hypothesis testing for identifying anomalous variables, offering theoretical insights, optimal sensing strategies, and practical algorithms.

Contribution

It introduces the use of mixed observations in hypothesis testing, characterizes error exponents, and develops algorithms for large-scale problems.

Findings

Mixed observations can improve error exponents over separate sampling.

Optimal sensing vectors can be explicitly constructed for Gaussian variables.

Proposed algorithms enable efficient large-scale hypothesis testing.

Abstract

In this paper, we study the problem of determining $k$ anomalous random variables that have different probability distributions from the rest $(n-k)$ random variables. Instead of sampling each individual random variable separately as in the conventional hypothesis testing, we propose to perform hypothesis testing using mixed observations that are functions of multiple random variables. We characterize the error exponents for correctly identifying the $k$ anomalous random variables under fixed time-invariant mixed observations, random time-varying mixed observations, and deterministic time-varying mixed observations. For our error exponent characterization, we introduce the notions of inner conditional Chernoff information and outer conditional Chernoff information. It is demonstrated that mixed observations can strictly improve the error exponents of hypothesis testing, over separate observations of individual random variables. We further characterize the optimal sensing vector maximizing the error exponents, which leads to explicit constructions of the optimal mixed observations in special cases of hypothesis testing for Gaussian random variables. These results show that mixed observations of random variables can reduce the number of required samples in hypothesis testing applications. In order to solve large-scale hypothesis testing problems, we also propose efficient algorithms - LASSO based and message passing based hypothesis testing algorithms.