Hypothesis Testing under Communication Constraints Theory and an Application in IoT

Abstract

Applications in the Internet of Things (IoT) often demand enabling low-compute devices to perform distributed inference and testing by communicating over a low bandwidth link. This gives rise to a plethora of new problems which may broadly be termed resource-constrained statistical inference problems. In this thesis, we consider two such problems. In the first part of the thesis, we study the following distributed hypothesis testing problem. Two parties observing sequences of uniformly distributed bits want to determine if their bits were generated independently or not. To that end, the first party communicates to the second. A simple communication scheme involves taking as few sample bits as determined by the sample complexity of independence testing and sending it to the second party. But is there a scheme that uses fewer bits of communication than the sample complexity, perhaps by observing more sample bits? We show that the answer to this question is in the affirmative. More generally, for any given joint distribution, we present a distributed independence test that uses linear correlation between functions of the observed random variables. Furthermore, we provide lower bounds for the general setting that use hypercontractivity and reverse hypercontractivity to obtain a measure change bound between the joint and the independent distributions. The resulting bounds are tight for both a binary symmetric source and a Gaussian symmetric source. The proposed scheme is then extended to handle high dimensional correlation testing with interactive communication, wherein one party observes a Gaussian vector X and the other party observes a jointly Gaussian scalar Y, and we seek to test if the norm of the vector of correlation between X and Y exceeds a given value or is it 0. We provide corresponding lower bounds to establish the optimality of the proposed scheme. Furthermore...