On the Error Exponent of Approximate Sufficient Statistics for M-ary Hypothesis Testing

TL;DR

This paper investigates the error exponent of approximate sufficient statistics in M-ary Gaussian signal detection, proposing a reduced set of statistics via sensing matrices and analyzing their performance compared to optimal detection.

Contribution

It introduces a construction of approximate sufficient statistics using sensing matrices with specific properties and analyzes their error exponents, extending classical detection theory to reduced-dimensional settings.

Findings

Error exponents increase linearly with compression ratio N/M.

Column-normalized group Hadamard matrices yield ensemble-tight bounds.

Approximate sufficient statistics can nearly match optimal detection performance.

Abstract

Consider the problem of detecting one of M i.i.d. Gaussian signals corrupted in white Gaussian noise. Conventionally, matched filters are used for detection. We first show that the outputs of the matched filter form a set of asymptotically optimal sufficient statistics in the sense of maximizing the error exponent of detecting the true signal. In practice, however, M may be large which motivates the design and analysis of a reduced set of N statistics which we term approximate sufficient statistics. Our construction of these statistics is based on a small set of filters that project the outputs of the matched filters onto a lower-dimensional vector using a sensing matrix. We consider a sequence of sensing matrices that has the desiderata of row orthonormality and low coherence. We analyze the performance of the resulting maximum likelihood (ML) detector, which leads to an achievable bound on the error exponent based on the approximate sufficient statistics; this bound recovers the original error exponent when N = M. We compare this to a bound that we obtain by analyzing a modified form of the Reduced Dimensionality Detector (RDD) proposed by Xie, Eldar, and Goldsmith [IEEE Trans. on Inform. Th., 59(6):3858-3874, 2013]. We show that by setting the sensing matrices to be column-normalized group Hadamard matrices, the exponents derived are ensemble-tight, i.e., our analysis is tight on the exponential scale given the sensing matrices and the decoding rule. Finally, we derive some properties of the exponents, showing, in particular, that they increase linearly in the compression ratio N/M.