Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework

TL;DR

This paper establishes fundamental limits on support recovery in probabilistic models using an information-theoretic approach, providing sharp bounds and thresholds for various models like linear and group testing.

Contribution

It introduces a unified information-theoretic framework for support recovery, deriving general bounds and sharp thresholds that improve upon previous results.

Findings

Matching scaling laws for measurement requirements

Sharp thresholds with constant factors identified

Conditions for both success and failure of recovery

Abstract

The support recovery problem consists of determining a sparse subset of a set of variables that is relevant in generating a set of observations, and arises in a diverse range of settings such as compressive sensing, and subset selection in regression, and group testing. In this paper, we take a unified approach to support recovery problems, considering general probabilistic models relating a sparse data vector to an observation vector. We study the information-theoretic limits of both exact and partial support recovery, taking a novel approach motivated by thresholding techniques in channel coding. We provide general achievability and converse bounds characterizing the trade-off between the error probability and number of measurements, and we specialize these to the linear, 1-bit, and group testing models. In several cases, our bounds not only provide matching scaling laws in the necessary and sufficient number of measurements, but also sharp thresholds with matching constant factors. Our approach has several advantages over previous approaches: For the achievability part, we obtain sharp thresholds under broader scalings of the sparsity level and other parameters (e.g., signal-to-noise ratio) compared to several previous works, and for the converse part, we not only provide conditions under which the error probability fails to vanish, but also conditions under which it tends to one.