Recovering Block-structured Activations Using Compressive Measurements

TL;DR

This paper investigates the detection and localization of weak, contiguous block activations in large matrices using limited noisy measurements, highlighting the roles of structure and adaptivity in measurement strategies.

Contribution

It demonstrates that adaptivity and contiguous structure significantly improve localization, providing precise tradeoffs and theoretical bounds for detection and localization tasks.

Findings

Adaptive measurements enhance localization accuracy.

Contiguous structure aids in reliable localization.

Theoretical bounds establish measurement requirements.

Abstract

We consider the problems of detection and localization of a contiguous block of weak activation in a large matrix, from a small number of noisy, possibly adaptive, compressive (linear) measurements. This is closely related to the problem of compressed sensing, where the task is to estimate a sparse vector using a small number of linear measurements. Contrary to results in compressed sensing, where it has been shown that neither adaptivity nor contiguous structure help much, we show that for reliable localization the magnitude of the weakest signals is strongly influenced by both structure and the ability to choose measurements adaptively while for detection neither adaptivity nor structure reduce the requirement on the magnitude of the signal. We characterize the precise tradeoffs between the various problem parameters, the signal strength and the number of measurements required to reliably detect and localize the block of activation. The sufficient conditions are complemented with information theoretic lower bounds.