Recovering Block-structured Activations Using Compressive Measurements
This addresses a specific problem in compressed sensing for structured signals, with incremental contributions to localization theory.
The paper tackles the problem of detecting and localizing a contiguous block of weak activation in a large matrix from noisy compressive measurements, showing that adaptivity and structure significantly reduce the required signal strength for localization but not for detection, with precise tradeoffs characterized between parameters.
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.