Recovering Block-structured Activations Using Compressive Measurements

We consider the problem of detection and localization of a small 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. However, contrary to results in compressed sensing, where it has been shown that neither adaptivity nor contiguous structure help much, we show that in our problem the magnitude of the weakest signals one can reliably localize is strongly influenced by both structure and the ability to choose measurements adaptively. We derive tight upper and lower bounds for the detection and estimation problems, under both adaptive and non-adaptive measurement schemes. 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.