Recovering block-structured activations using compressive measurements

We consider the problems of detection and support recovery of a contiguous block of weak activation in a large matrix, from noisy, possibly adaptively chosen, compressive (linear) measurements. We precisely characterize the tradeoffs between the various problem dimensions, the signal strength and the number of measurements required to reliably detect and recover the support of the signal, both for passive and adaptive measurement schemes. In each case, we complement algorithmic results with information-theoretic lower bounds. Analogous to the situation in the closely related problem of noisy compressed sensing, we show that for detection neither adaptivity, nor structure reduce the minimax signal strength requirement. On the other hand we show the rather surprising result that, contrary to the situation in noisy compressed sensing, the signal strength requirement to recover the support of a contiguous block-structured signal is strongly influenced by both the signal structure and the ability to choose measurements adaptively.