On joint recovery of sparse signals with common supports

This work is motivated by a distributed compressed sensing (DCS) scenario where multiple sensors independently perform compressed sensing and the sparse signals share a common support. The heterogeneous case is considered where the numbers of measurements and the noise levels at different sensors may be different. To analyse the performance, we focus on a probability model for sparse signals and use the state evolution tool developed for the approximate message passing (AMP) technique. In the noise free case, we are able to quantify the asymptotic rate region for exact recovery. The rate region has a shape that is significantly different from that by information theoretic analysis and provides a guidance for resource allocation in practice. It shows that an equal allocation of the number of measurements across sensors is strictly suboptimal. Finally, we also study the effect of the correlation among nonzero components from different sparse signals, which appears in many practical scenarios.