Hierarchical distributed compressive sensing for simultaneous sparse approximation and common component extraction

Simultaneous sparse approximation is a generalization of the standard sparse approximation, for simultaneously representing a set of signals using a common sparsity model. Distributed compressive sensing (DCS) framework has utilized simultaneous sparse approximation for generalizing compressive sensing to multiple signals. DCS finds the sparse representation of multiple correlated signals from compressive measurements using the common + innovation signal model. However, DCS is limited for joint recovery of a large number of signals since it requires large memory and computational time. In this paper, we propose a new hierarchical algorithm to implement the joint sparse recovery part of DCS more efficiently. The proposed approach is based on partitioning the input set and hierarchically solving for the sparse common component across these partitions. The numerical evaluation of the proposed method shows the decrease in computational time over DCS with an increase in reconstruction error. The proposed algorithm is evaluated for two different applications. In the first application, the proposed method is applied to video background extraction problem, where the background corresponds to the common sparse activity across frames. In the second application, a common network structure is extracted from dynamic functional brain connectivity networks.