Multi terminal probabilistic compressed sensing

In this paper, the `Approximate Message Passing' (AMP) algorithm, initially developed for compressed sensing of signals under i.i.d. Gaussian measurement matrices, has been extended to a multi-terminal setting (MAMP algorithm). It has been shown that similar to its single-terminal counterpart, the behavior of MAMP algorithm is fully characterized by a `State Evolution' (SE) equation for large block-lengths. This equation is used to obtain the rate-distortion curve of a multi-terminal memoryless source. It is observed that by spatially coupling the measurement matrices, the rate-distortion curve of MAMP algorithm undergoes a phase transition, where the measurement rate region corresponding to a low-distortion (approximately zero distortion) regime is fully characterized by the joint and the conditional Rényi information dimension (RID) of the multi-terminal source. This measurement rate region is very similar to the rate region of the Slepian-Wolf distributed source coding problem where the RID plays a role similar to the discrete entropy. Simulations are done to investigate the empirical behavior of MAMP algorithm. It is observed that simulation results match very well with the predictions of SE equation for reasonably large block-lengths.