Multi Terminal Probabilistic Compressed Sensing
This work addresses multi-terminal compressed sensing for distributed sources, offering an incremental extension of AMP with theoretical analysis but limited practical validation.
The paper extends the Approximate Message Passing (AMP) algorithm to a multi-terminal setting (MAMP) for compressed sensing, showing that its behavior is characterized by a State Evolution equation and that spatial coupling of measurement matrices leads to a phase transition in the rate-distortion curve, with the low-distortion regime defined by Renyi information dimensions.
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 has been 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 conditional Renyi 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 have been done to investigate the empirical behavior of MAMP algorithm. It is observed that simulation results match very well with predictions of SE equation for reasonably large block-lengths.