On the Invariance of Recovery Algorithms for Compressed Sensing based on Expectation-Consistent Approximate Inference

For compressed sensing iterative algorithms have been introduced, which use the inherent separation of the problem into a part defined by the channel observations and a part obeying the signal statistics. Expectation-consistent approximate inference allows a flexible separation into subproblems. This paper introduces a splitting, where the channel observations are partly considered together with the signal statistics, and examines the implications of this separation. We show that the respective recovery algorithm for compressed sensing is invariant under this separation for a suitably chosen initialization.