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Signal recovery from unitarily invariant measurements is investigated in thispaper. A message-passing algorithm is formulated on the basis of expectationpropagation (EP). A rigorous analysis is presented for the dynamics of thealgorithm in the large system limit, where both input and output dimensionstend to infinity while the compression rate is kept constant. The main resultis the justification of state evolution (SE) equations conjectured by Ma andPing. This result implies that the EP-based algorithm achieves theBayes-optimal performance that was originally derived via a non-rigorous toolin statistical physics and proved partially in a recent paper, when thecompression rate is larger than a threshold. The proof is based on an extensionof a conventional conditioning technique for the standard Gaussian matrix tothe case of the Haar matrix.

Rigorous Dynamics of Expectation-Propagation-Based Signal Recovery from Unitarily Invariant Measurements