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