Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices

In the phase retrieval problem one seeks to recover an unknown $n$ dimensional signal vector $\mathbf{x}$ from $m$ measurements of the form $y_i = |(\mathbf{A} \mathbf{x})_i|$ where $\mathbf{A}$ denotes the sensing matrix. A popular class of algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix $\mathbf{A}$ is generated by sub-sampling $n$ columns of a uniformly random (i.e. Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime ($m,n \rightarrow \infty, n/m \rightarrow \kappa$), the dynamics of the algorithm are given by a deterministic recursion known as the state evolution. For the special class of linearized message passing algorithms, we show that the state evolution is universal: it continues to hold even when $\mathbf{A}$ is generated by randomly sub-sampling columns of certain deterministic orthogonal matrices such as the Hadamard-Walsh matrix, provided the signal is drawn from a Gaussian prior

On the universality of noiseless linear estimation with respect to the measurement matrix

International audienceIn a noiseless linear estimation problem, one aims to reconstruct a vector x * from the knowledge of its linear projections y = Φx *. There have been many theoretical works concentrating on the case where the matrix Φ is a random i.i.d. one, but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case. Here we revisit this problematic through the prism of development of message passing methods, and consider not only the universality of the 1 transition, as previously addressed, but also the one of the optimal Bayesian reconstruction. We observed that the universality extends to the Bayes-optimal minimum mean-squared (MMSE) error, and to a range of structured matrices