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 notation="LaTeX">$\mathbf {x}$ from notation="LaTeX">$m$ measurements of form notation="LaTeX">$y_{i} = |(\mathbf {A} \mathbf {x})_{i}|$ , where {A}$ denotes sensing matrix. Many algorithms for this are based on approximate message passing. For these algorithms, it is known that if matrix generated by sub-sampling columns a uniformly random (i.e., Haar distributed) orthogonal matrix, in high asymptotic regime ( notation="LaTeX">$m,n \rightarrow \infty, n/m \kappa $ ), dynamics algorithm given deterministic recursion as state evolution. special class linearized message-passing we show evolution universal: continues hold even when randomly Hadamard-Walsh drawn Gaussian prior.