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

In the phase retrieval problem one seeks to recover an unknownndimensional signal vectorxfrommmeasurements of the formyᵢ = |(A x)ᵢ|, whereAdenotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrixAis generated by sub-samplingncolumns of a uniformly random (i.e., Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime (m,n → ∞, n/m → κ), the dynamics of the algorithm are given by a deterministic recursion known as the state evolution. For a special class of linearized message-passing algorithms, we show that the state evolution is universal: it continues to hold even whenAis generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, if the signal is drawn from a Gaussian prior.