Universality of Linearized Message Passing for Phase Retrieval With Structured Sensing Matrices - Dudeja, Rishabh; Bakhshizadeh, Milad (IEEE transactions on information theory, 2022)
In the phase retrieval problem one seeks to recover an unknown n dimensional signal vector x from m measurements of the form y_{i} = |(A x)_{i}| , where A denotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix 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 → ∞, 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 when A is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, if the signal is drawn from a Gaussian prior.