Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices
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In the phase retrieval problem one seeks to recover an unknown n dimensional signal vector 𝐱 from m measurements of the form y_i = |(𝐀𝐱)_i| where 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 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 the 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 certain deterministic orthogonal matrices such as the Hadamard-Walsh matrix, provided the signal is drawn from a Gaussian prior.
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