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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.Many algorithms for this problem are based on approximate message passing. Forthese algorithms, it is known that if the sensing matrix $\mathbf{A}$ isgenerated by sub-sampling $n$ columns of a uniformly random (i.e., Haardistributed) orthogonal matrix, in the high dimensional asymptotic regime ($m,n\rightarrow \infty, n/m \rightarrow \kappa$), the dynamics of the algorithm aregiven by a deterministic recursion known as the state evolution. For a specialclass of linearized message-passing algorithms, we show that the stateevolution is universal: it continues to hold even when $\mathbf{A}$ isgenerated by randomly sub-sampling columns of the Hadamard-Walsh matrix,provided the signal is drawn from a Gaussian prior.

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