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
$\mathbf {x}$
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. For these algorithms, it is known that if the sensing matrix
$\mathbf {A}$
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 \rightarrow \infty, n/m \rightarrow \kappa $
), 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
$\mathbf {A}$
is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, if the signal is drawn from a Gaussian prior.