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. 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 $\mathbf{A}$ 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 the 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 certain deterministic orthogonal matrices such as the Hadamard-Walsh matrix, provided the signal is drawn from a Gaussian prior.
翻译：在阶段检索问题中, 一个人试图从以$y_ i = {( mathbf{A}\ mathbff{x}}}_ 美元表示感测矩阵。 这个问题流行的算法类别以传递大致信息为基础。 对于这些算法, 已知的是, 如果感测矩阵 $\mathbf{A} 美元是用一个单一随机( e. haar 分布的) 或直方矩阵的一列的子抽样抽样抽样采集的, 美元 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = orgromas = = = = = = = 或 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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