论文标题
线性化消息的通用性通过结构化感应矩阵进行相检索
Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices
论文作者
论文摘要
在阶段检索问题中，人们试图从$ m $ y_i = |（\ mathbf {a} \ mathbf {x}）的表格中恢复未知的$ n $ dim二维信号向量$ \ mathbf {x} $。此问题的许多算法基于近似消息传递。对于这些算法，众所周知，如果传感矩阵$ \ mathbf {a} $是通过均匀随机的子采样$ n $列（即HAAR分布式）正交矩阵生成算法由称为状态进化的确定性递归给出。对于一类线性化消息的算法，我们表明状态进化是通用的：即使在hadamard-walsh矩阵的随机子采样柱中生成了$ \ mathbf {a} $，但前提是从高斯先验中得出信号。
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 ($m,n \rightarrow \infty, n/m \rightarrow κ$), 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 $\mathbf{A}$ is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, provided the signal is drawn from a Gaussian prior.
