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We study information theoretic limits of recovering an unknown $n$dimensional, complex signal vector $\mathbf{x}_\star$ with unit norm from $m$magnitude-only measurements of the form $y_i = |(\mathbf{A}\mathbf{x}_\star)_i|^2, \; i = 1,2 \dots , m$, where $\mathbf{A}$ is thesensing matrix. This is known as the Phase Retrieval problem and modelspractical imaging systems where measuring the phase of the observations isdifficult. Since in a number of applications, the sensing matrix has orthogonalcolumns, we model the sensing matrix as a subsampled Haar matrix formed bypicking $n$ columns of a uniformly random $m \times m$ unitary matrix. We studythis problem in the high dimensional asymptotic regime, where $m,n \rightarrow\infty$, while $m/n \rightarrow \delta$ with $\delta$ being a fixed number, andshow that if $m < (2-o_n(1))\cdot n$, then any estimator is asymptoticallyorthogonal to the true signal vector $\mathbf{x}_\star$. This lower bound issharp since when $m > (2+o_n(1)) \cdot n $, estimators that achieve a nontrivial asymptotic correlation with the signal vector are known from previousworks.

Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices