Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices
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We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x_ with unit norm from m magnitude-only measurements of the form y_i = m|(Ax_)_i|^2, i = 1,2 ... , m, where A is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking n columns of a uniformly random m × m unitary matrix. We study this problem in the high dimensional asymptotic regime, where m,n →∞, while m/n →δ with δ being a fixed number, and show that if m < (2-o_n(1))· n, then any estimator is asymptotically orthogonal to the true signal vector x_. This lower bound is sharp since when m > (2+o_n(1)) · n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.
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