Sample-Efficient Algorithms for Recovering Structured Signals from Magnitude-Only Measurements
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We consider the problem of recovering a signal x^* ∈R^n, from magnitude-only measurements y_i = |〈a_i,x^*〉| for i=[m]. Also called the phase retrieval, this is a fundamental challenge in bio-,astronomical imaging and speech processing. The problem above is ill-posed; additional assumptions on the signal and/or the measurements are necessary. In this paper we first study the case where the signal x^* is s-sparse. We develop a novel algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple; it combines the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that CoPRAM achieves a sample complexity of O(s^2 n) with Gaussian measurements a_i, matching the best known existing results; moreover, it demonstrates linear convergence in theory and practice. Additionally, it requires no extra tuning parameters other than signal sparsity s and is robust to noise. When the sorted coefficients of the sparse signal exhibit a power law decay, we show that CoPRAM achieves a sample complexity of O(s n), which is close to the information-theoretic limit. We also consider the case where the signal x^* arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of b and block sparsity k=s/b. For this problem, we design a recovery algorithm Block CoPRAM that further reduces the sample complexity to O(ks n). For sufficiently large block lengths of b=Θ(s), this bound equates to O(s n). To our knowledge, this constitutes the first end-to-end algorithm for phase retrieval where the Gaussian sample complexity has a sub-quadratic dependence on the signal sparsity level.
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