Sample-Efficient Algorithms for Recovering Structured Signals from Magnitude-Only Measurements

We consider the problem of recovering a signal $\mathbf{x}^* \in \mathbf{R}^n$, from magnitude-only measurements $yi = |\left\langle\mathbf{a}i,\mathbf{x}^{*\right\rangle|$} 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 $\mathbf{x}^*$ 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 $O(s^2\log n)$ with Gaussian measurements $\mathbf{a}_i$, 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 $O(s\log n)$, which is close to the information-theoretic limit. We also consider the case where the signal $\mathbf{x}^*$ 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 $O(ks\log n)$. For sufficiently large block lengths of $b=\Theta(s)$, this bound equates to $O(s\log n)$. 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.