Phase Retrieval Using Structured Sparsity: A Sample Efficient Algorithmic Framework

We consider the problem of recovering a signal $\mathbf{x}^* \in \mathbf{R}^n$, from magnitude-only measurements, $y_i = |\left\langle\mathbf{a}_i,\mathbf{x}^*\right\rangle|$ for $i=\{1,2,\ldots,m\}$. This is a stylized version of the classical phase retrieval problem, and is a fundamental challenge in bio-imaging systems, astronomical imaging, and speech processing. It is well known that the above problem is ill-posed, and therefore some additional assumptions on the signal and/or the measurements are necessary. In this paper, we first study the case where the underlying signal $\mathbf{x}^*$ is $s$-sparse. We develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple and be obtained via a natural combination of the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of $O(s^2 \log n)$ with Gaussian measurements $\mathbf{a}_i$, which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. Additionally, it requires no extra tuning parameters other than the signal sparsity level $s$. We then consider the case where the underlying 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 that we call 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.