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

We consider the problem of recovering a signal <inline-formula> <tex-math notation="LaTeX">$ \mathbf {x^{*}}\in \mathbb {R}^{n}$ </tex-math></inline-formula>, from magnitude-only measurements, <inline-formula> <tex-math notation="LaTeX">$y_{i} = \left |{ \left \langle{ { \mathbf {a}_{i}}, \mathbf {x^{*}} }\right \rangle }\right | $ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$i=\{1,2,\ldots ,m\}$ </tex-math></inline-formula>. This is a stylized version of the classical <italic>phase retrieval problem</italic>, and is a fundamental challenge in nano- and 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 consider the case where the underlying signal <inline-formula> <tex-math notation="LaTeX">$ \mathbf {x^{*}}$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-sparse. For this case, we develop a novel recovery algorithm that we call <italic>Compressive Phase Retrieval with Alternating Minimization</italic>, or <italic>CoPRAM</italic>. Our algorithm is simple and is 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 <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({{s^{2} \log n}}\right )$ </tex-math></inline-formula> with Gaussian measurements <inline-formula> <tex-math notation="LaTeX">$ { \mathbf {a}_{i}}$ </tex-math></inline-formula>, which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. An appealing feature of our algorithm is that it requires no extra tuning parameters other than the signal sparsity level <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>. Moreover, we show that our algorithm is robust to noise. The quadratic dependence of sample complexity on the sparsity level is sub-optimal, and we demonstrate how to alleviate this via <italic>additional</italic> assumptions beyond sparsity. First, we study the (practically) relevant case where the sorted coefficients of the underlying sparse signal exhibit a power law decay. In this scenario, we show that the CoPRAM algorithm achieves a sample complexity of <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({{s \log n}}\right )$ </tex-math></inline-formula>, which is close to the information-theoretic limit. We then consider the case where the underlying signal <inline-formula> <tex-math notation="LaTeX">$ \mathbf {x^{*}}$ </tex-math></inline-formula> arises from <italic>structured</italic> sparsity models. We specifically examine the case of <italic>block-sparse</italic> signals with uniform block size of <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> and block sparsity <inline-formula> <tex-math notation="LaTeX">$k=s/b$ </tex-math></inline-formula>. For this problem, we design a recovery algorithm that we call <italic>Block CoPRAM</italic> that further reduces the sample complexity to <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({{ks \log n}}\right )$ </tex-math></inline-formula>. For sufficiently large block lengths of <inline-formula> <tex-math notation="LaTeX">$b=\Theta (s)$ </tex-math></inline-formula>, this bound equates to <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({{s \log n}}\right )$ </tex-math></inline-formula>. To our knowledge, our approach constitutes the first family of <italic>linearly convergent</italic> algorithms for signal recovery from magnitude-only Gaussian measurements that exhibit a sub-quadratic dependence on the signal sparsity level.