Phase retrieval of low-rank matrices by anchored regression
Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show more »
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Publication Date:
NSF-PAR ID:
10302676
Journal Name:
Information and Inference: A Journal of the IMA
Volume:
10
Issue:
1
ISSN:
2049-8772
1. We tackle the problem of recovering a complex signal $\vx\in\mathbb{C}^n$ from quadratic measurements of the form $y_i=\vx^*\vA_i\vx$, where $\{\vA_i\}_{i=1}^m$ is a set of complex iid standard Gaussian matrices. This non-convex problem is related to the well understood phase retrieval problem where $\vA_i$ is a rank-1 positive semidefinite matrix. Here we study a general full-rank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior work either addresses the rank-1 case or focuses on real measurements. The several papers that address the full-rank complex case adopt the semidefinite relaxation approach and are thus computationally demanding. In this paper we propose a method based on the standard framework comprising a spectral initialization followed by iterative gradient descent updates. We prove that when the number of measurements exceeds the signal's length by some constant factor, a globally optimal solution can be recovered from complex quadratic measurements with high probability. Numerical experiments on simulated data corroborate our theoretical analysis.
2. In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (e.g.\ covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan $\sin \theta$ theorem is often used to bound the difference between the eigenvectors of a matrix $A$ and those of a perturbed matrix $\widetilde{A} = A + E$, in terms of $\ell_2$ norm. In this paper, we prove that when $A$ is a low-rank and incoherent matrix, the $\ell_{\infty}$ norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of $\sqrt{d_1}$ or $\sqrt{d_2}$ for left and right vectors, where $d_1$ and $d_2$ are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.
3. We study the low rank regression problem $\my = M\mx + \epsilon$, where $\mx$ and $\my$ are d1 and d2 dimensional vectors respectively. We consider the extreme high-dimensional setting where the number of observations n is less than d1+d2. Existing algorithms are designed for settings where n is typically as large as $\Rank(M)(d_1+d_2)$. This work provides an efficient algorithm which only involves two SVD, and establishes statistical guarantees on its performance. The algorithm decouples the problem by first estimating the precision matrix of the features, and then solving the matrix denoising problem. To complement the upper bound, we introduce new techniques for establishing lower bounds on the performance of any algorithm for this problem. Our preliminary experiments confirm that our algorithm often out-performs existing baselines, and is always at least competitive.