Algorithms and Theory for Robust PCA and Phase Retrieval

In this dissertation, we investigate two problems, both of which require the recovery of unknowns from measurements that are potentially corrupted by outliers. The first part focuses on the problem of \emph{robust principal component analysis} (PCA), which aims to recover an unknown low-rank matrix from a corrupted and partially-observed matrix. The robust PCA problem, originally nonconvex itself, has been solved via a convex relaxation based approach \emph{principal component pursuit} (PCP) in the literature. However, previous works assume that the sparse errors uniformly spread over the entire matrix and characterize the condition under which PCP guarantees exact recovery. We generalize these results by allowing non-uniform error corruptions over the low-rank matrix and characterize the conditions on the error corruption probability of each individual entry based on the local coherence of the low-rank matrix, under which correct recovery can be guaranteed by PCP. Our results yield new insights on the graph clustering problem beyond the relevant literature. The second part of the thesis studies the phase retrieval problem, which requires recovering an unknown vector from only its magnitude measurements. Differently from the first part, we solve this problem directly via optimizing nonconvex objectives. As the nonconvex objective is often constructed in such a way that the true vector is its global optimizer, the difficulty here is to design algorithms to find the global optimizer efficiently and provably. In order to solve this problem, we propose a gradient-like algorithm named reshaped Wirtinger flow (RWF). For random Gaussian measurements, we show that RWF enjoys linear convergence to a global optimizer as long as the number of measurements is on the order of the dimension of the unknown vector. This achieves the best possible sample complexity as well as the state-of-the-art computational efficiency. Moreover, we study the phase retrieval problem when the measurements are corrupted by adversarial outliers, which models situations with missing data or sensor failures. In order to resist possible observation outliers in an oblivious manner, we propose a novel median truncation approach to modify the nonconvex approach in both the initialization and the gradient descent steps. We apply the median truncation approach to the Poisson loss and the reshaped quadratic loss respectively, and obtain two algorithms \emph{median-TWF} and \emph{median-RWF}. We show that both algorithms recover the signal from a near-optimal number of independent Gaussian measurements, even when a constant fraction of measurements is corrupted. We further show that both algorithms are stable when measurements are corrupted by both sparse arbitrary outliers and dense bounded noises. We establish our results on the performance guarantee via the development of non-trivial concentration measures of the median-related quantities, which can be of independent interest

A Deterministic Theory for Exact Non-Convex Phase Retrieval

In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the convergence of the WF iterates to a true solution is attained geometrically under a single condition on the lifted forward model. As a result, a deterministic relationship between the accuracy of spectral initialization and the validity of {the regularity condition} is derived. In particular, we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant. This culminates into a sufficient condition that is equivalent to a restricted isometry-type property over rank-1, positive semi-definite matrices, and amounts to a less stringent requirement on the lifted forward model than those of prominent low-rank-matrix-recovery methods in the literature. We characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal-to-noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin