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**1 - 2**of**2**### Fast Algorithms for Robust PCA via Gradient Descent

"... Abstract We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observ ..."

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Abstract We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with r denoting rank and d dimension, we reduce the complexity from O(r 2 d 2 log(1/ε)) to O(rd 2 log(1/ε)) -a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than O(r 4 d log d log(1/ε)). Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where r is small compared to d, it also allows for near-linear-in-d run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

### A Geometric Analysis of Phase Retrieval

"... Abstract Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = |a * k x| for k = 1, . . . , m, is it possible to recover x ∈ C n (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in vario ..."

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Abstract Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = |a * k x| for k = 1, . . . , m, is it possible to recover x ∈ C n (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors a k 's are generic (i.i.d. complex Gaussian) and the number of measurements is large enough (m ≥ Cn log 3 n), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal x, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.