Low-rank matrix recovery from errors and erasures

By Yudong Chen, Ali Jalali, Sujay Sanghavi, Constantine Caramanis. 2011

171 citations recorded by the source corpus.

This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when a (natural) recently proposed method, based on convex optimization, succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. On the one hand, corollaries obtained by specializing this one single result in different ways recovers (upto poly-log factors) all the existing works in matrix completion, and sparse and low-rank matrix recovery. On the other hand, our results also provide the first guarantees for (a) deterministic matrix completion, and (b) recovery when we observe a vanishing fraction of entries of a corrupted matrix.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

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Citations in the corpus

Data Poisoning Attacks on Factorization-Based Collaborative Filtering (2016, background): “In terms of collaborative filtering/matrix completion, there is another line of established research that focuses on robust matrix completion, in which a small portion of elements or rows in the underlying low-rank matrix is assumed to be arbitrarily perturbed [18, 19, 20, 21].”
Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees (2015, methodology): “One line of work concerns the setting where the non-zero entries of S∗ are randomly located [19, 22], whereas another line of work focuses on deterministic models [20, 35, 1].”
Cognitive Internet of Things: A New Paradigm Beyond Connection (2014, background): “sparse components and , the problem can be further formulated as [38]”
Robust matrix completion (2014, background): “They proved exact recovery of the low-rank component under an incoherence condition on L 0 and some additional assumptions on the number of corrupted observations.”
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview (2018, background): “robust PCA [159] low-rank projection + thresholding [25] [7, 8, 213, 214] GD + thresholding [64, 215] alternating minimization [216] spectrum estimation — projected GD [217] [218, 219]”