Low-rank matrix recovery from errors and erasures

Abstract

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.

Convex graph invariant relaxations for graph edit distance

“…From the first three conditions, one can conclude that the pair (A, E * ) is an optimal solution of (P ) by a direct application of the KKT conditions. Uniqueness can be established by standard arguments regarding transverse intersections of the subspace Ω and the invariant spaces of G; see [6,Proposition 2] and [8,Lemma 6]. Conditions 1, 2, and 3 of this lemma essentially require that the subdifferential at a matrix specifying the edits with respect to the 1 norm has a nonempty intersection with the relative interior of the normal cone at an adjacency matrix representing G with respect to the Schur-Horn orbitope.…”

Improved graph clustering

“…where a fraction of the elements of the low-rank matrix are possibly arbitrarily modified, while others are untouched). Sparse and low-rank matrix decomposition using convex optimization was initiated by [24,25]; follow-up works [26,27] have the current state-of-the-art guarantees on this problem, and [28] applies it directly to graph clustering.…”

Dynamic Mode Decomposition for Robust PCA with Applications to Foreground/Background Subtraction in Video Streams and Multi-Resolution Analysis

“…Recovering low-rank and sparse matrices from incomplete or even corrupted observations is a common problem in many application areas, including statistics [1,9,51], bioinformatics [37], machine learning [28,47,49,52], computer vision [5,7,42,43,58], and signal and image processing [27,30,38]. In these areas, data often have high dimensionality, such as digital photographs and surveillance videos, which makes inference, learning, and recognition infeasible due to the "curse of dimensionality.…”