Robust Matrix Decomposition with Outliers

TL;DR

This paper investigates conditions for accurately decomposing a matrix into low-rank and sparse components using convex optimization, allowing for a higher number of outliers without assuming random outlier patterns.

Contribution

It provides improved recovery guarantees for matrix decomposition with outliers, without relying on randomness assumptions about outlier locations.

Findings

Stronger theoretical guarantees for exact recovery.

Allows more outliers than previous methods.

Does not assume random outlier distribution.

Abstract

Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identification, latent variable graphical modeling, and principal components analysis. We study conditions under which recovering such a decomposition is possible via a combination of $\ell_1$ norm and trace norm minimization. We are specifically interested in the question of how many outliers are allowed so that convex programming can still achieve accurate recovery, and we obtain stronger recovery guarantees than previous studies. Moreover, we do not assume that the spatial pattern of outliers is random, which stands in contrast to related analyses under such assumptions via matrix completion.