Robust and Compressive Matrix Decompositions

• Activity Number: 451
• Type: Topic Contributed
• Date/Time: Wednesday, August 1, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): John Wright*+
• Keywords: Low-rank recovery ; Matrix decomposition ; Sparsity ; Convex optimization

Abstract

We consider the problem of recovering a target matrix that is a superposition of a low-rank structure component and sparse error, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant low-rank recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers low-rank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of decomposing superpositions of structured matrices.