Sharp Recovery Bounds for Convex Demixing, with Applications

• Published in: Foundations of computational mathematics Vol. 14; no. 3; pp. 503 - 567
• Main Authors: McCoy, Michael B., Tropp, Joel A.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2014; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Complexity; Computer Science; Convex functions; Decoding; Demixing; Economics; Empirical analysis; Foundations; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical models; Mathematical optimization; Mathematical research; Mathematics; Mathematics and Statistics; Matrix; Matrix Theory; Numerical Analysis; Optimization; Signal processing; Statistics; Integral geometry; 60D05; Demixing; Sparsity; Convex optimization; 52B55; 52A22 (primary); 94B75 (secondary)
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-014-9191-2

Demixing refers to the challenge of identifying two structured signals given only the sum of the two signals and prior information about their structures. Examples include the problem of separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis, and the problem of decomposing an observed matrix into a low-rank matrix plus a sparse matrix. This paper describes and analyzes a framework, based on convex optimization, for solving these demixing problems, and many others. This work introduces a randomized signal model that ensures that the two structures are incoherent, i.e., generically oriented. For an observation from this model, this approach identifies a summary statistic that reflects the complexity of a particular signal. The difficulty of separating two structured, incoherent signals depends only on the total complexity of the two structures. Some applications include (1) demixing two signals that are sparse in mutually incoherent bases, (2) decoding spread-spectrum transmissions in the presence of impulsive errors, and (3) removing sparse corruptions from a low-rank matrix. In each case, the theoretical analysis of the convex demixing method closely matches its empirical behavior.