Sharp Recovery Bounds for Convex Demixing, with Applications

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 secon...

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Vydáno v:Foundations of computational mathematics Ročník 14; číslo 3; s. 503 - 567
Hlavní autoři: McCoy, Michael B., Tropp, Joel A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.06.2014
Springer Nature B.V
Témata:
Applications of Mathematics
Complexity
Computer Science
Decoding
Demixing
Economics
Empirical analysis
Foundations
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical models
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
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Shrnutí: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-014-9191-2