期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 52, 期 3, 页码 1030-1051出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2005.864420
关键词
algorithms; approximation methods; basis pursuit; convex program; linear regression; optimization methods; orthogonal matching pursuit; sparse representations
This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.
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