NDLI: Fast Solution of <formula formulatype=inline><tex>$ell _{1}$</tex> </formula>-Norm Minimization Problems When the Solution May Be Sparse

Content Provider	IEEE Xplore Digital Library
Author	Donoho, D.L. Tsaig, Y.
Copyright Year	1963
Abstract	The minimum lscr₁-norm solution to an underdetermined system of linear equations y=Ax is often, remarkably, also the sparsest solution to that system. This sparsity-seeking property is of interest in signal processing and information transmission. However, general-purpose optimizers are much too slow for lscr₁ minimization in many large-scale applications.In this paper, the Homotopy method, originally proposed by Osborne et al. and Efron et al., is applied to the underdetermined lscr₁-minimization problem min parxpar₁ subject to y=Ax. Homotopy is shown to run much more rapidly than general-purpose LP solvers when sufficient sparsity is present. Indeed, the method often has the following k-step solution property: if the underlying solution has only k nonzeros, the Homotopy method reaches that solution in only k iterative steps. This k-step solution property is demonstrated for several ensembles of matrices, including incoherent matrices, uniform spherical matrices, and partial orthogonal matrices. These results imply that Homotopy may be used to rapidly decode error-correcting codes in a stylized communication system with a computational budget constraint. The approach also sheds light on the evident parallelism in results on lscr₁ minimization and orthogonal matching pursuit (OMP), and aids in explaining the inherent relations between Homotopy, least angle regression (LARS), OMP, and polytope faces pursuit.
Sponsorship	IEEE Information Theory Society
Starting Page	4789
Ending Page	4812
Page Count	24
File Size	1945389
File Format	PDF
ISSN	00189448
Volume Number	54
Issue Number	11
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-11-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Matching pursuit algorithms Equations Signal processing Optimization methods Minimization methods Large-scale systems Iterative methods Iterative decoding Error correction codes Parallel processing underdetermined systems of linear equations basis pursuit $\ell _{1}$ minimization Homotopy methods Least Angle Regression (Lars) LASSO orthogonal matching pursuit polytope faces pursuit sparse representations
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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