Abstract
AbstractWe consider inexact linear equations y â Ίx where y is a given vector in ân, Ί is a given n Ă m matrix, and we wish to find x0,Ï” as sparse as possible while obeying ây â Ίx0,Ï”â2 †ϔ. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the đ1âminimization problem is convex and is considered tractable. We show that for most Ί, if the optimally sparse approximation x0,Ï” is sufficiently sparse, then the solution x1,Ï” of the đ1âminimization problem is a good approximation to x0,Ï”.We suppose that the columns of Ί are normalized to the unit đ2ânorm, and we place uniform measure on such Ί. We study the underdetermined case where m ⌠Ïn and Ï > 1, and prove the existence of Ï = Ï(Ï) > 0 and C = C(Ï, Ï) so that for large n and for all Ί's except a negligible fraction, the following approximate sparse solution property of Ί holds: for every y having an approximation ây â Ίx0â2 †ϔ by a coefficient vector x0 â âm with fewer than Ï Â· n nonzeros, This has two implications. First, for most Ί, whenever the combinatorial optimization result x0,Ï” would be very sparse, x1,Ï” is a good approximation to x0,Ï”. Second, suppose we are given noisy data obeying y = Ίx0 + z where the unknown x0 is known to be sparse and the noise âzâ2 †ϔ. For most Ί, noiseâtolerant đ1âminimization will stably recover x0 from y in the presence of noise z.We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments.Proof techniques include the use of almostâspherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices. © 2006 Wiley Periodicals, Inc.
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