Browsing by Author "DeVore, Ronald A."

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    Near Best Tree Approximation
    (2002-01-15) Baraniuk, Richard G.; DeVore, Ronald A.; Kyriazis, George; Yu, Xiang Ming; Center for Multimedia Communications (http://cmc.rice.edu/); Digital Signal Processing (http://dsp.rice.edu/)
    Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in teh appromant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [CDGO], [CDDD] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. The present paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a rpescribed approximation error. This accomplished in the case that the approximation error is measure in L2, or in the case p not equal to 2, in the Besove spaces, which is close to (but not the same as) Lp. Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.
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    A simple proof of the restricted isometry property for random matrices
    (2007-01-18) Baraniuk, Richard G.; Davenport, Mark A.; DeVore, Ronald A.; Wakin, Michael B.
    We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.