Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis |
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Authors: | Shahar Mendelson Alain Pajor Nicole Tomczak-Jaegermann |
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Affiliation: | (1) Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT, 0200, Australia;(2) Department of Mathematics, Technion, I.I.T, Haifa, 32000, Israel;(3) Laboratoire d’Analyse et Mathématiques Appliquées, Université de Marne-la-Vallée, 5 boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallee, Cedex 2, France;(4) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1 |
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Abstract: | We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L 2 sphere. The estimates are given in terms of the γ 2 functional with respect to the ψ 2 metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006 |
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Keywords: | Approximate reconstruction exact reconstruction empirical processes subgaussian operators asymptotic geometric analysis |
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