A Simple Proof of the Restricted Isometry Property for Random Matrices
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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.
KeywordsCompressed sensing Sampling Random matrices Concentration inequalities
Mathematics Subject Classification (2000)15N2 15A52 60F10 94A12 94A20
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- 1.Achlioptas, D.: Database-friendly random projections. In: Proc. ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, pp. 274–281, 2001 Google Scholar
- 5.Cohen, A., Dahmen, W., DeVore, R.: Compressed sensing and best k-term approximation. Preprint (2006) Google Scholar
- 6.Cohen, A., Dahmen, W., DeVore, R.: Near optimal approximation of arbitrary signals from highly incomplete measurements. Preprint (2007) Google Scholar
- 7.Dasgupta, S., Gupta, A.: An elementary proof of the Johnson–Lindenstrauss lemma. Tech. Report Technical report 99-006, U.C. Berkeley (March, 1999) Google Scholar
- 10.Gilbert, A., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse Fourier representations via sampling, 2005. In: ACM Symp. on Theoretical Computer Science, 2002 Google Scholar
- 12.Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: Symp. on Theory of Computing, pp. 604–613, 1998 Google Scholar
- 13.Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conf. in Modern Analysis and Probability, pp. 189–206, 1984 Google Scholar
- 14.Kashin, B.: The widths of certain finite dimensional sets and classes of smooth functions. Izvestia (41), 334–351 (1977) Google Scholar
- 19.Mendelson, S., Pajor, A., Tomczack-Jaegermann, N.: Reconstruction and subgaussian operators in asymptotic geometric analysis. Preprint (2006) Google Scholar