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A simple proof of the restricted isometry property for random matrices. (English) Zbl 1177.15015
Summary: We give a simple technique for verifying the restricted isometry property (as introduced by E. Candès and T. Tao [IEEE Trans. Inf. Theory 51, No. 12, 4203–4215 (2005)]) 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 [W. B. Johnson and J. Lindenstrauss, Contemp. Math. 26, 189–206 (1984; Zbl 0539.46017)]; 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 [B. S. Kashin, Izv. Akad. Nauk SSSR, Ser. Mat. 41, 334–351 (1977; Zbl 0354.46021)] (and their improvements due to Gluskin [A. Garnaev and E. D. Gluskin, Sov. Math., Dokl. 30, 200–204 (1984); translation from Dokl. Akad. Nauk SSSR 277, 1048–1052 (1984; Zbl 0588.41022)]) 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.

MSC:
15A22 Matrix pencils
60F10 Large deviations
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94A20 Sampling theory in information and communication theory
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