文件名称：A Simple Proof of the Restricted Isometry Property
文件大小：181KB
文件格式：PDF
更新时间：2012-10-30 15:09:06
Restricted Isometry Property
We give a simple technique for verifying the Restricted Isometry Property (as introduced by Cand`es 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, (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.