The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery [CRT06] . Informally, an m × n matrix satisfies RIP of order k in the ℓ p norm if Ax p ≈ x p for any vector x that is k-sparse, i.e., that has at most k non-zeros. The minimal number of rows m necessary for the property to hold has been extensively investigated, and tight bounds are known. Motivated by signal processing models, a recent work of Baraniuk et al [BCDH10] has generalized this notion to the case where the support of x must belong to a given model, i.e., a given family of supports. This more general notion is much less understood, especially for norms other than ℓ 2 . In this paper we present tight bounds for the model-based RIP property in the ℓ 1 norm. Our bounds hold for the two most frequently investigated models: tree-sparsity and block-sparsity. We also show implications of our results to sparse recovery problems.
The recent work of REF on model-based RIP-1 matrices (model expanders) also used a probabilistic construction.
On Model-Based RIP-1 Matrices
ArXiv
ArXiv
Mathematics Computer Science