The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery. 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 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.
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