Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
Robust recovery of signals from a structured union of subspaces
IEEE Transactions on Information Theory
Model-based compressive sensing
IEEE Transactions on Information Theory
Lower bounds for sparse recovery
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The Gelfand widths of lp-balls for 0
Journal of Complexity
K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance
Proceedings of the forty-third annual ACM symposium on Theory of computing
IEEE Transactions on Information Theory
Bayesian tree-structured image modeling using wavelet-domain hidden Markov models
IEEE Transactions on Image Processing
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The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery [5]. Informally, an m ×n matrix satisfies RIP of order k in the ℓp 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 [3] 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 ℓ2. In this paper we present tight bounds for the model-based RIP property in the ℓ1 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.