Abstract
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 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 [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.
The full version of this paper is available at http://arxiv.org/abs/1304.3604
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References
Open problems in data streams, property testing, and related topics (2011), http://people.cs.umass.edu/~mcgregor/papers/11-openproblems.pdf
Ba, K.D., Indyk, P., Price, E., Woodruff, D.P.: Lower Bounds for Sparse Recovery. In: SODA, pp. 1190–1197 (2010)
Baraniuk, R.G., Cevher, V., Duarte, M.F., Hegde, C.: Model-based compressive sensing. IEEE Transactions on Information Theory 56(4), 1982–2001 (2010)
Berinde, R., Gilbert, A.C., Indyk, P., Karloff, H.J., Strauss, M.J.: Combining geometry and combinatorics: A unified approach to sparse signal recovery. Allerton (2008)
Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59(8), 1207–1223 (2006)
Donoho, D.L.: Compressed sensing. IEEE Transactions on Information Theory 52(4), 1289–1306 (2006)
Eldar, Y., Mishali, M.: Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inform. Theory 55(11), 5302–5316 (2009)
Foucart, S., Pajor, A., Rauhut, H., Ullrich, T.: The Gelfand widths of lp-balls for 0 p ≤ 1. J. Complex. 26(6), 629–640 (2010)
Garnaev, A.Y., Gluskin, E.D.: On widths of the Euclidean ball. Sov. Math. Dokl. 30, 200–204 (1984)
Gilbert, A., Indyk, P.: Sparse recovery using sparse matrices. Proceedings of IEEE (2010)
Gluskin, E.D.: Norms of random matrices and widths of finite-dimensional sets. Math. USSR, Sb. 48, 173–182 (1984)
Indyk, P., Price, E.: K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance. In: STOC, pp. 627–636 (2011)
Kasin, B.S.: Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR, Izv. 11, 317–333 (1977)
Muthukrishnan, S.: Data streams: algorithms and applications. Found. Trends Theor. Comput. Sci. 1(2), 117–236 (2005)
Nachin, M.: Lower Bounds on the Column Sparsity of Sparse Recovery Matrices. MIT Undergraduate Thesis (2010)
Romberg, J., Choi, H., Baraniuk, R.: Bayesian tree-structured image modeling using wavelet-domain Hidden Markov Models. 10(7), 1056–1068 (2001)
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Indyk, P., Razenshteyn, I. (2013). On Model-Based RIP-1 Matrices. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_48
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DOI: https://doi.org/10.1007/978-3-642-39206-1_48
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