The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an m x n matrix satisfies RIP of order k for the L_p norm, if |Ax|_p is approximately |x|_p for every x with at most k non-zero coordinates. For every 1 <= p < infty we obtain almost tight bounds on the minimum number of rows m necessary for the RIP property to hold. Prior to this work, only the cases p = 1, 1 + 1/log(k), and 2 were studied. Interestingly, our results show that the case p=2 is a "singularity" point: the optimal number of rows m is Theta(k^p) for all p in [1, infty)-{2}, as opposed to Theta(k) for k=2. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.
@InProceedings{allenzhu_et_al:LIPIcs.SOCG.2015.451, author = {Allen-Zhu, Zeyuan and Gelashvili, Rati and Razenshteyn, Ilya}, title = {{Restricted Isometry Property for General p-Norms}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {451--460}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.451}, URN = {urn:nbn:de:0030-drops-51273}, doi = {10.4230/LIPIcs.SOCG.2015.451}, annote = {Keywords: compressive sensing, dimension reduction, linear algebra, high-dimensional geometry} }
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