License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.451
URN: urn:nbn:de:0030-drops-51273
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5127/
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Allen-Zhu, Zeyuan ; Gelashvili, Rati ; Razenshteyn, Ilya

Restricted Isometry Property for General p-Norms

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Abstract

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.

BibTeX - Entry

@InProceedings{allenzhu_et_al:LIPIcs:2015:5127,
  author =	{Zeyuan Allen-Zhu and Rati Gelashvili and Ilya Razenshteyn},
  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 =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5127},
  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}
}

Keywords: compressive sensing, dimension reduction, linear algebra, high-dimensional geometry
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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