License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.7
URN: urn:nbn:de:0030-drops-165695
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16569/
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Guruswami, Venkatesan ; Manohar, Peter ; Mosheiff, Jonathan

š“_p-Spread and Restricted Isometry Properties of Sparse Random Matrices

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Abstract

Random subspaces X of ā„āæ of dimension proportional to n are, with high probability, well-spread with respect to the š“ā‚‚-norm. Namely, every nonzero x āˆˆ X is "robustly non-sparse" in the following sense: x is Īµ ā€–xā€–ā‚‚-far in š“ā‚‚-distance from all Ī“ n-sparse vectors, for positive constants Īµ, Ī“ bounded away from 0. This "š“ā‚‚-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to X being a Euclidean section of the š“ā‚ unit ball. Explicit š“ā‚‚-spread subspaces of dimension Ī©(n), however, are unknown, and the best known explicit constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of certain sparse matrices.
Motivated by this, we study the spread properties of the kernels of sparse random matrices. We prove that with high probability such subspaces contain vectors x that are o(1)ā‹…ā€–xā€–ā‚‚-close to o(n)-sparse with respect to the š“ā‚‚-norm, and in particular are not š“ā‚‚-spread. This is strikingly different from the case of random LDPC codes, whose distance is asymptotically almost as good as that of (dense) random linear codes.
On the other hand, for p < 2 we prove that such subspaces are š“_p-spread with high probability. The spread property of sparse random matrices thus exhibits a threshold behavior at p = 2. Our proof for p < 2 moreover shows that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the š“_p norm, and in fact this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the š“ā‚ norm [Berinde et al., 2008]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of š“_p-RIP matrices for 1 ā‰¤ p < pā‚€, where 1 < pā‚€ < 2 is an absolute constant.

BibTeX - Entry

@InProceedings{guruswami_et_al:LIPIcs.CCC.2022.7,
  author =	{Guruswami, Venkatesan and Manohar, Peter and Mosheiff, Jonathan},
  title =	{{š“\underlinep-Spread and Restricted Isometry Properties of Sparse Random Matrices}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16569},
  URN =		{urn:nbn:de:0030-drops-165695},
  doi =		{10.4230/LIPIcs.CCC.2022.7},
  annote =	{Keywords: Spread Subspaces, Euclidean Sections, Restricted Isometry Property, Sparse Matrices}
}

Keywords: Spread Subspaces, Euclidean Sections, Restricted Isometry Property, Sparse Matrices
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022


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