Abstract
Random subspaces X of āāæ of dimension proportional to n are, with high probability, wellspread with respect to the šānorm. Namely, every nonzero x ā X is "robustly nonsparse" in the following sense: x is Īµ āxāāfar in šādistance from all Ī“ nsparse vectors, for positive constants Īµ, Ī“ bounded away from 0. This "šāspread" 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 šā unit ball. Explicit šāspread 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
āxāāclose to o(n)sparse with respect to the šānorm, and in particular are not šāspread. 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_pspread 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 šā norm [Berinde et al., 2008]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of š_pRIP 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\underlinepSpread and Restricted Isometry Properties of Sparse Random Matrices}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {7:17:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772419},
ISSN = {18688969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16569},
URN = {urn:nbn:de:0030drops165695},
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 