eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
7:1
7:17
10.4230/LIPIcs.CCC.2022.7
article
π_p-Spread and Restricted Isometry Properties of Sparse Random Matrices
Guruswami, Venkatesan
1
Manohar, Peter
2
Mosheiff, Jonathan
2
University of California, Berkeley, CA, USA
Carnegie Mellon University, Pittsburgh, PA, USA
Random subspaces X of ββΏ of dimension proportional to n are, with high probability, well-spread with respect to the πβ-norm. Namely, every nonzero x β X is "robustly non-sparse" in the following sense: x is Ξ΅ βxββ-far in πβ-distance from all Ξ΄ n-sparse vectors, for positive constants Ξ΅, Ξ΄ bounded away from 0. This "πβ-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 πβ unit ball. Explicit πβ-spread subspaces of dimension Ξ©(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)β
βxββ-close to o(n)-sparse with respect to the πβ-norm, and in particular are not πβ-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 π_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 π_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 πβ norm [Berinde et al., 2008]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of π_p-RIP matrices for 1 β€ p < pβ, where 1 < pβ < 2 is an absolute constant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.7/LIPIcs.CCC.2022.7.pdf
Spread Subspaces
Euclidean Sections
Restricted Isometry Property
Sparse Matrices