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$\ell_p$-Spread and Restricted Isometry Properties of Sparse Random Matrices

Venkatesan Guruswami, Peter Manohar, Jonathan Mosheiff

TL;DR

The paper analyzes the $\ell_p$-spread and RIP properties of subspaces formed as kernels of sparse random matrices. It shows a sharp dichotomy: kernels of sparse random matrices are not $\ell_2$-spread with high probability, yet they exhibit strong $\ell_p$-spread and $\ell_p$-RIP for $p<2$, derived from unique expansion in the underlying biregular bipartite graphs. The authors establish sharp singular-value bounds via a nomadic-walk/ Ihara–Bass framework and demonstrate that RIP for $p<2$ follows from vertex expansion; they also provide explicit constructions of $\ell_p$-RIP matrices using explicit expanders. Together, these results illuminate the limits of $\ell_2$-spread for sparse kernels and deliver explicit $\ell_p$-RIP matrices for a broad range of $p$ (up to a constant below 2). The work has implications for compressed sensing and dimensionality reduction with sparse measurements, showing a threshold behavior at $p=2$ and expanding the toolkit for explicit sparse RIP matrices.

Abstract

Random subspaces $X$ of $\mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $\ell_2$-norm. Namely, every nonzero $x \in X$ is "robustly non-sparse" in the following sense: $x$ is $\varepsilon \|x\|_2$-far in $\ell_2$-distance from all $δn$-sparse vectors, for positive constants $\varepsilon, δ$ bounded away from $0$. This "$\ell_2$-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 $\ell_1$ unit ball. Explicit $\ell_2$-spread subspaces of dimension $Ω(n)$, however, are unknown, and the best known constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors $x$ that are $o(1)\cdot \|x\|_2$-close to $o(n)$-sparse with respect to the $\ell_2$-norm, and in particular are not $\ell_2$-spread. On the other hand, for $p < 2$ we prove that such subspaces are $\ell_p$-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the $\ell_p$ norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the $\ell_1$ norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of $\ell_p$-RIP matrices for $1 \leq p < p_0$, where $1 < p_0 < 2$ is an absolute constant.

$\ell_p$-Spread and Restricted Isometry Properties of Sparse Random Matrices

TL;DR

The paper analyzes the -spread and RIP properties of subspaces formed as kernels of sparse random matrices. It shows a sharp dichotomy: kernels of sparse random matrices are not -spread with high probability, yet they exhibit strong -spread and -RIP for , derived from unique expansion in the underlying biregular bipartite graphs. The authors establish sharp singular-value bounds via a nomadic-walk/ Ihara–Bass framework and demonstrate that RIP for follows from vertex expansion; they also provide explicit constructions of -RIP matrices using explicit expanders. Together, these results illuminate the limits of -spread for sparse kernels and deliver explicit -RIP matrices for a broad range of (up to a constant below 2). The work has implications for compressed sensing and dimensionality reduction with sparse measurements, showing a threshold behavior at and expanding the toolkit for explicit sparse RIP matrices.

Abstract

Random subspaces of of dimension proportional to are, with high probability, well-spread with respect to the -norm. Namely, every nonzero is "robustly non-sparse" in the following sense: is -far in -distance from all -sparse vectors, for positive constants bounded away from . This "-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to being a Euclidean section of the unit ball. Explicit -spread subspaces of dimension , however, are unknown, and the best known constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors that are -close to -sparse with respect to the -norm, and in particular are not -spread. On the other hand, for we prove that such subspaces are -spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of -RIP matrices for , where is an absolute constant.

Paper Structure

This paper contains 35 sections, 40 theorems, 130 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Kernels of sparse matrices
  3. Our results
  4. Poor $\ell_2$-spread of sparse random matrices
  5. $\ell_p$-RIP and $\ell_p$-spread for $p < 2$
  6. Proof overview
  7. \ref{['mthm:ell2spreadneg']}: $\ker(A)$ in not $\ell_2$-spread
  8. \ref{['mthm:ExpansionToRIP']}: $\ell_p$-RIP for $p < 2$ from vertex expansion
  9. \ref{['mthm:singvalue']}: bounds on the singular values of $A$
  10. Organization
  11. Preliminaries
  12. Basic notation
  13. Biregular bipartite graphs and sparse matrices
  14. $\ell_p$ norms, compressibility, spread subspaces, and distortion
  15. Properties of random biregular graphs
  16. ...and 20 more sections

Key Result

Theorem 1

With high probability over $A$, there exists an $(m^{c},\frac{n^{-\Omega(\log(1/\alpha)/\log s)}}{1-\sqrt \alpha})$-$\ell_2$-compressible vector $x \in \ker(A)$, where $c < 1$ is an absolute constant. In particular, Moreover, there is a $\mathrm{poly}(n)$-time algorithm that, on input $A$, outputs such an $x$.

Theorems & Definitions (88)

  • Definition 1.1: $\ell_p$-spread
  • Theorem 1: Poor $\ell_2$-spread of $\ker(A)$
  • Theorem 2: Singular value bound
  • Theorem 3: Converse to \ref{['mthm:ell2spreadneg']}
  • Definition 1.2: $\ell_p$-RIP
  • Definition 1.3: Unique expanders
  • Theorem 4: $\ell_p$-RIP of expander graphs
  • Corollary 4: Good $\ell_p$-RIP and $\ell_p$-spread of $A$
  • Corollary 4: Explicit construction of $\ell_p$-RIP matrices
  • Theorem 5: Partial converse to \ref{['mthm:ellpspreadpos']}
  • ...and 78 more