[Paper Review] Investigate Invertibility of Sparse Symmetric Matrix

Wei Feng|arXiv (Cornell University)|Dec 10, 2017

Random Matrices and Applications10 references3 citations

TL;DR

This paper establishes that an $n \times n$ sparse symmetric random matrix $A$ with i.i.d. Bernoulli sparsity and sub-gaussian entries is invertible with high probability when the sparsity parameter $p \geq n^{-c}$ for some constant $c$ depending on the fourth moment of the entries. It proves a quantitative lower bound on the smallest singular value: $s_{\text{min}}(A) > \varepsilon \sqrt{p/n}$ with high probability, using a combinatorial approach and decoupling techniques adapted to symmetric sparsity.

ABSTRACT

In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n imes n$ sparse symmetric random matrix $A$ with $A_{ij} = δ_{ij} ξ_{ij}$ is invertible with high probability. Here, $δ_{ij}$s, $i\ge j$ are i.i.d. Bernoulli random variables with $\mathbb{P} \left(ξ_{ij}=1 ight) =p \ge n^{-c}$, $ξ_{ij}, i\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$ s_{ m min} (A) > \varepsilon \sqrt{\frac{p}{n}}. $$ with high probability.

Motivation & Objective

• To establish quantitative invertibility for sparse symmetric random matrices under general sub-gaussian entry assumptions.
• To extend non-sparse invertibility results to the sparse symmetric case using combinatorial and concentration techniques.
• To derive a high-probability lower bound on the smallest singular value $s_{\text{min}}(A)$ for such matrices.
• To address the challenge of symmetric sparsity, which complicates standard methods relying on i.i.d. non-symmetric structures.

Proposed method

• Model the matrix $A$ as $A_{ij} = \delta_{ij} \xi_{ij}$, where $\delta_{ij}$ are i.i.d. Bernoulli($p$) and $\xi_{ij}$ are i.i.d. sub-gaussian with mean 0, variance 1, and finite fourth moment.
• Condition on the sparsity pattern $\bm{\delta}$, treating $A$ as a random matrix with i.i.d. entries on a random support.
• Apply Gaussian concentration inequalities to control the operator norm $\|A\|$ conditionally on $\bm{\delta}$, using the $\sqrt{2}$-Lipschitz property of the norm under the Gaussian measure.
• Use a combinatorial lemma to control the norm of $Ax$ for sparse vectors $x$, adapted to symmetric sparsity.
• Apply Markov's inequality to the $pn$-th moment of $\|A\|$ to derive tail bounds on the operator norm.
• Combine concentration and moment bounds to show that $\|A\| \leq C_{\text{op}} \sqrt{pn}$ and $s_{\text{min}}(A) > \varepsilon \sqrt{p/n}$ with high probability.

Experimental results

Research questions

• RQ1 Can the invertibility of sparse symmetric random matrices be established with high probability under sub-gaussian entry assumptions?
• RQ2 What is the optimal lower bound on the smallest singular value $s_{\text{min}}(A)$ for such matrices?
• RQ3 How does the sparsity level $p$ affect the invertibility and condition number of symmetric random matrices?
• RQ4 Can the techniques used for non-Hermitian sparse matrices be adapted to the symmetric case?

Key findings

• For $p \geq n^{-c}$ with $c$ depending on the fourth moment of the entries, the matrix $A$ is invertible with high probability.
• The smallest singular value satisfies $s_{\text{min}}(A) > \varepsilon \sqrt{p/n}$ with high probability for any $\varepsilon > 0$.
• The operator norm $\|A\|$ is bounded by $C_{\text{op}} \sqrt{pn}$ with high probability.
• The condition number $\sigma(A) = s_{\text{max}}(A)/s_{\text{min}}(A)$ is bounded by $O(n)$ with high probability under the same sparsity condition.
• The result extends prior work on non-Hermitian sparse matrices to the symmetric case using a symmetric adaptation of the combinatorial method.
• The proof relies on conditioning on the sparsity pattern, moment estimates, and Gaussian concentration to control the operator norm and singular values.