[Paper Review] The smallest singular value of random combinatorial matrices

Tuan Tran|arXiv (Cornell University)|Jul 13, 2020

Random Matrices and Applications58 references6 citations

TL;DR

This paper establishes the first exponential bound on the singularity probability of random $ n \times n $ combinatorial matrices with i.i.d. rows of exactly $ n/2 $ zero entries. By introducing a novel Combinatorial Least Common Denominator (CLCD) and proving a small ball probability inequality for linear forms under random permutations, the authors derive a sharp tail bound: $ \mathbb{P}(s_n(Q_n) \leq \varepsilon / \sqrt{n}) \leq C\varepsilon + 2e^{-cn} $, which is optimal up to constants and implies $ \mathbb{P}(Q_n \text{ singular}) \leq 2e^{-cn} $.

ABSTRACT

Let $Q_n$ be a random $n imes n$ matrix with entries in $\{0,1\}$ whose rows are independent vectors of exactly $n/2$ zero components. We show that the smallest singular value $s_n(Q_n)$ of $Q_n$ satisfies \[ \mathbb{P}\Big\{s_n(Q_n)\le \frac{\varepsilon}{\sqrt{n}}\Big\} \le C\varepsilon + 2 e^{-cn} \quad \forall \varepsilon \ge 0, \] which is optimal up to the constants $C,c>0$. This improves on earlier results of Ferber, Jain, Luh and Samotij, as well as Jain. In particular, for $\varepsilon=0$, we obtain the first exponential bound in dimension for the singularity probability \[ \mathbb{P}\big\{Q_n \,\, ext{is singular}\big\} \le 2 e^{-cn}.\] To overcome the lack of independence between entries of $Q_n$, we introduce an arithmetic-combinatorial invariant of a pair of vectors, which we call a Combinatorial Least Common Denominator (CLCD). We prove a small ball probability inequality for the combinatorial statistic $\sum_{i=1}^{n}a_iv_{σ(i)}$ in terms of the CLCD of the pair $(a,v)$, where $σ$ is a uniformly random permutation of $\{1,2,\ldots,n\}$ and $a:=(a_1,\ldots,a_n), v:=(v_1,\ldots,v_n)$ are real vectors. This inequality allows us to derive strong anti-concentration properties for the distance between a fixed row of $Q_n$ and the linear space spanned by the remaining rows, and prove the main result.

Motivation & Objective

• To establish sharp tail bounds on the smallest singular value of random $ n \times n $ combinatorial matrices with exactly $ n/2 $ zero entries per row.
• To resolve the long-standing conjecture that the singularity probability decays exponentially, improving upon prior polynomial and sub-exponential bounds.
• To overcome the challenge of dependent entries in such matrices by introducing a new arithmetic-combinatorial invariant: the Combinatorial Least Common Denominator (CLCD).
• To develop a small ball probability inequality for linear forms $ \sum_{i=1}^n a_i v_{\sigma(i)} $, where $ \sigma $ is a random permutation, in terms of the CLCD of $ (\bm{a}, \bm{v}) $.
• To apply this inequality to control the distance from a fixed row to the span of the others, thereby proving the main result on the smallest singular value.

Proposed method

• Introduce the Combinatorial Least Common Denominator (CLCD) as a new invariant to measure arithmetic structure in pairs of real vectors $ (\bm{a}, \bm{v}) $, quantifying how well they can be approximated by rational combinations.
• Prove a small ball probability inequality: $ \mathbb{P}( |\sum_{i=1}^n a_i v_{\sigma(i)} - x| \leq \varepsilon ) \lesssim \varepsilon / \mathrm{CLCD} + e^{-c \cdot \mathrm{CLCD}^2} $, where $ \sigma $ is a uniform random permutation.
• Use the CLCD-based small ball inequality to derive anti-concentration bounds for the distance between a fixed row of $ Q_n $ and the linear span of the remaining rows.
• Apply these anti-concentration estimates to control the probability that the smallest singular value $ s_n(Q_n) $ is small, leading to the main tail bound.
• Establish the bound $ \mathbb{P}(s_n(Q_n) \leq \varepsilon / \sqrt{n}) \leq C\varepsilon + 2e^{-cn} $ for all $ \varepsilon \geq 0 $, with explicit constants $ C, c > 0 $.
• Extend the result to the more general case where row sums are $ d $ with $ \min(d, n-d) \gtrsim n $, though the paper focuses on $ d = n/2 $.

Experimental results

Research questions

• RQ1 What is the optimal small ball probability bound for the smallest singular value of random $ n \times n $ 0-1 matrices with exactly $ n/2 $ zeros per row?
• RQ2 Can an exponential bound on the singularity probability be established for such combinatorial matrices, despite the strong dependence between entries?
• RQ3 What arithmetic-combinatorial invariant can effectively capture the structure of vectors to enable anti-concentration in the presence of permutation-induced dependencies?
• RQ4 How can the inverse Littlewood-Offord theory be adapted to handle linear forms under random permutations rather than i.i.d. coefficients?
• RQ5 To what extent can the CLCD framework be extended to other models of exchangeable or dependent random matrices?

Key findings

• The paper establishes the first exponential bound on the singularity probability: $ \mathbb{P}(Q_n \text{ is singular}) \leq 2e^{-cn} $ for some $ c > 0 $, resolving a key open problem.
• The main tail bound $ \mathbb{P}(s_n(Q_n) \leq \varepsilon / \sqrt{n}) \leq C\varepsilon + 2e^{-cn} $ is optimal up to the constants $ C, c > 0 $, matching the behavior of i.i.d. Gaussian matrices.
• The Combinatorial Least Common Denominator (CLCD) is introduced as a novel invariant that captures the arithmetic structure of vector pairs under permutation, enabling new anti-concentration results.
• A small ball probability inequality is proven for $ \sum_{i=1}^n a_i v_{\sigma(i)} $, where $ \sigma $ is a uniform random permutation, in terms of the CLCD of $ (\bm{a}, \bm{v}) $, which is central to the proof.
• The result improves upon prior bounds: it strengthens Jain’s $ \mathbb{P}(s_n(Q_n) \leq \varepsilon / n^2) \leq C\varepsilon + 2e^{-n^{0.0001}} $ and Ferber et al.’s $ \mathbb{P}(s_n(Q_n) \leq n^{-D}) \leq n^{-C} $, achieving exponential decay.
• The framework is robust enough to extend to the general case where each row has $ d $ ones with $ \min(d, n-d) \gtrsim n $, suggesting broad applicability beyond the $ d = n/2 $ case.