The smallest singular value of signed random combinatorial matrices

Abstract: Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of ${-1,0,1}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c > 0$ such that for any $\varepsilon\ge0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting.

• The paper establishes that the probability of near-singularity is bounded by Cε + exp(–cn), ensuring robust invertibility for these matrices.
• It adapts and extends the Rudelson–Vershynin framework using novel combinatorial techniques, notably the Combinatorial Least Common Denominator (CLCD).
• The results imply that even with strong dependencies, the invertibility and condition number behave optimally, advancing both theory and practical applications.

The Smallest Singular Value of Signed Random Combinatorial Matrices

Introduction and Motivation

This paper investigates the quantitative invertibility properties of random matrices $M_n$ in which each row is uniformly sampled from all $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates (assuming even $n$). Such matrices interpolate between random substructures in random regular digraphs and signed ensemble models, but feature strong dependencies among the entries—deviating substantially from the standard i.i.d. paradigms.

The smallest singular value $s_n(M_n)$ is a central object in the non-asymptotic theory of random matrices. Its lower tail controls both invertibility and the condition number, which are crucial in numerical analysis, optimization, high-dimensional probability, and theoretical computer science. The singularity probability (i.e., $\mathbb{P}(s_n(M_n) = 0)$) is a stringent measure of random ensemble robustness. For classical models (e.g., i.i.d. Gaussian or Bernoulli entries), the celebrated bounds of von Neumann, Edelman, and Tikhomirov are known, but the extension of such results to dependent ensembles remains technically intricate due to the loss of independence and the emergence of highly correlated structures.

Main Results

The principal result establishes that for an $n \times n$ random matrix $M_n$ as described above, there exist constants $C, c > 0$ such that for all $\varepsilon \geq 0$,

$\{-1,0,1\}$0

In particular, the probability that $\{-1,0,1\}$1 is singular is exponentially small in $\{-1,0,1\}$2. This rate matches, up to constants, the optimal tail bounds known in the i.i.d. case, despite the globally coupled row structure.

The proof architecture adapts and extends the geometric approach of Rudelson and Vershynin to this setting, employing small ball probability techniques, structural inverse Littlewood–Offord theory, and combinatorial combinatorics on the dependency graph of the matrix. Notably, the dependence structure is tamed with the Combinatorial Least Common Denominator (CLCD), originally due to Tran, in a new signed setting, enabling anti-concentration estimates for highly structured linear forms arising in the combinatorial model.

Furthermore, the main theorem is shown to hold in the more general setting of row sparsity: for matrices where each row has exactly $\{-1,0,1\}$3 non-zero entries, as long as $\{-1,0,1\}$4 for some absolute constant.

Methodology and Technical Innovations

The proof exhibits the following structural decomposition, now canonical in modern random matrix theory:

• Compressible vs. Incompressible Vectors: The unit sphere is partitioned into compressible vectors (close to sparse vectors) and incompressible ones. For the former, net arguments and volumetric bounds suffice to show exponentially small probability for small singular values. For the latter, analytic invertibility bounds are required.
• Distance Problem and Littlewood–Offord Framework: The crux for incompressible vectors is to bound the probability that a random row is nearly contained in the span of the remaining rows, i.e., that $\{-1,0,1\}$5. This is controlled through estimates on the Lévy concentration function of linear forms $\{-1,0,1\}$6, leveraging anti-concentration phenomena.
• Combinatorial LCD and Small Ball Probabilities: The CLCD quantifies the arithmetic structure of the coefficients of the linear forms. By controlling the CLCD for vectors orthogonal to large subspaces, one obtains robust lower bounds for anti-concentration, even in the presence of strong row dependencies.
• Net Construction and Tensorization: Approximating nets are built for the relevant level sets in the CLCD filtration, carefully balancing cardinality against the anti-concentration gains. Tensorization arguments are then used to transfer small ball probability control from scalars to the Euclidean norm of random vectors.
• Dealing with Dependency: The key technical advance is the explicit modeling of the randomness of each row as a combination of a uniform selection of a support set (size $\{-1,0,1\}$7) and independent Rademacher signs assigned to the chosen coordinates, which allows for delicate probabilistic and arithmetic estimates.

Numerical Bound Summary

The main quantitative statement is: $\{-1,0,1\}$8 for all $\{-1,0,1\}$9 and sufficiently large $n/2$0. In particular, the singularity probability (for $n/2$1) is exponentially small.

Additionally, the operator norm is shown to satisfy

$n/2$2

for suitable $n/2$3.

For fixed vectors, the compressed and incompressible probabilities for small action under $n/2$4 are also exponentially small.

These results collectively mirror—and in significant aspects extend—the best known facts for i.i.d. ensembles to this non-product setting.

Theoretical and Practical Implications

This work extends the quantitative analysis of invertibility to highly nontrivial matrix ensembles where independence is broken by strong combinatorial row constraints. It demonstrates the robustness of the Rudelson–Vershynin framework and the flexibility of additive combinatorics techniques like the CLCD even in constraint-rich regimes.

From a theoretical standpoint, these results confirm the intuition that heavy dependency does not necessarily increase the chance of near-singularity, provided the global row structure retains sufficient "randomness" and arithmetic non-alignment. The use of combinatorial invariants (CLCD) is critical, indicating promising directions for further generalizations, for instance to regular bipartite graphs, random lifts, or other random combinatorial objects with signature constraints.

Practically, this provides assurance that randomized algorithms for large-scale, sparsely structured problems—where preservation of invertibility and bounds on the condition number are paramount—should retain favorable properties even under strong dependencies, provided the underlying combinatorics are sufficiently rich.

Speculation on Future Directions

• Extending CLCD-based analysis to broader ensembles: The methods invite extensions to random matrices modeling random regular digraphs, signed adjacency matrices, random graphs with prescribed degree sequences, or sparse random models with weak randomness.
• Universality: A compelling theoretical question is whether such exponential bounds hold universally for any sufficiently irreducible combinatorial ensemble.
• Connections to Robust Smoothed Analysis: Since the CLCD captures quantitative arithmetic structure, further work may yield smoothed analysis bounds for algorithms operating on random combinatorial data.
• Rates and Constants: Investigating exact constants, moderate deviation regimes, or tail refinements for the smallest singular value in combinatorially constrained matrices.

Conclusion

The paper gives a thorough analysis of the smallest singular value of signed random combinatorial matrices, establishing that the probability of nearly singular behavior is exponentially suppressed, on par with the optimal rates in the i.i.d. case. The technical core is an adaptation of geometric and additive combinatorics tools (notably the Combinatorial LCD) to the setting with domain-specific dependencies. These advances not only solve longstanding open questions in the theory of random matrices with dependent entries, but also open avenues for quantitative invertibility analyses in a broad class of structured random ensembles.