Average-case matrix discrepancy: satisfiability bounds (2410.17887v2)

Abstract: Given a sequence of $d \times d$ symmetric matrices ${\mathbf{W}i}{i=1}^n$, and a margin $\Delta > 0$, we investigate whether it is possible to find signs $(\epsilon_1, \dots, \epsilon_n) \in {\pm 1}^n$ such that the operator norm of the signed sum satisfies $|\sum_{i=1}^n \epsilon_i \mathbf{W}i|{\rm op} \leq \Delta$. Kunisky and Zhang (2023) recently introduced a random version of this problem, where the matrices ${\mathbf{W}i}{i=1}^n$ are drawn from the Gaussian orthogonal ensemble. This model can be seen as a random variant of the celebrated Matrix Spencer conjecture and as a matrix-valued analog of the symmetric binary perceptron in statistical physics. In this work, we establish a satisfiability transition in this problem as $n, d \to \infty$ with $n / d^2 \to \tau > 0$. First, we prove that the expected number of solutions with margin $\Delta=\kappa \sqrt{n}$ has a sharp threshold at a critical $\tau_1(\kappa)$: for $\tau < \tau_1(\kappa)$ the problem is typically unsatisfiable, while for $\tau > \tau_1(\kappa)$ the average number of solutions is exponentially large. Second, combining a second-moment method with recent results from Altschuler (2023) on margin concentration in perceptron-type problems, we identify a second threshold $\tau_2(\kappa)$, such that for $\tau>\tau_2(\kappa)$ the problem admits solutions with high probability. In particular, we establish that a system of $n = \Theta(d^2)$ Gaussian random matrices can be balanced so that the spectrum of the resulting matrix macroscopically shrinks compared to the semicircle law. Finally, under a technical assumption, we show that there exists values of $(\tau,\kappa)$ for which the number of solutions has large variance, implying the failure of the second moment method. Our proofs rely on establishing concentration and large deviation properties of correlated Gaussian matrices under spectral norm constraints.

• The paper establishes a sharp threshold for the expected number of balanced signings, showing exponential solution growth beyond a critical margin.
• It proves that for values above a second threshold, solutions exist with high probability for sets of Θ(d²) GOE matrices under operator norm limits.
• The findings advance the Matrix Spencer conjecture analysis and inspire algorithmic strategies for efficiently balancing large random matrices.

Overview of Average-Case Matrix Discrepancy: Satisfiability Bounds

The paper investigates the problem of average-case matrix discrepancy, where the aim is to balance a sequence of $d \times d$ symmetric matrices, specifically generated from the Gaussian Orthogonal Ensemble (GOE), such that the operator norm of the signed sum is bounded by a given margin $\Delta$. This paper presents insights into an average-case variant of the Matrix Spencer conjecture, a problem of significant interest in discrepancy theory and related fields.

Key Contributions

The paper's contributions can be categorized into two main results:

1. Sharp Threshold for Expected Solutions:
   • The authors establish a sharp threshold $\tau_1(\kappa)$ for the expected number of solutions with a margin $\Delta = \kappa \sqrt{n}$. Below this threshold, the problem is typically unsatisfiable, while above it, the number of solutions grows exponentially.
2. High Probability Existence of Solutions:
   • A second threshold $\tau_2(\kappa)$ is identified. For values of $\tau$ greater than $\tau_2(\kappa)$, the existence of solutions is shown with high probability, providing evidence that a set of $n = \Theta(d^2)$ matrices can be balanced effectively.

Theoretical and Practical Implications

The implications of these findings are profound for the field of discrepancy theory:

• Matrix Spencer Conjecture Insight:

The results offer a deeper understanding of the matrix discrepancy in an average-case scenario, shedding light on the feasibility of finding solutions that balance large sets of random matrices.

• Spectral Properties of Random Matrices:

The paper examines the large deviation properties and concentration inequalities for correlated Gaussian matrices under spectral norm constraints. This enhances existing knowledge and opens avenues for exploring the spectral characteristics of complex matrix ensembles.

Discussion and Future Directions

The exploration of matrix discrepancy from an average-case perspective provides a rich ground for further research:

• Enhancing Algorithmic Approaches:

The insights gained could inform the development of efficient algorithms to construct signings with low discrepancy for large matrices.

• Understanding Phase Transitions:

A comprehensive analysis of the phase diagram and satisfiability regions can help identify sharp thresholds and computational-to-statistical gaps in solving related problems.

• Extension to Other Ensembles:

While the current paper focuses on the GOE, extending the analysis to other types of random matrix ensembles could offer broader applicability and understanding of matrix discrepancy phenomena.

Conclusion

The paper makes significant strides in understanding the satisfiability bounds of average-case matrix discrepancy problems. Through rigorous analysis and proofs, it provides valuable insights into balancing random matrices and contributes to ongoing discourse in mathematical and computational fields regarding the feasibility of solutions under operator norm constraints. These findings pave the way for theoretical advancements and practical applications, particularly in areas intersecting with statistical physics and computational complexity.