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Average-case matrix discrepancy: satisfiability bounds

Antoine Maillard

TL;DR

This work analyzes average-case matrix discrepancy for Gaussian symmetric matrices by studying the satisfiability of signed sums with bounded operator norm. It derives sharp first-moment bounds via a large-deviation problem for constrained GOE matrices and identifies a second-moment threshold that delineates a SAT region, complemented by a proof that a macroscopic spectrum shrink can occur when balancing $\Theta(d^2)$ matrices. The authors also show that in parts of the phase diagram the second-moment method fails due to large fluctuations, suggesting a complex solution-space geometry beyond the symmetric binary perceptron. The results combine large-deviation theory, variational descriptions of constrained spectra, and log-Sobolev concentration for correlated Gaussian matrices, yielding a nuanced phase diagram with SAT and UNSAT regimes and several open questions about contiguity, freezing, and sharp thresholds.

Abstract

Given a sequence of $d \times d$ symmetric matrices $\{\mathbf{W}_i\}_{i=1}^n$, and a margin $Δ> 0$, we investigate whether it is possible to find signs $(ε_1, \dots, ε_n) \in \{\pm 1\}^n$ such that the operator norm of the signed sum satisfies $\|\sum_{i=1}^n ε_i \mathbf{W}_i\|_{\rm op} \leq Δ$. 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 $n / d^2 \to τ> 0$. First, we prove that the expected number of solutions with margin $Δ=κ\sqrt{n}$ has a sharp threshold at a critical $τ_1(κ)$: for $τ< τ_1(κ)$ the problem is typically unsatisfiable, while for $τ> τ_1(κ)$ 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 $τ_2(κ)$, such that for $τ>τ_2(κ)$ the problem admits solutions with high probability. In particular, we establish that a system of $n = Θ(d^2)$ 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 $(τ,κ)$ 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.

Average-case matrix discrepancy: satisfiability bounds

TL;DR

This work analyzes average-case matrix discrepancy for Gaussian symmetric matrices by studying the satisfiability of signed sums with bounded operator norm. It derives sharp first-moment bounds via a large-deviation problem for constrained GOE matrices and identifies a second-moment threshold that delineates a SAT region, complemented by a proof that a macroscopic spectrum shrink can occur when balancing matrices. The authors also show that in parts of the phase diagram the second-moment method fails due to large fluctuations, suggesting a complex solution-space geometry beyond the symmetric binary perceptron. The results combine large-deviation theory, variational descriptions of constrained spectra, and log-Sobolev concentration for correlated Gaussian matrices, yielding a nuanced phase diagram with SAT and UNSAT regimes and several open questions about contiguity, freezing, and sharp thresholds.

Abstract

Given a sequence of symmetric matrices , and a margin , we investigate whether it is possible to find signs such that the operator norm of the signed sum satisfies . Kunisky and Zhang (2023) recently introduced a random version of this problem, where the matrices 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 with . First, we prove that the expected number of solutions with margin has a sharp threshold at a critical : for the problem is typically unsatisfiable, while for 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 , such that for the problem admits solutions with high probability. In particular, we establish that a system of 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 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.

Paper Structure

This paper contains 28 sections, 30 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Setting and related literature
  3. Discrepancy theory
  4. Random vector discrepancy, and the symmetric binary perceptron
  5. Average-case matrix discrepancy
  6. Main results
  7. Setting of the problem
  8. Asymptotics of the first moment
  9. The satisfiability region
  10. Failure of the second moment method
  11. Discussion and consequences
  12. Sharp asymptotics of the first moment
  13. Large deviations results
  14. Solving the variational principle: proof of Proposition \ref{['prop:ldp_Wop']}
  15. Limiting spectral distribution of a norm-constrained Gaussian matrix
  16. ...and 13 more sections

Key Result

theorem 1

There exists $C > 0$ such that for all $n, d \geq 1$, and all $\bu_1, \cdots, \bu_n \in \bbR^d$ with $\|\bu_i\|_\infty \leq 1$ for all $i \in [n]$:

Figures (3)

  • Figure 1: Sketch of the satisfiable (SAT) and unsatisfiable (UNSAT) regimes in average-case matrix discrepancy, as proven by Theorems \ref{['thm:first_moment']} and \ref{['thm:second_moment']}. The border of the SAT region for $\kappa < 2$ is given by $\tau_2(\kappa)$, see Proposition \ref{['prop:tau2']}. Numerically, we find $\tau_2(\kappa \uparrow 2) \simeq 5.67$. For $\kappa >2$, the problem trivially admits a solution, see eq. \ref{['eq:kappa_gtr_2']}. The orange region is not characterized by our results, and remains open. The dotted purple line shows $\tau_\f(\kappa)$: according to Theorem \ref{['thm:fail_second_moment']}, for $\tau_1(\kappa) < \tau < \tau_\f(\kappa)$ the number of solutions has large expectation and variance, and the second moment method fails (see also Fig. \ref{['fig:fail_second_moment']}). The right plots show the limits $\kappa \downarrow 0$ (top) and $\kappa \uparrow 2$ (bottom). We emphasize that $\tau_1(\kappa), \tau_2(\kappa) \to +\infty$ as $\kappa \downarrow 0$.
  • Figure 2: Illustration of Theorem \ref{['thm:fail_second_moment']}. There exists a region of parameters (in purple) $\tau \in (\tau_1(\kappa), \tau_c(\kappa))$ for which the number of solutions $Z_\kappa$ to average-case matrix discrepancy satisfies both $\EE[Z_\kappa] \gg 1$ and $\EE[Z_\kappa^2] \gg (\EE[Z_\kappa])^2$, so that the second moment method fails at characterizing the feasibility of the problem.
  • Figure 3: $\rho_\kappa(x)$ of eq. \ref{['def:rhokappa_1']} for different values of $\kappa$. For $\kappa = 2$ one recovers the semicircle law.

Theorems & Definitions (39)

  • theorem 1: spencer1985six
  • theorem 2: turner2020balancing
  • theorem 3: Sharp threshold for the SBP aubin2019storageabbe2022proof
  • theorem 4: Semicircle law
  • theorem 5: Asymptotics of the first moment
  • proposition 1
  • theorem 6: Satisfiability region
  • theorem 7: Failure of the second moment method in part of the phase diagram
  • corollary 1
  • corollary 2: Sharp threshold sequence
  • ...and 29 more