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.

翻译：给定一个 $d \times d$ 对称矩阵序列 $\{\mathbf{W}_i\}_{i=1}^n$ 以及一个裕度 $\Delta > 0$，我们研究是否存在符号 $(\epsilon_1, \dots, \epsilon_n) \in \{\pm 1\}^n$，使得带符号和的算子范数满足 $\|\sum_{i=1}^n \epsilon_i \mathbf{W}_i\|_{\rm op} \leq \Delta$。Kunisky 与 Zhang (2023) 最近引入了该问题的一个随机版本，其中矩阵 $\{\mathbf{W}_i\}_{i=1}^n$ 从高斯正交系综中抽取。该模型可视为著名的 Matrix Spencer 猜想的随机变体，以及统计物理中对称二元感知机的矩阵值类比。在本工作中，我们建立了该问题在 $n, d \to \infty$ 且 $n / d^2 \to \tau > 0$ 时的可满足性转变。首先，我们证明具有裕度 $\Delta=\kappa \sqrt{n}$ 的解的期望数量在临界值 $\tau_1(\kappa)$ 处存在一个尖锐阈值：当 $\tau < \tau_1(\kappa)$ 时，问题通常不可满足；而当 $\tau > \tau_1(\kappa)$ 时，解的平均数量呈指数级增长。其次，结合二阶矩方法与 Altschuler (2023) 关于感知机类型问题中裕度集中性的最新结果，我们确定了第二个阈值 $\tau_2(\kappa)$，使得当 $\tau>\tau_2(\kappa)$ 时，问题以高概率存在解。特别地，我们证明了一个由 $n = \Theta(d^2)$ 个高斯随机矩阵构成的系统可以被平衡，使得所得矩阵的谱在宏观尺度上相较于半圆律收缩。最后，在一个技术性假设下，我们表明存在某些 $(\tau,\kappa)$ 值使得解的数量具有较大方差，这意味着二阶矩方法失效。我们的证明依赖于建立谱范数约束下相关高斯矩阵的集中性与大偏差性质。