Matrix Spencer：八个标准差即足够，以及稠密输入的近线性时间算法
Matrix Spencer: Eight Standard Deviations Suffice and an Almost-Linear Time Algorithm for Dense Input
- Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了 Matrix Spencer 猜想，即存在不一致性低于 8√n 的符号选择，并给出稠密输入下运行时间 n^{3+o(1)} 的随机算法，通过部分着色、无损编码和光滑谱障碍实现。
AI中文摘要：
Matrix Spencer 猜想断言：对于所有满足 $\\|A_i\\|\le1$ 的对称矩阵 $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$，存在符号 $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ 使得 $\\|\sum_{i=1}^n\varepsilon_iA_i\\|=O(\sqrt n)$。随机符号仅给出矩阵集中界 $O(\sqrt{n\log n})$，且该猜想此前仅在秩、块对角或 Frobenius 范数限制下已知。我们证明了该猜想：总存在一个不一致性低于 $8\sqrt n$ 的符号选择。我们还给出一个随机化算法，在实数算术模型中使用 $n^{3+o(1)}\operatorname{polylog}(1/p)$ 次算术运算，以不超过 $p$ 的失败概率找到不一致性低于 $12\sqrt n$ 的符号选择，该复杂度与稠密输入的规模 $n^3$ 匹配至次多项式因子。存在性证明是一个部分着色论证，其中包含一个新的估计：对于谱体 $\{x\in \mathbb{R}^n:\\|\sum_ix_iA_i\\|\le R\}$，通过在对角模型到非交换模型之间的矩阵加权 Poincaré 不等式插值对数配分函数，证明了其遗传高斯小球界。证明该猜想并将常数降至 $8$ 以下是不同的问题。部分着色论证在将高斯测度通过联合界转化为符号以及在小球估计半径远大于必要值时，两次损失了大因子。我们通过将高斯测度无损编码为符号来消除第一次损失，并通过具有认证系数的光滑谱障碍来消除第二次损失。算法将高斯点投影到谱体的光滑版本上，其导数是对单个 Gibbs 矩阵的迹。四种依次更廉价的维护该矩阵的方法将运行时间降至 $n^{3+o(1)}$。
英文摘要：
The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We also give a randomized algorithm that finds a signing of discrepancy below $12\sqrt n$ with failure probability at most $p$. The algorithm uses $n^{3+o(1)}\operatorname{polylog}(1/p)$ arithmetic operations in the real-arithmetic model. This matches the size $n^3$ of the dense input up to subpolynomial factors. In the other direction, we prove that for every $n$ there are collections of symmetric matrices such that every signing has discrepancy at least $(2-o(1))\sqrt{n}$. We present three different proofs of the matrix Spencer conjecture. The key to every proof is a hereditary small-ball estimate. This is a lower bound on the Gaussian measure of the spectral body $\{x\in \mathbb{R}^n:\|\sum_ix_iA_i\|\le R\}$ that holds for every subfamily of the matrices. The other ingredient turns that Gaussian measure into a partial signing. We give three approaches to obtain such a signing. The first one covers the cube by partially signed faces through Gaussian concentration with a constant $7\cdot10^9$. The second proof replaces the covering by a projection lemma with explicit parameters for a constant $156000$. The third proof turns Gaussian measure into signs by a lossless coding, with no union bound. It proves the estimate at the right radius with smooth spectral barriers and certified coefficients. It gives a constant below $7.88$. For algorithms, the main idea is to project Gaussian points onto a smoothed spectral body. The $n^{3+o(1)}$ time algorithm tracks the Gibbs matrix of that body across coordinate-descent steps with sketched increments and random refreshes.
