发表机构
ETH Zürich; Cheriton School of Computer Science, University of Waterloo; Johns Hopkins University（苏黎世联邦理工学院; 滑铁卢大学切里顿计算机科学学院; 约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种通过Fock空间上确定性算子范数控制来证明矩阵浓度不等式的方法，获得非交换Khintchine不等式的强化结果，并统一处理多种非交换随机变量模型。
AI 中文摘要
我们开发了一种证明矩阵浓度不等式的方法，该方法通过将随机矩阵与作用在适当Fock空间上的相关确定性算子进行识别，并控制这些算子及其在低阶子空间上的限制的范数。应用此方法，我们获得了非交换Khintchine不等式的新强化结果，改进了现有技术水平，特别是锐化了Bandeira、Boedihardjo和van Handel（2023）最近关于量化随机矩阵内在自由度的不等式。我们这些结果的证明基于相对简单的算子代数论证，既不涉及高斯插值，也不涉及迹矩的显式组合。此外，我们的技术同样适用于文献中感兴趣的几种非交换随机变量模型，例如由$q$-高斯和$\Gamma$-独立算子系统构造的算子级数，用相同的方法处理所有这些对象。我们获得了这些算子的新范数界，既包括非交换Khintchine不等式风格的，也包括Lehner算子范数公式风格的。
英文摘要
We develop a method for proving matrix concentration inequalities by identifying random matrices with associated deterministic operators acting on suitable Fock spaces and controlling norms of these operators and their restrictions to low-order subspaces. Applying this method, we obtain new strengthenings of the non-commutative Khintchine inequality that improve on the state of the art, in particular sharpening recent inequalities due to Bandeira, Boedihardjo, and van Handel (2023) quantifying intrinsic freeness of random matrices. Our proofs of these results are based on relatively simple operator algebra arguments and involve neither Gaussian interpolation nor explicit combinatorics of tracial moments. Further, our techniques apply equally well to several models of non-commutative random variables of interest in the literature, such as operator series constructed from $q$-Gaussian and $Γ$-independent systems of operators, treating all of these objects with the same method. We obtain new norm bounds for such operators both in the style of the non-commutative Khintchine inequality and in the style of Lehner's operator norm formula.
Comments31 pages
