论文标题
非线性基质浓度通过半群方法
Nonlinear Matrix Concentration via Semigroup Methods
论文作者
论文摘要
矩阵浓度不等式提供了有关随机矩阵与$ L_2 $运算符规范相关的可能性的信息。本文使用半群方法来得出尖锐的非线性基质不等式。特别是，这表明经典的Bakry-émery曲率标准意味着“矩阵Lipschitz”功能的Subgaussian浓度。该论点规避了需要开发log-sobolev不平等的矩阵版本，这是一种技术障碍，它阻止了以前在这种情况下推导矩阵浓度不平等的尝试。该方法统一并扩展了先前的大部分矩阵浓度工作。当应用于产品度量时，该理论将重现了由于Paulin等人而引起的基质Efron-Stein不平等。它还处理具有均匀阳性RICCI曲率的Riemannian歧管上的基质值函数。
Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. In particular, it is shown that the classic Bakry-Émery curvature criterion implies subgaussian concentration for "matrix Lipschitz" functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron-Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.
