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2021 Nonlinear matrix concentration via semigroup methods
De Huang, Joel A. Tropp
Electron. J. Probab. 26: 1-31 (2021). DOI: 10.1214/20-EJP578

Abstract

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $\ell _{2}$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical 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.

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De Huang. Joel A. Tropp. "Nonlinear matrix concentration via semigroup methods." Electron. J. Probab. 26 1 - 31, 2021. https://doi.org/10.1214/20-EJP578

Information

Received: 30 June 2020; Accepted: 25 December 2020; Published: 2021
First available in Project Euclid: 7 January 2021

Digital Object Identifier: 10.1214/20-EJP578

Subjects:
Primary: 46N30, 60B20
Secondary: 46L53, 60J25

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Vol.26 • 2021
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