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Nonlinear matrix concentration via semigroup methods

Huang, De and Tropp, Joel A. (2021) Nonlinear matrix concentration via semigroup methods. Electronic Journal of Probability, 26 . Art. No. 8. ISSN 1083-6489.

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Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the ℓ₂ 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.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Huang, De0000-0003-4023-9895
Tropp, Joel A.0000-0003-1024-1791
Additional Information:Creative Commons Attribution 4.0 International License. Submitted to EJP on June 30, 2020, final version accepted on December 25, 2020. We thank Ramon van Handel for his feedback on an early version of this manuscript. He is responsible for the observation and proof that matrix Poincaré inequalities are equivalent with scalar Poincaré inequalities, and we are grateful to him for allowing us to incorporate these ideas. We appreciate the thoughtful feedback from the anonymous referees, which has helped us streamline this paper. DH was funded by NSF grants DMS-1907977 and DMS-1912654. JAT gratefully acknowledges funding from ONR awards N00014-17-12146 and N00014-18-12363, and he would like to thank his family for their support in these difficult times.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-17-12146
Office of Naval Research (ONR)N00014-18-12363
Subject Keywords:Bakry–Émery criterion; concentration inequality; functional inequality; Markov process; matrix concentration; local Poincaré inequality; semigroup
Classification Code:2010 Mathematics Subject Classification. Primary: 60B20, 46N30. Secondary: 60J25, 46L53
Record Number:CaltechAUTHORS:20201218-154427353
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Official Citation:Huang, De; Tropp, Joel A. Nonlinear matrix concentration via semigroup methods. Electron. J. Probab. 26 (2021), paper no. 8, 31 pp. doi:10.1214/20-EJP578.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107215
Deposited By: George Porter
Deposited On:21 Dec 2020 15:42
Last Modified:04 Feb 2021 19:03

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