Huang, De and Tropp, Joel A. (2021) From Poincaré inequalities to nonlinear matrix concentration. Bernoulli, 27 (3). pp. 1724-1744. ISSN 1350-7265. doi:10.3150/20-BEJ1289. https://resolver.caltech.edu/CaltechAUTHORS:20201218-154430753
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Abstract
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
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| Additional Information: | © 2021 ISI/BS. Received: 1 June 2020; Revised: 1 October 2020; Published: August 2021. First available in Project Euclid: 10 May 2021. Ramon Van Handel offered valuable feedback on a preliminary version of this work, and we are grateful to him for the proof of Proposition 2.4. 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. | ||||||||||
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| Subject Keywords: | concentration inequality, functional inequality, Markov process, matrix concentration, Poincaré inequality, semigroup | ||||||||||
| Issue or Number: | 3 | ||||||||||
| Classification Code: | 2010 Mathematics Subject Classification. Primary: 60B20, 46N30. Secondary: 60J25, 46L53 | ||||||||||
| DOI: | 10.3150/20-BEJ1289 | ||||||||||
| Record Number: | CaltechAUTHORS:20201218-154430753 | ||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20201218-154430753 | ||||||||||
| Official Citation: | De Huang. Joel A. Tropp. "From Poincaré inequalities to nonlinear matrix concentration." Bernoulli 27 (3) 1724-1744, August 2021. https://doi.org/10.3150/20-BEJ1289 | ||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||
| ID Code: | 107217 | ||||||||||
| Collection: | CaltechAUTHORS | ||||||||||
| Deposited By: | George Porter | ||||||||||
| Deposited On: | 21 Dec 2020 15:40 | ||||||||||
| Last Modified: | 09 Jun 2021 20:01 |
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