Published October 15, 2019
| Submitted
Journal Article
Open
A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices
- Creators
- Huang, De
Chicago
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Abstract
In this paper we prove the concavity of the k-trace functions, A↦(Tr_k[exp(H+lnA)])^(1/k), on the convex cone of all positive definite matrices. Tr_k[A] denotes the k_(th) elementary symmetric polynomial of the eigenvalues of A. As an application, we use the concavity of these k-trace functions to derive tail bounds and expectation estimates on the sum of the k largest (or smallest) eigenvalues of a sum of random matrices.
Additional Information
© 2019 Published by Elsevier. Received 30 November 2018, Accepted 13 June 2019, Available online 17 June 2019. The research was in part supported by the NSF Grant DMS-1613861. The author would like to thank Joel A. Tropp for providing deep insights and rich materials in theories of random matrices and multilinear algebra. The author also gratefully appreciates the inspiring discussions with Thomas Y. Hou, Florian Schaefer, Shumao Zhang and Ka Chun Lam during the development of this paper. The kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where the early ideas of this work started, is gratefully acknowledged.Attached Files
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Additional details
- Eprint ID
- 96487
- Resolver ID
- CaltechAUTHORS:20190617-153352760
- NSF
- DMS-1613861
- Created
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2019-06-17Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field