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Published August 2021 | Submitted
Journal Article Open

From Poincaré inequalities to nonlinear matrix concentration


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.

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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August 20, 2023
August 20, 2023