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.
Funding Statement
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.
Acknowledgment
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.
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
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