Nonlinear matrix concentration via semigroup methods
| Autor: | Joel A. Tropp, De Huang |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Pure mathematics Matrix (mathematics) 60J25 matrix concentration FOS: Mathematics Bakry–Émery criterion Markov process Concentration inequality 46L53 local Poincaré inequality Ricci curvature Mathematics 60B20 Semigroup 46N30 concentration inequality Probability (math.PR) Riemannian manifold Lipschitz continuity functional inequality semigroup Mathematics::Differential Geometry Statistics Probability and Uncertainty Primary: 60B20 46N30. Secondary: 60J25 46L53 Operator norm Random matrix Mathematics - Probability |
| Zdroj: | Electron. J. Probab. |
| Popis: | Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. In particular, it is shown that the classic Bakry-\'Emery 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. |
| Databáze: | OpenAIRE |
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