Matrix Poincaré inequalities and concentration

Richard Aoun, Marwa Banna, Pierre Youssef

Research output: Contribution to journalArticlepeer-review

Abstract

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

Original languageEnglish (US)
Article number107251
JournalAdvances in Mathematics
Volume371
DOIs
StatePublished - Sep 16 2020

Keywords

  • Functional inequalities
  • Matrix Poincaré inequalities
  • Matrix concentration inequalities
  • Matrix inequalities
  • Strong Rayleigh measures

ASJC Scopus subject areas

  • Mathematics(all)

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