TY - JOUR

T1 - Matrix Poincaré inequalities and concentration

AU - Aoun, Richard

AU - Banna, Marwa

AU - Youssef, Pierre

PY - 2020/9/16

Y1 - 2020/9/16

N2 - 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.

AB - 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.

KW - Functional inequalities

KW - Matrix Poincaré inequalities

KW - Matrix concentration inequalities

KW - Matrix inequalities

KW - Strong Rayleigh measures

UR - http://www.scopus.com/inward/record.url?scp=85085889296&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85085889296&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2020.107251

DO - 10.1016/j.aim.2020.107251

M3 - Article

AN - SCOPUS:85085889296

SN - 0001-8708

VL - 371

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107251

ER -