  We study the deviation inequality for a sum of high-dimensional random
matrices and operators with dependence and arbitrary heavy tails. There is an
increase in the importance of the problem of estimating high-dimensional
matrices, and dependence and heavy-tail properties of data are among the most
critical topics currently. In this paper, we derive a dimension-free upper
bound on the deviation, that is, the bound does not depend explicitly on the
dimension of matrices, but depends on their effective rank. Our result is a
generalization of several existing studies on the deviation of the sum of
matrices. Our proof is based on two techniques: (i) a variational approximation
of the dual of moment generating functions, and (ii) robustification through
truncation of eigenvalues of matrices. We show that our results are applicable
to several problems such as covariance matrix estimation, hidden Markov models,
and overparameterized linear regression models.
