Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle

We consider the deviation inequalities for the sums of independent $d$ by $d$ random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension $d$, but rather on the effective rank. In an elementary and unified manner, we show the following results: 1) A deviation bound for the sums of independent positive-semi-definite matrices. This result complements the dimension-free bound of Koltchinskii and Lounici [Bernoulli, 2017] on the sample covariance matrix in the sub-Gaussian case. 2) A new bound for truncated covariance matrices that is used to prove a dimension-free version of the bound of Adamczak, Litvak, Pajor and Tomczak-Jaegermann [Journal Of Amer. Math. Soc., 2010] on the sample covariance matrix in the log-concave case. 3) Dimension-free bounds for the operator norm of the sums of random tensors of rank one formed either by sub-Gaussian or by log-concave random vectors. This complements the result of Gu\'{e}don and Rudelson [Adv. in Math., 2007]. 4) A non-isotropic version of the result of Alesker [Geom. Asp. of Funct. Anal., 1995] on the deviation of the norm of sub-exponential random vectors. 5) A dimension-free lower tail bound for sums of positive semi-definite matrices with heavy-tailed entries, sharpening the bound of Oliveira [Prob. Th. and Rel. Fields, 2016]. Our approach is based on the duality formula between entropy and moment generating functions. In contrast to the known proofs of dimension-free bounds, we avoid Talagrand's majorizing measure theorem, as well as generic chaining bounds for empirical processes. Some of our tools were pioneered by O. Catoni and co-authors in the context of robust statistical estimation.