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

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