Variational Principle and applications to covariance estimates.

The goal of this presentation is to give an overview of the recent paper "Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle", by Nikita Zhivotovskiy. Motivated by the classical empirical covariance estimation problem, he introduces a new method to bound the supremum of random processes, using the so-called Duality Formula between the Kullback-Leibler divergence and the log-Generating Function. As a result, he improves the classical covariance estimates via the effective rank of the covariance matrix. If time permits, we will also state the second result of this paper, which concerns random tensors and the log-concave case.