This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses mathematical statistics wide range settings, from distribution-free to distribution-dependent, sub-Gaussian sub-exponential, sub-Gamma, and sub-Weibull random variables, the mean maximum concentration. provides results these settings with some fresh new results. Given increasing popula...

Now we come to one of the most important and inspiring realizations of freeness. Up to now we have realized free random variables by generators of the free group factors or by creation and annihilation operators on full Fock spaces. This was not very surprising because our definition of freeness was modelled according to such situations. But there are objects from a quite different mathematical...

Given n (discrete or continuous) random variables Xi, the (2 n − 1)-dimensional vector obtained by evaluating the joint entropy of all non-empty subsets of {X1,. .. , Xn} is called an entropic vector. Determining the region of entropic vectors is an important open problem with many applications in information theory. Recently, it has been shown that the entropy regions for discrete and continuo...

We consider the composite log-determinant optimization problem, arising from the `1 regularized Gaussian maximum likelihood estimator of a sparse inverse covariance matrix, in a high-dimensional setting with a very large number of variables. Recent work has shown this estimator to have strong statistical guarantees in recovering the true structure of the sparse inverse covariance matrix, or alt...

Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, called Gaussian free fields, which arise in semi-supervised learning and computer ...