Session: Technical Session II

Date & Time: Friday, May 22 — 1:30 PM–3:45 PM

Talk Title

Regressions in Covariances, Dependencies and Graphs

Abstract

We present an overview of the role of regression as a unifying theme in modeling dependencies using covariates and regularization. The plan is to mimic the success of regressions in modeling the mean vector (first moments) and achieve almost the same for the covariance matrix (second moments) of a random vector and eventually its distribution. It is recognized that the road to modeling covariances using covariates is bumpier and more perilous than that for the mean; positivity and the large number of parameters are the two major obstacles. Two directions that regression can help are the covariance regression or simply writing a covariance matrix as linear combinations of known matrices (covariates), and the hidden regression idea which makes it possible to view columns of a matrix or its transform as parameters of certain regressions and estimate them iteratively one column at a time. The latter is at the heart of the regularized Gaussian likelihood or the graphical Lasso algorithm which does not accommodate covariates. While covariances, their inverses and graphs have played significant roles in exploring linear dependencies for a long time, only lately copulas have assumed similar roles in modeling dependence in the presence of nonlinearity, extreme and long-tail behavior. The importance of the order among the variables in proposing non-iterative, scalable algorithms with guaranteed positive-definiteness for sparse covariance estimation will be pointed out.