Principal Components of Cumulants

• Activity Number: 587
• Type: Invited
• Date/Time: Thursday, August 2, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Lek-Heng Lim*+ and Jason Morton
• Companies: The University of Chicago and Penn State University
• Keywords: cumulants ; principal components ; skewness ; kurtosis ; symmetric tensors ; PCA

Abstract

Multivariate Gaussian data are completely characterized by their mean and covariance but higher-order cumulants are unavoidable in non-Gaussian data. For univariate data, cumulants are well-studied as skewness and kurtosis. For multivariate data, they are tensor-valued analogs of covariance matrix capturing higher-order dependence. We shall argue that multivariate cumulants should be analyzed via their principal components -- natural generalizations of the usual principal components of a covariance matrix and best viewed as a subspace selection method based on higher-order interactions the way PCA obtains varimax subspaces. Stochastic gradient descent on Grassmannians permits us to estimate principal subspaces of cumulants of very high dimensions.

Gian-Carlo Rota famously said that "Even today, the statistical theory of cumulants wears a halo of mystery that we still are a long way from dispelling. We do not hesitate to predict that cumulants will soon be inserted in the mainstream of mathematics." That was in 1986 and Rota's prediction did not materialize -- cumulants are still as mysterious. We hope our approach would be a step towards dispelling this mystery.