Title: Algebraic Models for Multilinear Dependence

Speaker: Jason Morton, Stanford University

Date/Time: Thursday, April 16, 2009, 9:00 am - 10:00 am (PST), 10:00 am - 11:00 am (MST)

Location: Building 915, Room S101 (CA), video conferenced to NM, Building CSRI, Room 95

Brief Abstract: Gaussian data is completely characterized by its mean and covariance.

However, modern data are almost always non-Gaussian and higher-order statistics such as cumulants are inevitable. For univariate data, the third and fourth cumulants are scalar-valued and relatively well-studied as skewness and kurtosis. But for multivariate data, these cumulants are tensor-valued, higher-order analogs of the covariance matrix capturing higher-order dependence in the data. In addition to their relative obscurity, there are few effective methods for analyzing these cumulant tensors. We propose a technique along the lines of Principal Component Analysis and Independent Component Analysis to analyze multivariate, non-Gaussian data motivated by the multilinear algebraic properties of cumulants. Our method relies on finding the principal cumulant components that account for most of the variation in all higher-order cumulants, in the same manner PCA obtains varimax components for Gaussian data. Estimation is performed by maximization over a Grassmannian, using only standard matrix operations.

Applications include multi-moment portfolio optimization and image dimension reduction.

CSRI POC: Heidi Ammerlahn, (925) 294-2234 and Tammy Kolda, (925) 294-4769