Principal Cumulant Component Analysis

Multivariate Gaussian data is completely characterized by its mean and covariance, yet modern non-Gaussian data makes higher-order statistics such as cumulants inevitable. For univariate data, the third and fourth scalar-valued cumulants are relatively well-studied as skewness and kurtosis. 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 principal cumulant components that account for most of the variation in all higher-order cumulants, just as PCA obtains varimax components. An efficient algorithm based on limited-memory quasi-Newton maximization over a Grassmannian, using only standard matrix operations, may be used to find the principal cumulant components. Numerical experiments include forecasting higher portfolio moments and image dimension reduction.