Modeling Higher-Order Dependence With Cumulant Tensors

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The second cumulant tensor of a multivariate distribution is itscovariance matrix, which provides a partial description of itsdependence structure (complete in the Gaussian case). Innumerablesuccessful statistical methods are based on analyzing the covariancematrix, e.g. imposing rank restrictions as in principal componentanalysis or zeros in its inverse as in Gaussian graphical models.Moreover, the covariance matrix plays a critical role in optimizationin finance and other areas involving optimization of risky payoffs,since it is the bilinear form yielding the variance of a linearcombination of variables. For multivariate, non-Gaussian data, thecovariance matrix is an incomplete description of the dependencestructure. Cumulant tensors are the multivariate generalization ofunivariate skewness and kurtosis and the higher-order generalizationof covariance matrices, and allow a more complete description ofdependence. The research investigates a number of problems in theory,estimation, algorithms, and applications around modeling higher-ordernon-Gaussian dependence with cumulant tensors.Data arising from modern applications like computer vision, finance,and computational biology are rarely well described by a normaldistribution, though analysis often proceeds as if they were. Forexample, one cause of the financial crisis and the damage it did tomany investors was an over-reliance on the variance-based riskmeasures appropriate primarily for normal distributions. This canallow risk to be in a sense hidden in the higher-order structure,where it can be ignored or even made worse by application oftraditional risk metrics. Cumulant tensors provide a promising avenuefor modeling higher-order dependence. Success in developing thesemodels will have broad impacts in the analysis of real-world data withcomplex dependence, particularly in modeling and managing financialrisk and in dimension reduction, and could help improve the robustnessof parts of the financial system.