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