Modeling Higher-Order Dependence With Cumulant Tensors

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.