Principal Components of Cumulants

Monday, March 26, 2012 - 3:30pm - 4:15pm
Keller 3-180
Lek-Heng Lim (University of Chicago)
Multivariate Gaussian data are completely characterized by their mean and
covariance but higher-order cumulants are unavoidable in non-Gaussian
data. For univariate data, these are well-studied via 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. We argue that multivariate cumulants may be
studied via their principal components, defined in a manner analogous to
the usual principal components of a covariance matrix. It is best viewed
as a subspace selection method that accounts for higher-order dependence
the way PCA obtains varimax subspaces. A variant of stochastic gradient
descent on the Grassmannian permits us to estimate principal components of
cumulants of any order in excess of 10,000 dimensions readily on a laptop
computer. This is joint work with Jason Morton.
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