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Geometry Seminar

Date: December 3, 2010

Time: 4:00PM - 5:00PM

Location: MILN 216

Speaker: Lek-Heng Lim, U. Chicago


Title: On the Geometry of Cumulants

Abstract: Gian-Carlo Rota famously said that "Even today, the statistical theory of cumulants wears a halo of mystery that we still are a long way from dispelling. We do not hesitate to predict that cumulants will soon be inserted in the mainstream of mathematics." That was in 1986 and Rota's prediction did not materialize --- cumulants are still as mysterious as they were a quarter century ago.

We would like to propose an explanation for this: Too much has been focus on the combinatorics of cumulants and too little on its geometry. In this talk, we would like to discuss the geometry underlying cumulants and examine two unusual ways to analyze cumulants akin to principal components analysis: (1) decomposing a homogeneous forms into a linear combination of powers of linear forms; (2) decomposing a symmetric hypermatrix into a multilinear combination of points on a Stiefel manifold. In the latter, one may identify "principal cumulant components" that simultaneously account for variations in all cumulants via optimization over a single Grassmannian.