Modes of Gaussian Mixtures and an Inequality for the Distance Between Curves in Space

Abstract

This dissertation studiess high dimensional problems from a low dimensional perspective. First, we explore rectifiable curves in high-dimensional space by using the Fréchet distance between and total curvatures of the two curves to bound the difference of their lengths. We create this bound by mapping the curves into R^2 while preserving the length between the curves and increasing neitherthe total curvature of the curves nor the Fr\'echet distance between them. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner for dimensions greater than three and it generalizesa result by F\'ary and Chakerian.In the second half of the dissertation, we analyze Gaussian mixtures. In particular, we consider the sum of n Gaussians, where each Gaussian is centered at the vertex of a regular n-simplex. Fixing the width of the Guassians and varying the diameter of the simplex from zero to infinity by increasing a parameter that we call the scale factor, we find the window of scale factors for which the Gaussian mixture has more modes, or local maxima, than components of the mixture.We see that the extra mode created is subtle, but can be higher than the modes closer to the vertices of the simplex. In addition, we prove that all critical points are located on a set of one-dimensional lines (axes) connecting barycenters of complementary faces ofthe simplex.