Ph. D. Defense

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

Speaker:Brittany Fasy
brittany at
Date: Monday, June 11, 2012
Time: 12:00pm - 2:00pm
Location: North 311, Duke
Carlo Tomasi, John Harer, Hubert Bray


This dissertation studies high-dimensional problems from a low-dimensional perspective. First, we bound the difference between two rectifiable curves in high-dimensional space by using the Fréchet distance between and total curvatures of the two curves. We create this bound by mapping the curves into R^3 while preserving the lengths of the curves and increasing neither the total curvature of the curves nor the Fréchet 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 generalizes a result by Fáry and Chakerian.

In the second half of the dissertation, we analyze Gaussian mixtures. In particular, we consider the sum of n + 1 identical isotropic Gaussians, where each Gaussian is centered at the vertex of a regular n-simplex. We prove that all critical points are located on one-dimensional lines (axes) connecting barycenters of complementary faces of the simplex. Fixing the width of the Gaussians 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. We analyze these modes using the one-dimensional axes that contain the critical points. We see that the extra mode created is subtle, but becomes more pronounced as the dimension increases.

Advisor(s): Herbert Edelsbrunner