Ph. D. Defense
Modes of Gaussian Mixtures and an Inequality for the Distance between Curves in Space
Speaker:  Brittany Fasy
brittany at cs.duke.edu 
Date: 
Monday, June 11, 2012 
Time: 
12:00pm  2:00pm 
Location: 
North 311, Duke 

Carlo Tomasi, John Harer, Hubert Bray 


Abstract
This dissertation studies highdimensional problems from a lowdimensional perspective. First, we bound the diﬀerence between two rectiﬁable curves in highdimensional 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 CohenSteiner 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 nsimplex. We prove that all critical points are located on onedimensional 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 onedimensional 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