Shapes of Gaussian Mixture Models
joint work with Herbert Edelsbrunner and Günter Rote.
Extended Abstract: SoCG 2012
To appear in: DCG
It has been an open question whether the sum of finitely many isotropic
Gaussian kernels in n >=2 dimensions can have more modes than kernels,
until in 2003 Carreira-Perpinan and Williams exhibited
n+1 isotropic Gaussian kernels in R^n with n+2 modes.
We give a detailed analysis of this example,
showing that it has exponentially many critical points and
that the resilience of the extra mode grows like the square root of the dimension.
In addition, we exhibit finite configurations of isotropic
Gaussian kernels with superlinearly many modes.
The Difference of Lengths of Curves in R^n
Acta Sci. Math. (Szeged)
We bound the difference in length of two curves in terms
of their total curvatures and the Fréchet distance.
The bound is independent of the dimension of the ambient
Euclidean space, it improves upon a bound by Cohen-Steiner
and Edelsbrunner, and it generalizes a result by
Prelim: Discovering Metrics and Scale Space
adviser: Dr. Herbert Edelsbrunner
26 July 2010
Researchers across many fields are interested in the topological
and geometric properties of shapes in Euclidean space.
We describe three open questions related to metrics and scale space.
First, we present a result that bounds the difference of lengths of curves
in Euclidean space by a function of the total curvature and the Fréchet
distance between the curves. Although this result improves upon another
inequality that additionally depends on the dimension in which the curves
are embedded, it is still unknown if the new inequality is tight. Second,
we explore the deep structure, or scale space, of images. The scale space
of an image is a family of related images obtained through convolution
with the Gaussian kernel. We ask how to extend this idea to the study of
a more general class of functions. Finally, we introduce the
vineyard--a collection of curves in R^3--associated with a homotopy.
In particular, we use the heat equation homotopy, which arises from scale
space, to measure the distance between two functions. We ask how to
quantify the relationship between this metric and other metrics between
Research Initiation Project
adviser: Dr. Herbert Edelsbrunner
Persistence homology is a tool used to classify topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. In this project, I am concerned with using persistence to characterize the difference between two functions f, g : S R, where S is a topological space embedded in R^n. Furthermore, I assume that we have a homotopy between the functions f and g. By stacking the persistence diagrams, we create a vineyard of curves that connect the points in the diagram for f with the points in the points in the diagram for g. The choice of homotopy will affect the resulting vineyard. I investigated the changing behavior of the persistence diagrams when the heat equation is used to create a homotopy between the functions f and g.
adviser: Dr. Samuel Smith
A basic problem in mathematics is the classification problem. In group theory we have the Classification Problem for Groups: Given a collection G of groups, classify the groups G in G up to isomorphism. And in homotopy theory, we have the Homotopy Classification Problem for Spaces: Given a collection T of topological spaces, classify the spaces X in T up to homotopy equivalence.
We show that in some cases, these classification problems are actually equivalent. In addition, we give some examples classifying groups by the centralizers and classifying the path components of function spaces.
Computer Science Education
adviser: Dr. Stephen Cooper
I created virtual worlds using a 3D interactive animation environment,
Alice. This environment had
previously been used as a program visualization tool for introductory computer
science classes. The
worlds I created enable use of this tool at the intermediate programming level by
introducing the concepts of lists and arrays. This research
has provoked interest in expanding the use of Alice to higher level CS