Dr. Zosso's "Studying the Shape of Data"
- Monday, April 16, 2018 from 4:10pm to 5:00pm
- Wilson Hall, 1-144 - view map
In this talk I will rst provide a lay-man's random walk from geometric measure
theory to integral geometry of convex bodies, loosely based on Gian-Carlo Rota's 1997 AMS
colloquium lecture. An important result is the following: the volume of a convex body A
thickened (dilated) by p is a polynomial in p (Steiner formula). Its coefficients are called
Minkowski functionals, mixed volumes, or Quermassintegrals (for different normalizations);
they can be identified with properties such as volume, perimeter, mean width, etc. of the
initial body, and fully characterize A from a geometric probability point of view.
I will then sketch how we now would like to use well-established extensions of the above to
the convex ring (finite unions of convex bodies) as a starting point for further inquiry into
studying the shape of data: Consider a data set in the form of a point cloud in some poten-
tially high-dimensional feature space, where individual points are samples from a nice un-
derlying structure that we are trying to uncover (think: samples from a donut in 3-space--
gotta have this donut). We can generate a finite union of convex bodies parametrized by
that data set by associating a ball of radius r with each data point. By estimating the
volume (with multiplicities) of this ball-cloud, for different thickenings p, we can determine
the Quermassintegrals through polynomial fitting (at scale r).
We anticipate that such a tool will be useful in generic data science as well as in some
concrete low-dimensional "shape-from-samples" problems, such as the shape description of
biolms or the characterization of percolation.