Dr. Zosso's "Studying the Shape of Data"
When:
 Monday, April 16, 2018 from 4:10pm to 5:00pm
Where:
 Wilson Hall, 1144  view map
Description:

In this talk I will rst provide a layman's random walk from geometric measure
theory to integral geometry of convex bodies, loosely based on GianCarlo 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 wellestablished 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 highdimensional 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 3space
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 ballcloud, 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 lowdimensional "shapefromsamples" problems, such as the shape description of
biolms or the characterization of percolation. Contact:
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