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Title: Reverse VC calculations: using
statistics in model theory
Dr. David Lippel
Haverford College, Haverford, PA
Abstract: Let F be a family of sets; for example, F is the family
of all triangles in the plane, or the family of all spheres in space
(or, more exotically, a semi-algebraic family in p-adic n-space).
The Vapnik-Chervonenkis (VC) dimension of F is a measure of
the combinatorial complexity of F. Once you know the VC
dimension of F, theorems from computational geometry, like
the Epsilon-Net Theorem, give nice geometric consequences
for F. After introducing VC dimension and the Epsilon-Net
Theorem, I will discuss a statistical strategy for reversing the
flow of information in this theorem. Instead of starting with
knowledge of the VC dimension, we merely hypothesize
"dimension=d" for some value d. Then, we observe the
geometric behavior of F using computer experiments and
compare the observed behavior with the behavior that is
predicted by the theorems (under the hypothesis
"dimension=d"). If our observed results have sufficiently low
probability (conditioned on "dimension=d"), then we can reject
the hypothesis "dimension=d" with a high degree of
confidence. Ultimately, we hope to use such methods to shed
light on conjectures about VC dimension (more properly, VC
density) in both the reals and the p-adics.
This project is joint work with Deirdre Haskell and Nigel PynnCoates.
Department of
February 28th,
3:45 p.m.
204 Morgan Hall
Refreshments will be
served at 3:30 p.m.