Events for 04/26/2021 from all calendars

Geometry Seminar

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Frank Sottile, Texas A&M University

Title: Euclidean distance degree of hypersurfaces via mixed volume

Abstract: The Euclidean distance degree (EDD) of a variety X in Rⁿ measures the algebraic complexity of computing the point of X closest to a general point u in Rⁿ . It is the number of critical points of the complexified distance function from u to X . Known formulas involve polar classes of the conormal variety to X or Chern classes of X.

 In this talk, I will discuss formulas of a different character, when X is a hypersurface whose defining equation is general given its Newton polytope. In this case, the EDD is shown to be the mixed volume of the critical point equations. This uses Bernstein's Other Theorem, which is of independent interest. We give an interesting closed formula for the EDD when the Newton polytope is a rectangular parallelepiped. This is joint work with Paul Breiding and James Woodcock.

Industrial and Applied Math

Time: 5:00PM - 6:00PM

Location: Zoom

Speaker: Dr. Boris Hanin, Princeton University

Title: Optimization and Generalization in Overparameterized Models

Abstract: Modern machine learning models, such as neural networks, have a number of theoretically puzzling but empirically robust properties. Chief among them are: (a) neural networks are trained on datasets which are much smaller than the total number of model parameters; (b) training proceeds by empirical risk minimization via a first order method from a random starting point and, despite the non-convexity of the risk, typically returns a global minimizer; (c) this minimizer of the risk not only fits interpolates the data precisely but also performs well on unseen data (i.e. generalizes). The purpose of this talk is to introduce these fascinating properties and give some basic intuitions for why they might be possible. The emphasis will be on heuristics rather than on precise theorems.