Felix Rydell: The Infinite ED Discriminant

Abstract

Given an algebraic variety and a data point, the Euclidean distance problem is to find the closest point on the variety. It is known that to each generic data point, the number of critical points associated to the optimization problem is fixed and finite; this number is called the Euclidean distance degree. The set of data points where the number of critical points differs from the Euclidean distance degree is a variety called the ED discriminant. The data points that have infinitely many critical data points is a subvariety of the ED discriminant called the infinite ED discriminant. In this talk we explore how to compute this object using computer algebra systems and we classify what surfaces in three-dimensional space have a one-dimensional infinite ED discriminant. This talk is based on joint work Emil Horobet.