# Euclidean distance degrees and nearest point problems

Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer. An alternative method is to determine the critical points of the objective function on the model.

In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model's Zariski closure is a topological invariant called the Euclidean Distance (ED) Degree.

In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover, I will describe a topological method for determining Euclidean distance degree and a numerical algebraic geometry approach for determining critical points of the squared Euclidean distance function.

*Jose Israel Rodriguez is a Van Vleck Visiting Professor at the University of Wisconsin - Madison. He received his graduate degree from UC Berkeley and has a wide range of interests, including algebraic statistics and numerical methods in algebraic geometry.*