Nearest Points on Toric Varieties

TL;DR

This paper computes the Euclidean distance degree of projective toric varieties and develops algorithms to find the closest points on these varieties to given data points, extending existing formulas for $A$-discriminants.

Contribution

It extends the formula for the degree of the $A$-discriminant to compute the Euclidean distance degree of projective toric varieties and introduces algorithms for nearest point computation.

Findings

Derived a formula for the Euclidean distance degree of toric varieties.

Developed reliable algorithms for nearest point computation on real toric varieties.

Extended Matsui and Takeuchi's formula to broader classes of varieties.

Abstract

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.