The Euclidean distance degree of an algebraic variety

TL;DR

This paper develops a computational algebraic geometry framework to analyze the Euclidean distance degree of algebraic varieties, quantifying the number of critical points of the squared distance function for generic points.

Contribution

It introduces a theoretical approach and computational tools for determining the Euclidean distance degree of varieties, with applications to low rank matrices and related problems.

Findings

The Euclidean distance degree counts critical points of squared distance functions.

Tools for exact computation of Euclidean distance degree are developed.

Application to varieties like low rank matrices demonstrates practical utility.

Abstract

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.