Euclidean distance degree defect of singular projective varieties

TL;DR

This paper introduces a topological formula to compute the defect in Euclidean distance degrees of complex projective varieties, extending previous smooth case results and aiding in algebraic complexity analysis.

Contribution

It provides a new constructible and topological approach to calculate the ED degree defect for any complex projective variety, broadening applicability.

Findings

Derived a topological formula for ED degree defect

Extended previous smooth case results to singular varieties

Facilitated broader computation of ED degrees in applications

Abstract

The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.