Defect of Euclidean distance degree

TL;DR

This paper investigates the gap between two invariants measuring the algebraic complexity of nearest point problems on complex projective varieties, introducing a new method to compute this defect using Singularity Theory.

Contribution

It introduces a novel approach to compute the defect of Euclidean distance degree for smooth complex projective varieties using classical Singularity Theory techniques.

Findings

Computed the defect of ED degree for certain varieties.

Established a new method for calculating ED degrees.

Provided insights into the algebraic complexity of optimization problems.

Abstract

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering, statistics, and data science, have a significant gap between these two numbers. We call this difference the defect of the ED degree of an algebraic variety. In this paper we compute this defect by classical techniques in Singularity Theory, thereby deriving a new method for computing ED degrees of smooth complex projective varieties.