The numerical algebraic geometry of bottlenecks

TL;DR

This paper develops a numerical homotopy method to compute bottlenecks on algebraic varieties, which are critical for understanding geometric properties like reach and have applications in optimization and computer graphics.

Contribution

It introduces an optimal-path homotopy for approximating all isolated bottlenecks on smooth algebraic varieties, with bounds related to Euclidean distance degree.

Findings

Homotopy method efficiently computes bottlenecks

Bounds on bottleneck count in terms of Euclidean distance degree

Applications in distance computation and reach estimation

Abstract

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety $X \subseteq \mathbb{C}^n$ are the lines in $\mathbb{C}^n$ which are normal to $X$ at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini and Macaulay2.