The Grassmann distance complexity

TL;DR

This paper introduces Grassmann Distance Complexity (GDC), a measure of the difficulty in solving nearest point problems on Grassmannians, extending concepts from Euclidean Distance Degree to a more complex geometric setting.

Contribution

It defines GDC for subanalytic sets in Grassmannians, establishes fundamental properties, and provides bounds and conditions for critical points, including a nonlinear Eckart-Young theorem.

Findings

GDC quantifies complexity of nearest point problems in Grassmannians.

Bounds for GDC are established for algebraic varieties.

A nonlinear Eckart-Young theorem characterizes critical points.

Abstract

Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept quantifies the complexity of solving the nearest point problem for subanalytic sets in the Grassmannian, using the intrinsic Riemannian distance. Unlike the Euclidean case, the Grassmannian distance is neither smooth nor semialgebraic, and its study requires using Lipschitz critical point theory and o-minimal geometry. We establish fundamental properties of GDC, including computable bounds for real algebraic varieties and conditions ensuring the finiteness of critical points. Our results also include a nonlinear version of the classical Eckart-Young theorem, which characterizes critical points of the distance function from a generic $k$-plane to simple Schubert varieties.