Packing Lines, Planes, etc.: Packings in Grassmannian Space

TL;DR

This paper explores optimal arrangements of subspaces in Euclidean space using computational methods and a new sphere-based reformulation, with applications to data visualization.

Contribution

It introduces a novel sphere-based reformulation for packing subspaces, enabling the identification of many optimal packings and improving visualization techniques.

Findings

Computed new packings for modest N, n, m values

Reformulated the packing problem using sphere representations

Proved many packings are optimal

Abstract

This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension (m-1)(m+2)/2, which provides a (usually) lower-dimensional representation than the Pluecker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's "Grand Tour" method.