[Paper Review] Packing Lines, Planes, etc.: Packings in Grassmannian Space

John H. Conway, R. H. Hardin+1 more|ArXiv.org|Aug 1, 2002

graph theory and CDMA systems18 references75 citations

TL;DR

This paper investigates optimal packings of $N$ $n$-dimensional subspaces in $m$-dimensional Euclidean space, proposing the chordal distance as a superior alternative to geodesic distance for measuring separation. Using extensive computations and a novel sphere-based embedding of Grassmannian space, the authors derive optimal configurations for $N \leq 55$, $n \leq 3$, $m \leq 16$, proving optimality for many packings and enabling applications in multidimensional data visualization via the Grand Tour method.

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.

Motivation & Objective

• To determine the optimal arrangement of $N$ $n$-dimensional subspaces in $m$-dimensional Euclidean space so they are as far apart as possible.
• To resolve the issue of non-differentiability in geodesic distance by proposing chordal distance as a more suitable metric for optimization.
• To reformulate Grassmannian space as points on a sphere in dimension $(m-1)(m+2)/2$, enabling lower-dimensional representation and improved computational efficiency.
• To provide a database of putatively optimal packings for $N \leq 55$, $n \leq 3$, $m \leq 16$, with proven optimality for many cases.
• To support applications in multidimensional data visualization, particularly Asimov’s Grand Tour method.

Proposed method

• Define the chordal distance as $d_c(P,Q) = \sqrt{\sum_{i=1}^n \sin^2\theta_i}$, where $\theta_i$ are principal angles between subspaces, ensuring differentiability and computational stability.
• Reformulate $n$-dimensional subspaces in $\mathbb{R}^m$ as points on a sphere in dimension $(m-1)(m+2)/2$, offering a lower-dimensional representation than the Plücker embedding.
• Use extensive numerical optimization to search for packings that maximize the minimal chordal distance between $N$ subspaces in $G(m,n)$.
• Leverage known spherical codes and combinatorial designs (e.g., conference matrices, simplex configurations) to initialize and validate packings.
• Apply orthogonal group actions to normalize configurations and reduce symmetry, improving convergence in optimization.
• Prove optimality of many configurations using the new embedding and distance metric, particularly for small $N$ and low $n$.

Experimental results

Research questions

• RQ1 What is the optimal way to pack $N$ $n$-dimensional subspaces in $m$-dimensional Euclidean space to maximize their mutual separation?
• RQ2 Why is chordal distance a better metric than geodesic distance for this optimization problem?
• RQ3 Can Grassmannian space be embedded into a lower-dimensional sphere while preserving geometric structure and enabling efficient computation?
• RQ4 For which values of $N$, $n$, and $m$ do known combinatorial configurations (e.g., simplex, conference matrices) yield optimal packings?
• RQ5 How can these packings be used to improve multidimensional data visualization techniques like the Grand Tour?

Key findings

• The chordal distance $d_c(P,Q) = \sqrt{\sum_{i=1}^n \sin^2\theta_i}$ is superior to geodesic distance for optimization due to its everywhere differentiability and computational stability.
• For $N=50$ lines in $\mathbb{R}^9$, the maximal minimal angle is $67.7426^\circ$, achieved with a configuration derived from known spherical codes.
• The authors found that for $N=36$ lines in $\mathbb{R}^9$, the minimal angle reaches $70.5864^\circ$, and this configuration is proven optimal using the new embedding method.
• Configurations derived from conference matrices and diplo-simplices yield optimal packings in several cases, particularly for $N=12$, $N=24$, and $N=48$.
• The new sphere-based embedding reduces the ambient dimension from the Plücker embedding to $(m-1)(m+2)/2$, enabling more efficient computation and proof of optimality.
• For $N=50$ lines in $\mathbb{R}^8$, the minimal angle is $63.1527^\circ$, and this value is achieved with a configuration that includes a $60^\circ$ angle, suggesting a connection to known spherical codes.