Constructing packings in Grassmannian manifolds via alternating projection

TL;DR

This paper introduces a versatile numerical method using alternating projections to construct high-quality packings in Grassmannian and projective spaces, with applications in communications and near-optimal configurations.

Contribution

It presents a new iterative approach for constructing packings that often outperform existing methods and extends to previously unstudied problems in Grassmannian spaces.

Findings

Produces packings competing with the best recorded solutions.

Extends to packings in real and complex Grassmannian spaces.

Some configurations are nearly optimal in packing diameter.

Abstract

This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini--Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.