Experimental Mathematics

Constructing Packings in Grassmannian Manifolds via Alternating Projection

I. S. Dhillon, R. W. Heath, Jr., T. Strohmer, and J. A. Tropp

Full-text: Open access

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 in which 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.

Article information

Source
Experiment. Math., Volume 17, Issue 1 (2008), 9-35.

Dates
First available in Project Euclid: 18 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.em/1227031894

Mathematical Reviews number (MathSciNet)
MR2410113

Zentralblatt MATH identifier
1155.52304

Subjects
Primary: 51N15: Projective analytic geometry 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31]

Keywords
Combinatorial optimization packing projective spaces Grassmannian spaces Tammes’ Problem

Citation

Dhillon, I. S.; Heath, R. W.; Strohmer, T.; Tropp, J. A. Constructing Packings in Grassmannian Manifolds via Alternating Projection. Experiment. Math. 17 (2008), no. 1, 9--35. https://projecteuclid.org/euclid.em/1227031894


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