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.
- Topics:
- Tight Frames
Download: pdf
Citation
- Constructing Packings in Grassmannian Manifolds via Alternating Projections (pdf, software)
I. Dhillon, R. Jr., T. Strohmer, J. Tropp.
Experimental Mathematics 17(1), pp. 9-35, 2008.
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