Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.0535

Constructing packings in Grassmannian manifolds via alternating projection

I. S. Dhillon·R. W. Heath Jr·T. Strohmer·J. A. Tropp
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

I. S. Dhillon, R. W. Heath Jr, T. Strohmer, J. A. Tropp. 2007-09-04. Constructing packings in Grassmannian manifolds via alternating projection. https://arxiv.org/abs/0709.0535

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pullback theorem and rigidity for Sobolev mappings on Carnot groups

This paper establishes a pullback theorem via mollification to extend the rigidity theory of Sobolev mappings between Carnot groups to the low-integrability regime where the Sobolev exponent $p$ is less than the homogeneous dimension $ν_1$ of the source group. The core technical achievement is the rigorous analysis of the convergence of mollified approximations $f_\varepsilon$ for a mapping $f \in W^{1,p}$, demonstrating that the pullbacks of left-invariant differential forms converge appropriately. This allows for the recovery of more properties of Pansu differentiable mappings. The main results are: (1) A generalization of the rigidity theorem to the range $p<ν_1$ for $f\in W^{1,p}$. (2) When $p>Q$, such mappings are shown to be locally Hölder continuous with exponent $1-\frac{Q}{p}$, where $p>Q$ and $Q=\max \left\{\text{homogeneous dim of}\ G_i\right\}$. (3) with the stratified structure of contact Sobolev mappings, we generalize the non-embedding theorem to contact Sobolev mappings.

math.MG↗
arXiv2026

Coarse geometry of metric measure spaces

Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence for metric measure spaces and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called weighted $\ell^\infty$ homology, for large-scale doubling metric measure spaces. We prove that this homology is invariant under measured coarse equivalence and that the vanishing of its zeroth homology is equivalent to weighted non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures.

math.MG↗
arXiv2026

No Prime Tiling of an Isosceles Triangle

No isosceles triangle can be cut into a prime number (greater than three) of congruent triangles, and only an equilateral triangle can be cut into three congruent triangles.

math.MG↗