Packing Lines, Planes, etc.: Packings in Grassmannian Spaces

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 Plücker 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.

Volume Growth and General Rate Quantization on Grassmann Manifolds

“…Most works discuss the equal dimensional quantization (p = q) only. The Rankin bound in G n,p (R) is obtained in [6] when the code size is large. The Gilbert-Varshamov and Hamming bounds on G n,p (L) are calculated in [7] when p is fixed and n is asymptotically large.…”

“…Roughly speaking, a quantization is a representation of a source: it maps an element in G n,q (L) (the source) into a subset C ⊂ G n,p (L), which is often discrete and referred to as a code. While most works assume that p = q [6]- [10], we are interested in a more general case where p may not necessarily equal to q; thus the term unequal dimensional quantization. The performance limit of quantization is given * This work is supported by NSF Grant DMS-0505680 and Thomson Inc.…”

Explicit Geometric Integration of Polynomial Vector Fields

“…Based on the data for packing points in Grassmannians in [3,21], it seems likely that such configurations will be optimal for certain choices of n and d. The drawbacks are that (i) the discrete subgroups of the classical groups are not known in general, (ii) such a subgroup can act transitively only on a very restricted set of numbers of points s, and (iii) such configurations may be singular.…”

“…The dimension of the sphere can be changed, and Conway, Hardin, and Sloane [3] have considered arranging s lines (or, more generally, k-planes) through the origin in R n , that is, choosing s points in the Grassmannian G(n, k) of k-planes. Both the sphere S n and Grassmannian G(n, k) are homogeneous spaces, and their homogeneity seems to be essential to the problem's appeal-arranging points on a square would not seem so important.…”

“…For example, the 3 quadratics x 2 , (x + y) 2 , and y 2 are all pure powers (i.e., functions of k = 1 variable) and span the quadratics in two variables, while the 4 pure cubics x 3 , (x + y) 3 , (x − y) 3 , and y 3 span all cubics in two variables. We shall see that these 4 cubics are the 'best possible' in a certain sense, while the 3 quadratics are not.…”