Constructing Packings in Grassmannian Manifolds Via Alternating Projection

Constructing Packings in Grassmannian Manifolds Via Alternating Projection

Constructing Packings in Grassmannian Manifolds via Alternating Projection I. S. Dhillon, R. W. Heath Jr., T. Strohmer, and J. A. Tropp CONTENTS This paper describes a numerical method for finding good pack- 1. Introduction ings in Grassmannian manifolds equipped with various met- 2. Packing in Grassmannian Manifolds rics. This investigation also encompasses packing in projective 3. Alternating Projection for Chordal Distance spaces. In each case, producing a good packing is equivalent 4. Bounds on the Packing Diameter to constructing a matrix that has certain structural and spectral 5. Experiments properties. By alternately enforcing the structural condition and 6. Discussion then the spectral condition, it is often possible to reach a matrix 7. Tammes’ Problem that satisfies both. One may then extract a packing from this Acknowledgments matrix. References 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 diam- eters that are nearly optimal. 1. INTRODUCTION Let us begin with the standard facetious example. Imag- ine that several mutually inimical nations build their cap- ital cities on the surface of a featureless globe. Being concerned about missile strikes, they wish to locate the closest pair of cities as far apart as possible. In other words, what is the best way to pack points on the sur- face of a two-dimensional sphere? This question, first discussed by the Dutch biologist Tammes [Tammes 30], is the prototypical example of packing in a compact metric space. It has been stud- ied in detail for the last 75 years. More recently, re- 2000 AMS Subject Classification: Primary 51N15, 52C17 searchers have started to ask about packings in other compact spaces. In particular, several communities have Keywords: Combinatorial optimization, packing, projective spaces, Grassmannian spaces, Tammes’ Problem investigated how to arrange subspaces in a Euclidean c A K Peters, Ltd. 1058-6458/2008 $ 0.50 per page Experimental Mathematics 17:1, page 9 10 Experimental Mathematics, Vol. 17 (2008), No. 1 space so that they are as distinct as possible. An equiv- 1.2 Alternating Projection alent formulation is to find the best packings of points We will attempt to solve the feasibility problem (1–1) in a Grassmannian manifold. This problem has appli- in Grassmannian manifolds equipped with a number of cations in quantum computing and wireless communica- different metrics, but the same basic algorithm applies tions. There has been theoretical interest in subspace in each case. Here is a high-level description of our ap- packing since the 1960s [T´oth 65], but the first detailed proach. numerical study appears in a 1996 paper of Conway, First, we show that each configuration of subspaces is Hardin, and Sloane [Conway et al. 96]. associated with a block Gram matrix whose blocks con- The aim of this paper is to describe a flexible numer- trol the distances between pairs of subspaces. Then we ical method that can be used to construct packings in prove that a configuration solves the feasibility problem Grassmannian manifolds equipped with several different (1–1) if and only if its Gram matrix possesses both a metrics. The rest of this introduction provides a formal structural property and a spectral property. The overall statement of abstract packing problems, and it offers an algorithm consists of the following steps. overview of our approach to solving them. 1. Choose an initial configuration and construct its ma- 1.1 Abstract Packing Problems trix. Although we will be working with Grassmannian mani- 2. Alternately enforce the structural condition and the folds, it is more instructive to introduce packing problems spectral condition in hope of reaching a matrix that M in an abstract setting. Let be a compact metric space satisfies both. endowed with the distance function distM.Thepacking diameter of a finite subset X is the minimum distance be- 3. Extract a configuration of subspaces from the output tween some pair of distinct points drawn from X . That matrix. is, def In our work, we choose the initial configuration ran- packM(X ) = min distM(xm,xn). m=n domly and then remove similar subspaces from it with a simple algorithm. One can imagine more sophisticated In other words, the packing diameter of a set is the diam- approaches to constructing the initial configuration. eter of the largest open ball that can be centered at each Flexibility and ease of implementation are the major point of the set without encompassing any other point. advantages of alternating projection. This article demon- (It is also common to study the packing radius,whichis strates that appropriate modifications of this basic tech- half the diameter of this ball.) An optimal packing of nique allow us to construct solutions to the feasibility N X points is an ensemble that solves the mathematical problem in Grassmannian manifolds equipped with var- program ious metrics. Some of these problems have never been max packM(X ), studied numerically, and the experiments point toward |X |=N intriguing phenomena that deserve theoretical attention. where |·| returns the cardinality of a finite set. The op- Moreover, we believe that the possibilities of this method timal packing problem is guaranteed to have a solution have not been exhausted and that it will see other appli- because the metric space is compact and the objective is cations in the future. a continuous function of the ensemble X . Alternating projection does have several drawbacks. This article focuses on a feasibility problem closely con- It may converge very slowly, and it does not always yield nected with optimal packing. Given a number ρ, the goal a high level of numerical precision. In addition, it may is to produce a set of N points for which not deliver good packings when the ambient dimension or the number of subspaces in the configuration is large. packM(X ) ≥ ρ. (1–1) 1.3 Motivation and Related Work This problem is notoriously difficult to solve because it This work was motivated by applications in electrical en- is highly nonconvex, and it is even more difficult to de- gineering. In particular, subspace packings solve certain termine the maximum value of ρ for which the feasibility extremal problems that arise in multiple-antenna com- problem is soluble. This maximum value of ρ corresponds munication systems [Zheng and Tse 02, Hochwald et al. to the diameter of an optimal packing. 00, Love et al. 04]. This application requires complex Dhillon et al.: Constructing Packings in Grassmannian Manifolds via Alternating Projection 11 Grassmannian packings that consist of a small number 1.5 Outline of Article of subspaces in an ambient space of low dimension. Our Here is a brief overview of this article. In Section 2, we algorithm is quite effective in this parameter regime. The develop a basic description of Grassmannian manifolds resulting packings fill a significant gap in the literature, and present some natural metrics. Section 3 explains since existing tables consider only the real case [Sloane why alternating projection is a natural algorithm for pro- 04a]. See Section 6.1 for additional discussion of the wire- ducing Grassmannian packings, and it outlines how to less application. apply this algorithm for one specific metric. Section 4 The approach to packing via alternating projection gives some theoretical upper bounds on the optimal di- was discussed in a previous publication [Tropp et al. ameter of packings in Grassmannian manifolds. Section 05], but the experiments were limited to a single case. 5 describes the outcomes of an extensive set of numeri- We are aware of several other numerical methods that cal experiments and explains how to apply the algorithm can be used to construct packings in Grassmannian to other metrics. Section 6 offers some discussion and manifolds [Conway et al. 96, Trosset 01, Agrawal et conclusions. Appendix 7 explores how our methodology al. 01]. These techniques rely on ideas from nonlinear applies to Tammes’ problem of packing on the surface of programming. a sphere. Our experiments resulted in tables of packing diame- ters. We did not store the configurations produced by the 1.4 Historical Interlude algorithm. The Matlab code that produced these data is The problem of constructing optimal packings in various available on request from [email protected]. metric spaces has a long and lovely history. The most These tables and figures are intended only to describe famous example may be Kepler’s conjecture that an op- the results of our experiments; it is likely that many of the timal packing of spheres in three-dimensional Euclidean packing diameters could be improved with additional ef- space1 locates them at the points of a face-centered cu- fort. In all cases, we present the results of calculations for bic lattice. For millennia, greengrocers have applied this the stated problem, even if we obtained a better packing theorem when stacking oranges, but it has only been by solving a different problem. For example, a complex established rigorously within the last few years [Hales packing should always improve on the corresponding real 04]. Packing problems play a major role in modern com- packing. If the numbers indicate otherwise, it just means munications because error-correcting codes may be in- that the complex experiment yielded an inferior result. terpreted as packings in the Hamming space of binary As a second example, the optimal packing diameter must strings [Cover and Thomas 91].

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