Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

[en] We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R-n. In these formulas, p-planes are represented as the column space of n x p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

Disciplines :

Mathematics

Author, co-author :

Absil, P.-A.

Mahony, R.

Sepulchre, Rodolphe ; Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation

Language :

English

Title :

Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

Publication date :

January 2004

Journal title :

Acta Applicandae Mathematicae

ISSN :

0167-8019

eISSN :

1572-9036

Publisher :

Kluwer Academic Publ, Dordrecht, Netherlands

Volume :

Issue :

Pages :

199-220

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://www.springerlink.com/content/q6356j236hpn3579/

Available on ORBi :

since 21 December 2009

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Scopus citations^®

256

Scopus citations^®
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234

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199

Bibliography

Absil, P.-A.: Invariant subspace computation: A geometric approach, Ph.D. thesis, Faculté des Sciences Appliquées, Université de Liège, Secrétariat de la FSA, Chemim des Chevreuils 1 (Bât. B52), 4000 Lièe, Belgium, 2003.
Absil, P.-A., Mahoney, R., Sepulchre, R. and Van Dooren, P.: A Grassmann-Rayleigh quotient iteration for computing invariant subspaces, SIAM Rev. 44(1) (2002), 57-73.
Absil, P.-A., Sepulchre, R., Van Dooren, P. and Mahony, R.: Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Anal. Appl., to appear.
Absil, P.-A. and Van Dooren, P.: Two-sided Grassmann-Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl. (2002), submitted.
Björk, Å. and Golub, G. H.: Numerical methods for computing angles between linear subspaces, Math. Comp. 27 (1973), 579-594.
Boothby, W. M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1975.
Chatelin, F.: Simultaneous Newton's iteration for the eigenproblem, Computing 5(Suppl.) (1984), 67-74.
Chavel, I.: Riemannian Geometry - A Modern Introduction, Cambridge Univ. Press, 1993.
Common, P. and Golub, G. H.: Tracking a few extreme singular values and vectors in signal processing, Proc. IEEE 78(8) (1990), 1327-1343.
Demmel, J. W.: Three methods for refining estimates of invariant subspaces, Computing 38 (1987), 43-57.
Dennis, J. E. and Schnabel, R. B.: Numerial Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983.
do Carmo, M. P.: Riemannian Geometry, Birkhäuser, 1992.
Doolin, B. F. and Martin, C. F.: Introduction to Differential Geometry for Engineers, Monographs and Textbooks in Pure and Applied Mathematics 136, Marcel Deckker, Inc., New York, 1990.
Edelman, A., Arias, T. A. and Smith, S. T.: The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20(2) (1998), 303-353.
Ferrer, J., García, M. I. and Puerta, F.: Differentiable families of subspaces, Linear Algebra Appl. 199 (1994), 229-252.
Gabay, D.: Minimizing a differentiable function over a differential manifold, J. Optim. Theory Appl. 37(2) (1982), 177-219.
Golub, G. H. and Van Loan, C. F.: Matrix Computations, 3rd edn, The Johns Hopkins Univ. Press, 1996.
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, Oxford, 1978.
Helmke, U. and Moore, J. B.: Optimization and Dynamical Systems, Springer, 1994.
Karcher, H.: Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30(5) (1977), 509-541.
Kendall, W. S.: Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence, Proc. London Math. Soc. 61(2) (1990), 371-406.
Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. 1, 2, Wiley, 1963.
Leichtweiss, K.: Zur riemannschen Geometrie in grassmannschen Mannigfaltigkeiten, Math. Z. 76 (1961), 334-366.
Luenberger, D. G.: Optimization by Vector Space Methods, Wiley, Inc., 1969.
Lundström, E. and Eldén, L.: Adaptive eigenvalue computations using Newton's method on the Grassmann manifold, SIAM J. Matrix Anal. Appl., 23(3) (2002), 819-839.
Machado, A. and Salavessa, I.: Grassmannian manifolds as subsets of Euclidean spaces, Res. Notes in Math. 131 (1985), 85-102.
Mahony, R. E.: The constrained Newton method on a Lie group and the symmetric eigenvalue problem, Linear Algebra Appl. 248 (1996), 67-89.
Mahony, R. and Manton, J. H.: The geometry of the Newton method on non-compact Lie groups, J. Global Optim. 23(3) (2002), 309-327.
Nomizu, K.: Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.
Ran, A. C. M. and Rodman, L.: A class of robustness problems in matrix analysis, In: D. Alpay. I. Gohberg, and V. Vinnikov (eds), Interpolation Theory, Systems Theory and Related Topics, The Harry Dym Anniversary Volume, Operator Theory: Advances and Applications 134, Birkhäuser, 2002, pp. 337-383.
Simoncini, V. and Elden, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT 42(1) (2002), 159-182.
Smith, S. T.: Geometric optimization methods for adaptive filtering, Ph.D. thesis, Division of Applied Sciences, Harvard University, Cambridge, MA, 1993.
Smith, S. T.: Optimization techniques on Riemannian manifolds, In: A. Bloch (ed.), Hamiltonian and Gradient Flows, Algorithms and Control, Fields Institute Communications, Vol. 3, Amer. Math. Soc., 1994, pp. 113-136.
Stewart, G. W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev. 15(4) (1973), 727-764.
Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Acad. Publ., Dordrecht, 1994.
Wong, Y.-C.: Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 589-594.
Woods, R. P.: Characterizing volume and surface deformations in an atlas framework: Theory, applications, and implementation, NeuroImage 18(3) (2003), 769-788.