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A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

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Abstract

Given data points p 0,…,p N on a closed submanifold M of ℝn and time instants 0=t 0<t 1<⋅⋅⋅<t N =1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as “regular” as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in ℝn and on the unit sphere.

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References

  1. P.A. Absil, K.A. Gallivan, Accelerated line-search and trust-region methods, SIAM J. Numer. Anal. 47(2), 997–1018 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. P.A. Absil, R. Mahony, B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim. 6(2), 531–547 (2005).

    Article  MathSciNet  Google Scholar 

  3. P.A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds (Princeton University Press, Princeton, 2008).

    MATH  Google Scholar 

  4. P.A. Absil, J. Trumpf, R. Mahony, B. Andrews, All roads lead to Newton: feasible second-order methods for equality-constrained optimization, Technical report UCL-INMA-2009.024 (2009).

  5. C. Altafini, The de Casteljau algorithm on SE(3), in Nonlinear Control in the Year 2000 (2000), pp. 23–34.

    Chapter  Google Scholar 

  6. D.P. Bertsekas, Nonlinear Programming (Athena Scientific, Belmont, 1995).

    MATH  Google Scholar 

  7. W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, revised 2nd edn. (Academic Press, San Diego, 2003).

    Google Scholar 

  8. M. Camarinha, F. Silva Leite, P. Crouch, Splines of class C k on non-Euclidean spaces, IMA J. Math. Control Inf. 12(4), 399–410 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  9. M.P. do Carmo, Riemannian Geometry. Mathematics: Theory & Applications (Birkhäuser Boston, Boston, 1992). Translated from the second Portuguese edition by Francis Flaherty.

    Google Scholar 

  10. I. Chavel, Riemannian Geometry, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 98 (Cambridge University Press, Cambridge, 2006). A modern introduction.

    Book  MATH  Google Scholar 

  11. P. Crouch, G. Kun, F. Silva Leite, The de Casteljau algorithm on the Lie group and spheres, J. Dyn. Control Syst. 5, 397–429 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Crouch, F. Silva Leite, Geometry and the dynamic interpolation problem, in Proc. Am. Control Conf. (Boston, 26–29 July, 1991), pp. 1131–1136.

    Google Scholar 

  13. P. Crouch, F. Silva Leite, The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces, J. Dyn. Control Syst. 1(2), 177–202 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Dyn, Linear and nonlinear subdivision schemes in geometric modeling, in Foundations of Computational Mathematics, Hong Kong, 2008. London Math. Soc. Lecture Note Ser., vol. 363 (Cambridge University Press Cambridge, 2009), pp. 68–92.

    Google Scholar 

  15. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, 3rd edn. Universitext (Springer, Berlin, 2004).

    Book  MATH  Google Scholar 

  16. K. Hüper, F. Silva Leite, On the geometry of rolling and interpolation curves on S n, SO n , and Grassmann manifolds, J. Dyn. Control Syst. 13(4), 467–502 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Jakubiak, F. Silva Leite, R.C. Rodrigues, A two-step algorithm of smooth spline generation on Riemannian manifolds, J. Comput. Appl. Math. 194(2), 177–191 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  18. P.E. Jupp, J.T. Kent, Fitting smooth paths to spherical data, J. R. Stat. Soc. Ser. C 36(1), 34–46 (1987).

    MathSciNet  MATH  Google Scholar 

  19. H. Karcher, Riemannian center of mass and mollifier smoothing, Commun. Pure Appl. Math. 30(5), 509–541 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Klassen, A. Srivastava, Geodesic between 3D closed curves using path straightening, in European Conference on Computer Vision, ed. by A. Leonardis, H. Bischof, A. Pinz (eds.), (2006), pp. 95–106.

    Google Scholar 

  21. A. Kume, I.L. Dryden, H. Le, Shape-space smoothing splines for planar landmark data, Biometrika 94(3), 513–528 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Lazard, J. Tits, Domaines d’injectivité de l’application exponentielle, Topology 4, 315–322 (1965/1966).

    Article  MathSciNet  Google Scholar 

  23. J.M. Lee, Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176 (Springer, New York, 2007).

    Google Scholar 

  24. A. Linnér, Symmetrized curve-straightening, Differ. Geom. Appl. 18(2), 119–146 (2003).

    Article  MATH  Google Scholar 

  25. L. Machado, F.S. Leite, K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst. 16(1), 121–148 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Machado, F. Silva Leite, Fitting smooth paths on Riemannian manifolds, Int. J. Appl. Math. Stat. 4(J06), 25–53 (2006).

    MathSciNet  MATH  Google Scholar 

  27. L. Machado, F. Silva Leite, K. Hüper, Riemannian means as solutions of variational problems, LMS J. Comput. Math. 9, 86–103 (2006) (electronic).

    MathSciNet  MATH  Google Scholar 

  28. J.W. Milnor, Morse Theory (Princeton University Press, Princeton, 1963).

    MATH  Google Scholar 

  29. L. Noakes, G. Heinzinger, B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inf. 6(4), 465–473 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  30. B. O’Neill, Semi-Riemannian Geometry. Pure and Applied Mathematics, vol. 103 (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1983).

    MATH  Google Scholar 

  31. R.S. Palais, Morse theory on Hilbert manifolds, Topology 2, 299–340 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Popiel, L. Noakes, Bézier curves and C 2 interpolation in Riemannian manifolds, J. Approx. Theory 148(2), 111–127 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Samir, P.A. Absil, A. Srivastava, E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Tech. Rep. UCL-INMA-2009.023-v3, Université catholique de Louvain (2010).

  34. T. Shingel, Interpolation in special orthogonal groups, IMA J. Numer. Anal. 29(3), 731–745 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Smyrlis, V. Zisis, Local convergence of the steepest descent method in Hilbert spaces, J. Math. Anal. Appl. 300(2), 436–453 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  36. A.J. Tromba, A general approach to Morse theory, J. Differ. Geom. 12(1), 47–85 (1977).

    MathSciNet  MATH  Google Scholar 

  37. J. Wallner, E. Nava Yazdani, P. Grohs, Smoothness properties of Lie group subdivision schemes. Multiscale Model. Simul. 6(2), 493–505 (2007) (electronic).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. ISIT, Clermont Université, BP 10448, 63000, Clermont-Ferrand, France

    Chafik Samir

  2. ICTEAM Institute and Centre for Systems Engineering and Applied Mechanics (CESAME), Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    P.-A. Absil

  3. Department of Statistics, Florida State University, Tallahassee, FL, 32306, USA

    Anuj Srivastava

  4. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Eric Klassen

Authors
  1. Chafik Samir
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  2. P.-A. Absil
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  3. Anuj Srivastava
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  4. Eric Klassen
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Correspondence to P.-A. Absil.

Additional information

Communicated by Nira Dyn and Michael Todd.

This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. This research was supported in part by AFOSR FA9550-06-1-0324 and ONR N00014-09-10664.

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Samir, C., Absil, PA., Srivastava, A. et al. A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds. Found Comput Math 12, 49–73 (2012). https://doi.org/10.1007/s10208-011-9091-7

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  • DOI: https://doi.org/10.1007/s10208-011-9091-7

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