A discrete regression method on manifolds and its application to data on SO(n)

Nicolas Boumal, P. A. Absil

Research output: Chapter in Book/Report/Conference proceedingConference contribution

21 Scopus citations

Abstract

The regression problem of fitting a "smooth", discrete curve to data points on a Riemannian manifold is formulated here as an unconstrained, finite-dimensional optimization problem. Smoothness of a discrete curve, seen as a sequence of close points on the manifold, is assessed and encouraged by a regularity term in the objective function. This term is built upon a generalization of finite differences to manifolds introduced in this work. Tuning of the balance between fitting and regularity (or energy-efficiency) is achieved by adjusting two parameters. The proposed framework is described in detail and is then applied to the special orthogonal group SO(n), i.e., the set of rotations in ℝn. A Riemannian version of the nonlinear conjugate gradient method is used to minimize the resulting objective. To this end, an explicit formula for the derivative of the matrix logarithm is derived, yielding explicit formulas for the gradient of the objective. Numerical results are presented and show that smooth curves in SO(n) can be obtained in a few hundred iterations with the proposed algorithm.

Original languageEnglish (US)
Title of host publicationProceedings of the 18th IFAC World Congress
PublisherIFAC Secretariat
Pages2284-2289
Number of pages6
Edition1 PART 1
ISBN (Print)9783902661937
DOIs
StatePublished - Jan 1 2011
Externally publishedYes

Publication series

NameIFAC Proceedings Volumes (IFAC-PapersOnline)
Number1 PART 1
Volume44
ISSN (Print)1474-6670

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering

Keywords

  • Conjugate gradient methods
  • Differential geometric methods
  • Discrete time
  • Finite differences
  • Interpolation algorithms
  • Least-squares approximation
  • Non-parametric regression
  • Rotation

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  • Cite this

    Boumal, N., & Absil, P. A. (2011). A discrete regression method on manifolds and its application to data on SO(n). In Proceedings of the 18th IFAC World Congress (1 PART 1 ed., pp. 2284-2289). (IFAC Proceedings Volumes (IFAC-PapersOnline); Vol. 44, No. 1 PART 1). IFAC Secretariat. https://doi.org/10.3182/20110828-6-IT-1002.00542