TY - GEN

T1 - A discrete regression method on manifolds and its application to data on SO(n)

AU - Boumal, Nicolas

AU - Absil, P. A.

N1 - Funding Information:
★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. The first author is a F.R.S.-FNRS Research Fellow.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - Conjugate gradient methods

KW - Differential geometric methods

KW - Discrete time

KW - Finite differences

KW - Interpolation algorithms

KW - Least-squares approximation

KW - Non-parametric regression

KW - Rotation

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U2 - 10.3182/20110828-6-IT-1002.00542

DO - 10.3182/20110828-6-IT-1002.00542

M3 - Conference contribution

AN - SCOPUS:84866763119

SN - 9783902661937

T3 - IFAC Proceedings Volumes (IFAC-PapersOnline)

SP - 2284

EP - 2289

BT - Proceedings of the 18th IFAC World Congress

PB - IFAC Secretariat

ER -