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 -