A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.

[1]  Andreas Arvanitoyeorgos,et al.  An Introduction to Lie Groups and the Geometry of Homogeneous Spaces , 2003 .

[2]  F. W. Warner.  Foundations of Differentiable Manifolds and Lie Groups , 1971 .

[3]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[4]  Frank L. Lewis,et al.  Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities , 1987 .

[5]  Maher Moakher,et al.  Means and Averaging in the Group of Rotations , 2002, SIAM J. Matrix Anal. Appl..

[6]  I. Chavel.  Riemannian Geometry: Subject Index , 2006 .

[7]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[8]  H. Karcher,et al.  Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems , 1974 .

[9]  Sigurdur Helgason.  Elementary differential geometry , 2001 .

[10]  Marcello Pelillo,et al.  Energy Minimization Methods in Computer Vision and Pattern Recognition , 2003, Lecture Notes in Computer Science.

[11]  W. Kendall.  Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence , 1990 .

[12]  Jorge S. Marques,et al.  A HMM approach to the estimation of random trajectories on manifolds , 2002, Signal Process..

[13]  G. Gal'perin.  A concept of the mass center of a system of material points in the constant curvature spaces , 1993 .

[14]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[15]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[16]  Jorge S. Marques,et al.  Shape Tracking Using Centroid-Based Methods , 2001, EMMCVPR.

[17]  Jonathan H. Manton,et al.  Optimization algorithms exploiting unitary constraints , 2002, IEEE Trans. Signal Process..