Abstract
Averaging over the Lie group SO(p) of special orthogonal matrices has several applications in the neural network field. The problem of averaging over the group SO(3) has been studied in details and, in some specific cases, it admits a closed form solution. Averaging over a generic-dimensional group SO(p) has also been studied recently, although the common formulation in terms of Riemannian mean leads to a matrix-type non-linear problem to solve, which, in general, may be tackled via iterative algorithms only. In the present paper, we propose a novel formulation of the problem that gives rise to a closed form solution for the average SO(p)-matrix.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cardoso, J.A.R.: An Explicit Formula for the Matrix Logarithm. South African Optometrist 64, 80–83 (2005)
Ferreira, R., Xavier, J.: Hessian of the Riemannian Squared-Distance Function on Connected Locally Symmetric Spaces with Applications. In: Proc. of the 7th Portuguese Conference on Automatic Control (Controlo 2006), Special session on control, optimization and computation (2006)
Fiori, S.: A Theory for Learning by Weight Flow on Stiefel-Grassman Manifold. Neural Computation 13, 1625–1647 (2001)
Fiori, S.: Quasi-Geodesic Neural Learning Algorithms Over the Orthogonal Group: A Tutorial. Journal of Machine Learning Research 6, 743–781 (2005)
Fiori, S., Tanaka, T.: An Algorithm to Compute Averages on Matrix Lie Groups. IEEE Trans. on Signal Processing 57, 4734–4743 (2009)
Manning, R.S., Bulman, G.B.: Stability of an Elastic Rod Buckling into a Soft Wall. Proc. of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, 2423–2450 (2005)
Moakher, M.: Means and Averaging in the Group of Rotations. SIAM Journal of Matrix Analysis and Applications 24, 1–16 (2002)
Prentice, M.J.: Fitting Smooth Paths to Rotation Data. The Journal of the Royal Statistical Society Series C (Applied Statistics) 36, 325–331 (1987)
Rao, R.P.N., Ruderman, D.L.: Learning Lie Groups for Invariant Visual Perception. Advances in Neural Information Processing Systems (NIPS) 11, 810–816 (1999)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, 2nd edn., vol. 1. Perish Press, Berkeley (1979)
Warmuth, M.K.: A Bayes Rule for Density Matrices. In: Advances in Neural Information Processing Systems (NIPS 2005), vol. 18. MIT Press, Cambridge (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fiori, S. (2010). A Closed-Form Solution to the Problem of Averaging over the Lie Group of Special Orthogonal Matrices. In: Zhang, L., Lu, BL., Kwok, J. (eds) Advances in Neural Networks - ISNN 2010. ISNN 2010. Lecture Notes in Computer Science, vol 6063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13278-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-13278-0_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13277-3
Online ISBN: 978-3-642-13278-0
eBook Packages: Computer ScienceComputer Science (R0)