Summary
This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in ℝn to data which can be solved using principal component analysis. We show how the fitting problem can be reduced for biinvariant distances to a generalized mean calculation on an homogeneous space. For biinvariant Riemannian distances we provide an algorithm based on the Karcher mean gradient algorithm. We illustrate our approach by some examples on SO(n).
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Acknowledgments
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 research was initiated during the postdoctoral stay of the first author at the University of Liège.
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Lageman, C., Sepulchre, R. (2010). Optimal Data Fitting on Lie Groups: a Coset Approach. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds) Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12598-0_15
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DOI: https://doi.org/10.1007/978-3-642-12598-0_15
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