Optimal Data Fitting on Lie Groups: a Coset Approach

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).