 
Summary: ALGORITHM COMPARISON FOR KARCHER MEAN COMPUTATION OF
ROTATION MATRICES AND DIFFUSION TENSORS
Quentin Rentmeesters, P.A. Absil
Universit´e catholique de Louvain,
Department of Mathematical Engineering,
B1348 LouvainlaNeuve, Belgium
ABSTRACT
This paper concerns the computation, by means of gra
dient and Newton methods, of the Karcher mean of a finite
collection of points, both on the manifold of 3 × 3 rotation
matrices endowed with its usual biinvariant metric and on
the manifold of 3 × 3 symmetric positive definite matrices
endowed with its usual affine invariant metric. An explicit
expression for the Hessian of the Riemannian squared dis
tance function of these manifolds is given. From this, a con
dition on the step size of a constant step gradient method
that depends on the data distribution is derived. These ex
plicit expressions make a more efficient implementation of
the Newton method possible and it is shown that the Newton
method outperforms the gradient method in some cases.
