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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,
B-1348 Louvain-la-Neuve, 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 bi-invariant 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.
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