Paper ID sheet UCL-INMA-2011.007

Title

Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors

Authors

Quentin Rentmeesters, P.-A. Absil

Abstract

This paper concerns the computation, by means of gradient and Newton methods, of the Karcher mean of a finite collection of points, both on the manifold of 3 x 3 rotation matrices endowed with its usual bi-invariant metric and on the manifold of 3 x 3 symmetric positive definite matrices endowed with its usual affine invariant metric. An explicit expression for the Hessian of the Riemannian squared distance function of these manifolds is given. From this, a condition on the step size of a constant step gradient method that depends on the data distribution is derived. These explicit 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.

Key words

steepest descent; Newton's method; Karcher mean; Fréchet mean; Riemannian center of mass

Status

Proceedings of the 19th European Signal Processing Conference (EUSIPCO-2011), Barcelona, Spain, August 29 - September 2, 2011, pp. 2229-2233, 2011. ISSN: 2076-1465

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BibTeX entry

@INPROCEEDINGS{RenAbs2011,
  author = "Quentin Rentmeesters and P.-A. Absil",
  title = "Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors",
  year = 2011,
  booktitle = "Proceedings of the 19th European Signal Processing Conference (EUSIPCO 2011)",
  pages = "2229-2233",
  publisher = "EURASIP",
  issn = "2076-1465",
}