Abstract
We provide algorithms for computing the Karcher mean of positive definite semi-infinite quasi-Toeplitz matrices. After showing that the power mean of quasi-Toeplitz matrices is a quasi-Toeplitz matrix, we obtain a first algorithm based on the fact that the Karcher mean is the limit of a family of power means. A second algorithm, that is shown to be more effective, is based on a generalization to the infinite-dimensional case of a reliable algorithm for computing the Karcher mean in the finite-dimensional case. Numerical tests show that the Karcher mean of infinite-dimensional quasi-Toeplitz matrices can be effectively approximated with a finite number of parameters.
The work of the first and second authors is partly supported by the INdAM through a GNCS project.
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Notes
- 1.
Recently, the term “Karcher mean” for matrices is falling out of use, in favor of “Riemannian center of mass” or “matrix geometric mean”. We kept the former here, because we deal with operators, with no underlying Riemannian structure.
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Bini, D.A., Iannazzo, B., Meng, J. (2021). Algorithms for Approximating Means of Semi-infinite Quasi-Toeplitz Matrices. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_45
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