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arXiv · 2305.18996

The barycenter in free nilpotent Lie groups and its application to iterated-integrals signatures

Marianne Clausel·Joscha Diehl·Raphael Mignot·Leonard Schmitz·Nozomi Sugiura·Konstantin Usevich
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Abstract

We establish the well-definedness of the barycenter (in the sense of Buser and Karcher) for every integrable measure on the free nilpotent Lie group of step $L$ (over $\mathbb{R}^d$). We provide two algorithms for computing it, using methods from Lie theory (namely, the Baker-Campbell-Hausdorff formula) and from the theory of Gr\"obner bases of modules. Our main motivation stems from measures induced by iterated-integrals signatures, and we calculate the barycenter for the signature of the Brownian motion.

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Marianne Clausel, Joscha Diehl, Raphael Mignot, Leonard Schmitz, Nozomi Sugiura, Konstantin Usevich. 2023-05-30. The barycenter in free nilpotent Lie groups and its application to iterated-integrals signatures. https://arxiv.org/abs/2305.18996

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