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

2023/05/30 by Marianne Clausel, Clausel, Marianne, Joscha Diehl +9 · 1 citation
Mathematics · #13P10 #15A69 (Secondary) #60J65 #60L10 (Primary) 22E25 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Random Matrices and Applications #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2305.18996

openalex publication_date 2023/05/30 · openalex created_date 2023/06/01 · openalex updated_date 2026/07/28

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 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 Gröbner 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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