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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

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 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.

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

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 (over ). 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.
Paper Structure (28 sections, 28 theorems, 181 equations, 1 figure, 6 tables)

This paper contains 28 sections, 28 theorems, 181 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Free Lie algebras and iterated-integral signatures
  4. Baker-Campbell-Hausdorff formula
  5. Basis of the truncated Lie algebra and its dual
  6. BCH in the truncated Lie algebra
  7. The barycenter in the nilpotent Lie group
  8. Definition and properties
  9. Key lemma
  10. Proof of the main theorem
  11. Reducing the number of terms with Gröbner bases
  12. Reducing the number of terms with an antisymmetrized BCH formula
  13. Recursive updates of group means
  14. Algorithm using updates in the ambient space
  15. Main result
  16. ...and 13 more sections

Key Result

Theorem 1

$\mathsf{IIS}_{\le L}( X )$ is an element of $\mathcal{G}_{\le L}( \mathbb{R}^d )$.

Figures (1)

  • Figure 1: On the left the mean of the signatures $\mathsf{IIS}(X^{(i)})$ of three paths is assumed to be given by the signature $\mathsf{IIS}(M)$ of some path $M$. If we attach to all paths $X^{(i)}$ a new path segment $Y$, since the mean is right invariant, this corresponds to attaching that path segment to $M$. On the right, we see the analogous visualization for left invariance. Due to Chen's identity, attaching path corresponds to the group product.

Theorems & Definitions (79)

  • Theorem 1: bib:FV2010
  • Theorem 2: Baker-Campbell-Hausdorff formula
  • Example 3
  • Definition 4
  • Example 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Definition 8: bib:BK1981,bib:PL2020
  • ...and 69 more