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Continuous-time filtering in Lie groups: estimation via the Fr{é}chet mean of solutions to stochastic differential equations

Magalie Bénéfice, Marc Arnaudon, Audrey Giremus

TL;DR

The paper addresses continuous-time filtering on Lie groups by computing the Fréchet mean ${\mathscr E}_t$ of the SDE solution ${X_t}$ to obtain a minimal-variance estimator for predicting the state between observations. It derives a McKean–Vlasov-type equation ${\mathrm d}{\mathscr E}_t = {\mathscr E}_t h_t dt$ with $h_t$ expressed through derivatives of the log map and Hessians, and specializes to $G = SO(3)$ to obtain explicit drift and covariance evolution formulas for the estimation error. Simulations on $SO(3)$ validate the proposed approach against Gauss-Newton on Lie groups, showing close agreement in rotation estimates. The method emphasizes minimal modeling assumptions and preserves geometric structure, with potential extensions to broader homogeneous and Riemannian manifolds and to posterior updates using measurements.

Abstract

We compute the Fréchet mean $\mathscr{E}_t$ of the solution $X_{t}$ to a continuous-time stochastic differential equation in a Lie group. It provides an estimator with minimal variance of $X_{t}$. We use it in the context of Kalman filtering and more precisely to infer rotation matrices. In this paper, we focus on the prediction step between two consecutive observations. Compared to state-of-the-art approaches, our assumptions on the model are minimal.

Continuous-time filtering in Lie groups: estimation via the Fr{é}chet mean of solutions to stochastic differential equations

TL;DR

The paper addresses continuous-time filtering on Lie groups by computing the Fréchet mean of the SDE solution to obtain a minimal-variance estimator for predicting the state between observations. It derives a McKean–Vlasov-type equation with expressed through derivatives of the log map and Hessians, and specializes to to obtain explicit drift and covariance evolution formulas for the estimation error. Simulations on validate the proposed approach against Gauss-Newton on Lie groups, showing close agreement in rotation estimates. The method emphasizes minimal modeling assumptions and preserves geometric structure, with potential extensions to broader homogeneous and Riemannian manifolds and to posterior updates using measurements.

Abstract

We compute the Fréchet mean of the solution to a continuous-time stochastic differential equation in a Lie group. It provides an estimator with minimal variance of . We use it in the context of Kalman filtering and more precisely to infer rotation matrices. In this paper, we focus on the prediction step between two consecutive observations. Compared to state-of-the-art approaches, our assumptions on the model are minimal.

Paper Structure

This paper contains 6 sections, 4 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prerequisites: Fréchet means on Lie groups
  3. Computing the Fréchet mean of a stochastic differential equation
  4. Proof of Theorem \ref{['T2']}
  5. Simulations
  6. Conclusion

Key Result

proposition 1

If the support of the random variable $X$ is included in a regular geodesic ball of $G$, then ${\mathscr E}(X)$ is the unique minimizer in ${\mathscr H}(X)$ of $g\mapsto {\mathbb E}\left[ \rho^2(g,X)\right]$, where $\rho$ is the Riemannian distance in $G$. Moreover, we have In other words, ${\mathscr E}(X)$ is the Fréchet mean of $X$. Consequently, ${\mathscr E}(X)$ is the estimator of $X$ minimi

Figures (1)

  • Figure 1: Comparison between the empirical and the proposed Fréchet mean.

Theorems & Definitions (8)

  • definition 1
  • definition 2
  • proposition 1
  • theorem 1
  • proof
  • theorem 2
  • theorem 3
  • proof