Learning Lie Group Generators from Trajectories

TL;DR

This work tackles the inverse exponential problem on matrix Lie groups by learning to recover a constant Lie algebra generator $\xi$ from discretized trajectories. It constructs a data-driven pipeline that converts trajectories into sequences of Lie algebra displacements $\delta_t$, normalizes them, and feeds them to a shallow MLP that regresses $\xi$ in $\mathfrak{g}$. The approach is evaluated across SE(2), SE(3), SO(3), and SL(2,$\mathbb{R}$), showing accurate generator recovery with low error and robustness to moderate noise, and revealing some numerical considerations for non-compact or sensitive groups. The results suggest that inverse exponential maps can be learned effectively with simple architectures, highlighting potential uses of Lie-algebraic priors in geometric learning and robotics. A companion GitHub repository provides benchmarks and tools for reproduction and extension to additional Lie groups and real-world data.

Abstract

This work investigates the inverse problem of generator recovery in matrix Lie groups from discretized trajectories. Let $G$ be a real matrix Lie group and $\mathfrak{g} = \text{Lie}(G)$ its corresponding Lie algebra. A smooth trajectory $γ($t$)$ generated by a fixed Lie algebra element $ξ\in \mathfrak{g}$ follows the exponential flow $γ($t$) = g_0 \cdot \exp(t ξ)$. The central task addressed in this work is the reconstruction of such a latent generator $ξ$ from a discretized sequence of poses $ \{g_0, g_1, \dots, g_T\} \subset G$, sampled at uniform time intervals. This problem is formulated as a data-driven regression from normalized sequences of discrete Lie algebra increments $\log\left(g_{t}^{-1} g_{t+1}\right)$ to the constant generator $ξ\in \mathfrak{g}$. A feedforward neural network is trained to learn this mapping across several groups, including $\text{SE(2)}, \text{SE(3)}, \text{SO(3)}, and \text{SL(2,$\mathbb{R})$}$. It demonstrates strong empirical accuracy under both clean and noisy conditions, which validates the viability of data-driven recovery of Lie group generators using shallow neural architectures. This is Lie-RL GitHub Repo https://github.com/Anormalm/LieRL-on-Trajectories. Feel free to make suggestions and collaborations!

Figures (4)

• Figure 1: MSE Loss, Trajectory Comparison and of predicted and ground truth trajectories in $\mathrm{SE}(2)$.
• Figure 2: MSE Loss, Trajectory Comparison and of predicted and ground truth trajectories in $\mathrm{SE}(3)$.
• Figure 3: MSE Loss, Trajectory Comparison and of predicted and ground truth trajectories in $\mathrm{SO}(3)$.
• Figure 4: MSE Loss, Trajectory Comparison and of predicted and ground truth trajectories in $SL(2,\mathbb{R})$.