Learning Lie Group Generators from Trajectories

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 ${\hat{I}}^3($t$)$ generated by a fixed Lie algebra element $\hat{I}¾\in \mathfrak{g}$ follows the exponential flow ${\hat{I}}^3($t$)=g_0\cdot \exp (t\hat{I}¾)$. The central task addressed in this work is the reconstruction of such a latent generator $\hat{I}¾$ 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 $\hat{I}¾\in \mathfrak{g}$. 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!

7 pages, 12 figures