Learning Lie Group Generators from Trajectories

• URL: http://arxiv.org/abs/2504.03220v1
• Date: Fri, 04 Apr 2025 07:08:59 GMT
• Title: Learning Lie Group Generators from Trajectories
• Authors: Lifan Hu,
• Abstract summary: This work investigates the inverse problem of generator recovery in matrix Lie groups from discretized trajectories.<n>A feedforward neural network is trained to learn this mapping across several groups.<n>It demonstrates strong empirical accuracy under both clean and noisy conditions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\gamma($t$)$ generated by a fixed Lie algebra element $\xi \in \mathfrak{g}$ follows the exponential flow $\gamma($t$) = g_0 \cdot \exp(t \xi)$. The central task addressed in this work is the reconstruction of such a latent generator $\xi$ 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 $\xi \in \mathfrak{g}$. 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!

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