Learning Lie Group Generators from Trajectories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912309264252928 |
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| author | Hu, Lifan |
| author_facet | Hu, Lifan |
| contents | 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! |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning Lie Group Generators from Trajectories Hu, Lifan Machine Learning 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! |
| title | Learning Lie Group Generators from Trajectories |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2504.03220 |