用神经网络从轨迹中恢复李群生成元，适用于机器人、视觉等场景
Learning Lie Group Generators from Trajectories
- 通过离散轨迹的李代数增量学习生成元映射
- 在干净和噪声条件下均实现高精度恢复
- 适合对李群轨迹建模感兴趣的科研与工程人员
本文研究从离散轨迹中逆向恢复矩阵李群的生成元问题。设 $G$ 为实矩阵李群，$ rak{g} = ext{Lie}(G)$ 为其对应的李代数。由固定李代数元素 $ξ o rak{g}$ 生成的光滑轨迹 $γ(t)$ 满足指数流 $γ(t) = g_0 \ullet \exp(t ξ)$。核心任务是从均匀采样得到的位姿序列 $\{g_0, g_1, \dots, g_T\} \subset G$ 中重建隐含生成元 $ξ\in\frak{g}$。该问题被建模为从归一化的离散李代数增量序列 $\log(g_t^{-1} g_{t+1})$ 到常量生成元 $ξ$ 的数据驱动回归。采用前馈神经网络在多个群上训练，包括 $\text{SE(2)}$, $\text{SE(3)}$, $\text{SO(3)}$, 以及 $\text{SL(2,\mathbb{R})}$。实验表明，在干净和噪声条件下均表现出强鲁棒性与高精度，验证了使用浅层神经网络实现李群生成元数据驱动恢复的可行性。代码已开源：https://github.com/Anormalm/LieRL-on-Trajectories。
原文摘要 · Abstract (English)
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!
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