用机器人逆运动学验证扩散模型能否学好数据几何结构。
Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics
- 在已知几何的逆运动学任务中训练扩散模型，直接检验其对内在维度的捕捉能力。
- 模型从得分函数恢复的内在维度与理论值完全一致，6-DoF和7-DoF均成立。
- 隐空间线性插值仍保持约束满足，说明模型还学到了更深层几何结构。
近期研究认为扩散模型能恢复训练数据流形中的几何结构，但证据多来自自然图像，其底层几何未知。本文在逆运动学（IK）这一几何可解析的设定下研究该问题：每个任务约束定义一个配置空间流形，其内在维数已知，可作为评估模型学习几何的真值。针对6-DoF UR5和7-DoF Franka机器人，我们在七类约束家族上训练单一条件扩散模型，覆盖从离散解分支到自运动流形的多种情形。实验表明，模型从得分函数恢复的内在维度，与对应约束流形的理论自由度完全匹配。此外，隐空间线性插值生成的解仍紧密贴近目标约束流形，表明模型不仅学习了维数，还捕获了约束族的几何结构。约束逆运动学为研究扩散模型学习的内在几何提供了可控实验环境。
原文摘要 · Abstract (English)
Recent studies suggest that diffusion models can recover geometric structure in the data manifolds they are trained on, yet the supporting evidence has so far come mostly from natural-image data, where the underlying geometry itself is unknown. We study this question in a setting where the geometry is analytically tractable: constrained inverse kinematics (IK). Each task-space constraint defines a configuration-space manifold with known intrinsic dimension, giving direct ground truth for evaluating the geometry learned by the model. For each of the 6-DoF UR5 and 7-DoF Franka, we train a single conditional diffusion model across seven constraint families, spanning solution manifolds from discrete IK branches to self-motion manifolds. Our empirical results reveal that the intrinsic dimension recovered from the model's score function matches the analytical degrees of freedom of the corresponding constraint manifold across both robots. Moreover, linear interpolation in the latent space leads to generated solutions that remain close to the appropriate constraint manifold, indicating that the learned representation further captures geometric structure of the constraint family beyond intrinsic dimension alone. Constrained IK therefore offers a controlled setting for studying the intrinsic geometry learned by diffusion models.
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