Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics
Provides a controlled benchmark to evaluate whether diffusion models learn geometric structure, addressing a gap in prior work that relied on natural images with unknown geometry.
The paper shows that diffusion models can recover the intrinsic geometry of data manifolds in a controlled setting (constrained inverse kinematics), where the true manifold dimension is known. For two robots, the intrinsic dimension estimated from the model's score function matches the analytical degrees of freedom across seven constraint families.
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.