Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics

2026-06-24

AI Summary

This work investigates whether diffusion models can learn and recover the intrinsic geometric structure of data manifolds without prior knowledge. To this end, the authors construct a controlled experimental setting based on constrained inverse kinematics (IK), where task-space constraints define configuration manifolds with known dimensions and analytical solutions. Conditional diffusion models are trained in this setting and evaluated on UR5 and Franka robotic platforms across seven families of constraints. Through analyses of score functions, linear interpolation in latent space, and intrinsic dimension estimation, the study demonstrates that the recovered intrinsic dimensions align closely with theoretical degrees of freedom, and interpolated trajectories remain near the constraint manifolds. These findings provide the first empirical validation that diffusion models can effectively learn complex geometric structures and exhibit a meaningful geometric inductive bias.

Abstract

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.

Problem

• diffusion models
• intrinsic geometry
• constrained inverse kinematics
• data manifolds
• intrinsic dimension

Innovation

• diffusion models
• constrained inverse kinematics
• intrinsic geometry
• manifold learning
• score function