# Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics

arXiv: https://arxiv.org/abs/2606.26408 · HTML: https://mlsanity.com/paper/2606.26408/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 17 of its 32 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 17 of 17)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [IKFlow: Generating Diverse Inverse Kinematics Solutions](https://mlsanity.com/paper/2111.08933/) (2021)
- [Exploring the Representation Manifolds of Stable Diffusion Through the Lens of Intrinsic Dimension](https://mlsanity.com/paper/2302.09301/) (2023)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)
- [Generative Graphical Inverse Kinematics](https://mlsanity.com/paper/2209.08812/) (2022)
- [NODE IK: Solving Inverse Kinematics with Neural Ordinary Differential Equations for Path Planning](https://mlsanity.com/paper/2209.00498/) (2022)
- [How Well do Diffusion Policies Learn Kinematic Constraint Manifolds?](https://mlsanity.com/paper/2510.01404/) (2025)
- [IKDiffuser: a Diffusion-based Generative Inverse Kinematics Solver for Kinematic Trees](https://mlsanity.com/paper/2506.13087/) (2025)

## Abstract (arXiv)

Recent studies suggest that diffusion models can recover geometric structure in the data 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.