[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81690-en":3,"doc-seo-81690-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81690,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics","Recent studies suggest diffusion models can recover geometric structure on the data manifolds they are trained on, but most evidence comes from natural-image settings where geometry is not analytically known. This work studies constrained inverse kinematics, where each task-space constraint defines a configuration-space manifold with known intrinsic dimension and topology. Single conditional diffusion models are trained for UR5 (6-DoF) and Franka (7-DoF) across seven constraint families. The recovered intrinsic dimension matches analytic degrees of freedom, and latent-space linear interpolation yields solutions that remain close to the correct constraint manifolds.","Exploring the Intrinsic Geometry of Diffusion Models with Constrained Inverse Kinematics  \nMiguel Angel Rogel Garcia∗ , Phone Thiha Kyaw∗ , and Jonathan Kelly Institute for Aerospace Studies, University of Toronto, Toronto, Canada {miguel.rogel,phone.thiha, [jonathan.kelly](jonathan.kelly}@robotics.utias.utoronto.ca)[}](jonathan.kelly}@robotics.utias.utoronto.ca)[@robotics.utias.utoronto.ca](jonathan.kelly}@robotics.utias.utoronto.ca)  \n∗Equal contribution.  \narXiv :2606 .26408v2 [ cs .RO] 9 Jul 2026  \nFig. 1: Diffusion models capture the geometry of constraint manifolds in inverse kinematics. Given a constraint family anda target pose, we analyze the learned diffusion model along two axes: estimating the intrinsic dimension of the model’s learned distribution and the behaviour of linear interpolation between samples in latent space. The estimated intrinsic dimension matches the analytical dimension of the underlying constraint manifold. Latent interpolation stays close to the constraint manifold, further indicating that the model has encoded the manifold’s geometry.  \nAbstract—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.  \nI. INTRODUCTION  \nDiffusion models have become the dominant generative framework for high-dimensional data. Although they are trained only to denoise, a growing line of work reveals that their internal representations recover surprisingly rich geometric structure. The score function aligns with the normal bundle of the data manifold and recovers its intrinsic dimension [30], which itself varies with the conditioning input [15] . Pullback metrics on feature spaces yield semantically meaningful tangent directions [22] . Riemannian metrics built from the score admit geodesics that remain on the data manifold [8, 2] . Taken together, these findings suggest that diffusion models can encode aspects of the geometry of the distributions on which they are trained.  \nMuch of this evidence has been collected on natural-image manifolds, where the intrinsic dimension itself is an estimated quantity [24, 4] and the topology is recovered only indirectly. We complement this line of work by studying the constrained  \ninverse kinematics setting. Here, the data manifold is given analytically across a parametric family, as the IK solution set forms a constraint manifold of known dimension and topology. This manifold varies smoothly with a small conditioning vector, and we ask how much of that variation is reflected in the geometry of the trained model’s latent space.  \nWhile prior generative IK solvers have focused on sample quality [1, 21, 3, 18, 31], we instead treat constrained IK asa controlled setting for analyzing diffusion-model geometry. Closest to","cbCaislZFQzVc95d","https://ap.wps.com/l/cbCaislZFQzVc95d","pdf",7085554,5,1,21,"English","en",105,"# Introduction\n# Background\n## Constraint manifolds and their intrinsic geometry","[{\"question\":\"What makes constrained inverse kinematics a suitable testbed for studying diffusion-model geometry?\",\"answer\":\"Each task-space constraint defines a configuration-space constraint manifold with known intrinsic dimension and topology, providing analytical ground truth for evaluation.\"},{\"question\":\"How do the authors validate that the diffusion model learns the correct intrinsic dimension?\",\"answer\":\"They estimate intrinsic dimension from the model’s score function and compare it with the analytically known degrees of freedom of the underlying constraint manifold for each constraint type.\"},{\"question\":\"What does linear interpolation in the diffusion model’s latent space reveal?\",\"answer\":\"Generated solutions from latent-space linear interpolation stay close to the corresponding constraint manifold, indicating the learned representation captures manifold geometry beyond intrinsic dimension 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makes constrained inverse kinematics a suitable testbed for studying diffusion-model geometry?","Question",{"text":76,"@type":77},"Each task-space constraint defines a configuration-space constraint manifold with known intrinsic dimension and topology, providing analytical ground truth for evaluation.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How do the authors validate that the diffusion model learns the correct intrinsic dimension?",{"text":81,"@type":77},"They estimate intrinsic dimension from the model’s score function and compare it with the analytically known degrees of freedom of the underlying constraint manifold for each constraint type.",{"name":83,"@type":74,"acceptedAnswer":84},"What does linear interpolation in the diffusion model’s latent space reveal?",{"text":85,"@type":77},"Generated solutions from latent-space linear interpolation stay close to the corresponding constraint manifold, indicating the learned representation captures manifold geometry beyond 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