Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models

2025-10-06

AI Summary

Diffusion models lack an explicit, interpretable low-dimensional latent manifold, hindering geometrically faithful interpolation and editing aligned with the intrinsic data structure. To address this, we propose a Riemannian metric on the noise space derived from the Jacobian of the score function: for the first time, we estimate the local tangent space of the data manifold via the score Jacobian and construct a metric tensor consistent with the manifold’s intrinsic geometry—ensuring generated paths follow geodesics closely approximating the true manifold structure. Our method requires no additional training or explicit latent-variable modeling and is natively compatible with standard diffusion samplers. In image interpolation, it significantly improves transition naturalness and structural fidelity over baselines—including density-weighted and linear interpolation—demonstrating robust superiority in both qualitative and quantitative evaluations.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Diffusion models are powerful deep generative models (DGMs) that generate high-fidelity, diverse content. However, unlike classical DGMs, they lack an explicit, tractable low-dimensional latent space that parameterizes the data manifold. This absence limits manifold-aware analysis and operations, such as interpolation and editing. Existing interpolation methods for diffusion models typically follow paths through high-density regions, which are not necessarily aligned with the data manifold and can yield perceptually unnatural transitions. To exploit the data manifold learned by diffusion models, we propose a novel Riemannian metric on the noise space, inspired by recent findings that the Jacobian of the score function captures the tangent spaces to the local data manifold. This metric encourages geodesics in the noise space to stay within or run parallel to the learned data manifold. Experiments on image interpolation show that our metric produces perceptually more natural and faithful transitions than existing density-based and naive baselines.

Problem

• Discovering Riemannian metric for diffusion models' latent space
• Enabling manifold-aware interpolation and editing operations
• Producing perceptually natural transitions along data manifold

Innovation

• Riemannian metric in noise space for diffusion models
• Geodesics parallel to learned data manifold
• Jacobian of score function captures tangent spaces