Extracting the Data Manifold from Diffusion Models via a Score-Based Non-Conformal Riemannian Metric

Diffusion models generate high-quality samples, but unlike VAEs or GANs they do not provide an explicit low-dimensional latent space that parameterizes the data manifold. This makes manifold-aware operations, such as geometrically faithful interpolation and guidance that stays on the learned manifold, difficult to formulate. We propose a training-free Riemannian metric on the noise space derived from the score Jacobian. The metric exploits its spectral structure, which separates tangent and normal directions of the data manifold, and therefore encourages paths and guidance updates to remain tangential to the manifold. It gives a single geometric tool for global geodesic interpolation and local correction of classifier-free guidance. Experiments on synthetic data, image and video interpolation, and text-to-image guidance show that the proposed metric preserves manifold geometry better than density-based alternatives.