Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models

Diffusion models are powerful deep generative models, butunlike classical models, they lack an explicit low-dimensional latent spacethat parameterizes the data manifold. This absence makes it difficultto perform manifold-aware operations, such as geometrically faithful in-terpolation or conditional guidance that respects the learned manifold.We propose a training-free Riemannian metric on the noise space, de-rived from the Jacobian of the score function. The key insight is thatthe spectral structure of this Jacobian separates tangent and normaldirections of the data manifold; our metric leverages this separation to en-courage paths to stay tangential to the manifold rather than drift towardhigh-density regions. To validate that our metric faithfully captures themanifold geometry, we examine it from two complementary angles. First,geodesics under our metric yield perceptually more natural interpolationsthan existing methods on synthetic, image, and video frame datasets.Second, the tangent–normal decomposition induced by our metric pre-vents classifier-free guidance from deviating off the manifold, improvinggeneration quality while preserving text-image alignment.