When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis

Score learning is a powerful framework for modeling complex distributions, with applications in generative modeling and Bayesian inverse problems. Focusing on the low-temperature regime (σ → 0), we uncover a sharp separation between two tasks: recovering the data distribution requires much higher score accuracy than recovering the data manifold. Our analysis shows that the leading-order term Θ(σ−2) depends only on the manifold support, allowing manifold recovery with error o(σ−2), while exact distribution recovery requires the much more stringent error of o(1). This insight enables efficient manifold sampling, including a simple diffusion-based algorithm that produces the uniform distribution on the manifold, and relaxes score-accuracy requirements for Bayesian inverse problems.