From Score Approximation to Distribution Approximation in Score-Based Diffusion Models

TL;DR - This paper proves that accurate neural-network approximation of a diffusion model’s score function yields a quantitatively close generated distribution in KL divergence. It connects classical score approximation guarantees to end-to-end generative accuracy.

• The KL-error bound depends explicitly on score approximation error, the diffusion noise schedule, and terminal prior mismatch.
• A residual error remains when the forward process’s terminal distribution differs from the reverse process’s initialization prior.
• The proof combines Hornik’s universal approximation theorem, Girsanov’s theorem on path space, and the data processing inequality.
• The analysis is approximation-theoretic rather than based on finite-sample statistics or structural assumptions about the data.