Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine

원문 초록

Diffusion models generate high-dimensional data with remarkable quality, yet how training discovers and exploits low-dimensional manifold structure remains incompletely understood. We analyze a two-stage collapse-and-refine procedure motivated by the small-noise score: its singular normal component identifies manifold projection as the leading denoising target, while the remaining correction encodes density and geometry. Under suitable assumptions, we prove that annealed mean-field training asymptotically recovers the population denoiser and quantify the resulting geometric alignment error. With the learned representation fixed, residual score learning reduces to convex random-feature regression. We establish generalization bounds that separate residual learning error, propagated projection error, and geometric approximation bias, with statistical exponents governed by the intrinsic dimension. Under additional structural assumptions, we further derive an end-to-end sampling guarantee. Motivated by this analysis, we develop Score-induced Latent Diffusion (SiLD), a practical two-stage framework combining low-noise denoising representation learning with latent diffusion, without requiring KL regularization of the latent representation. Experiments on Stacked MNIST and molecular generation benchmarks demonstrate competitive generation quality relative to architecture-matched VAE-based latent diffusion baselines.