Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data

Researchers developed new theoretical guarantees for score-based diffusion models that better reflect real-world data structures. The analysis shows these models can adapt to intrinsic low-dimensional geometry and avoid the curse of dimensionality through convergence rates based on Wasserstein dimension rather than ambient dimension.

• →New finite-sample error bounds for diffusion models work under milder conditions than previous analyses, requiring only finite moments without compact support assumptions.
• →Convergence rates depend on the intrinsic Wasserstein dimension rather than ambient dimension, demonstrating natural adaptation to data geometry.
• →The theoretical framework bridges diffusion model analysis with GANs and optimal transport theory.
• →Results apply to all Wasserstein-p distances and extend to distributions with unbounded support.
• →The work provides more optimistic convergence guarantees that better match empirical success of diffusion models.