Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality
This provides a theoretical foundation for the efficiency of DDPMs in generative AI, showing optimal adaptivity to low-dimensional data structures, though it is incremental as it builds on prior work and aligns with concurrent research.
The paper demonstrates that denoising diffusion probabilistic models (DDPMs) achieve nearly linear iteration complexity with the intrinsic dimension k of data distributions, which is optimal for KL divergence, addressing the gap between theoretical guarantees and practical efficiency.
The denoising diffusion probabilistic model (DDPM) has emerged as a mainstream generative model in generative AI. While sharp convergence guarantees have been established for the DDPM, the iteration complexity is, in general, proportional to the ambient data dimension, resulting in overly conservative theory that fails to explain its practical efficiency. This has motivated the recent work Li and Yan (2024a) to investigate how the DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. We strengthen this line of work by demonstrating, in some sense, optimal adaptivity to unknown low dimensionality. For a broad class of data distributions with intrinsic dimension $k$, we prove that the iteration complexity of the DDPM scales nearly linearly with $k$, which is optimal when using KL divergence to measure distributional discrepancy. Notably, our work is closely aligned with the independent concurrent work Potaptchik et al. (2024) -- posted two weeks prior to ours -- in establishing nearly linear-$k$ convergence guarantees for the DDPM.