Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions (2409.18804v2)

Abstract: Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning. In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t. the Kullback-Leibler divergence, and $O(\sqrt{D})$ w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

• The paper introduces a theoretical framework linking DDPMs with Gaussian Processes to achieve convergence rates independent of the ambient dimension.
• It establishes high-probability bounds for the score function and derives convergence rates for both KL divergence and Wasserstein distance based on intrinsic dimensions.
• The study proposes a novel discretization scheme that reduces sampling complexity from O(d^4 log^4 D) to O(d^3 log^2 D), enhancing practical implementation.

Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

The paper "Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions" by Iskander Azangulov, George Deligiannidis, and Judith Rousseau presents a detailed theoretical paper on the behavior of Denoising Diffusion Probabilistic Models (DDPMs) under the manifold hypothesis in high-dimensional spaces. This investigation is primarily concerned with understanding how these models perform when the data distribution lies on a lower-dimensional manifold within an ambient high-dimensional space.

Theoretical Foundations

The manifold hypothesis posits that real-world high-dimensional data often resides near a much lower-dimensional manifold. The authors aim to explore the implications of this hypothesis for the convergence and performance of DDPMs, which have shown empirical success in various applications such as image, audio, and video generation. The framework builds on the theory of Gaussian Processes and provides new convergence rates for diffusion models that are independent of the ambient dimension.

Key Contributions

1. Framework and Convergence Rates:
   • The paper introduces a new theoretical framework connecting DDPMs to Gaussian Processes, deriving convergence rates for learning the score function and sampling using DDPMs.
   • Specifically, it is shown that under the manifold hypothesis, DDPMs achieve rates independent of the ambient dimension in terms of learning the score function. For the Kullback-Leibler divergence, the rates are independent of the ambient dimension, and for the Wasserstein distance, the rates scale as $O(\sqrt{D})$.
2. High-Probability Bounds:
   • New high-probability bounds on the score function $s(t, x)$ are derived, demonstrating that these bounds depend only on the intrinsic dimension $d$, log-complexity $C_{\log}$, and not on the ambient dimension $D$.
   • These bounds are established through concentration results on the maximum of Gaussian processes, showing that the noise added during the forward diffusion process is almost orthogonal to the manifold when $D$ is large.
3. Discretization Scheme:
   • The authors propose a novel discretization scheme improving the sampling complexity from previous $O(d^4 \log^4 D)$ to $O(d^3 \log^2 D)$. The analysis relies on precise high-probability bounds on the score function obtained in the previous sections.
4. Minimax Manifold Estimator:
   • The paper enhances the existing minimax manifold estimator by significantly reducing the dimensionality of the optimization problem. The proposed estimator has an error rate independent of $D$ and can be efficiently implemented in practice.

Results and Implications

The results indicate that diffusion models are robust under the manifold hypothesis, providing near-optimal convergence rates that only depend on the intrinsic dimension $d$ rather than the ambient dimension $D$. This has practical significance as it suggests that high-dimensional generative modeling can be effective even when the underlying data distribution lies on a low-dimensional manifold.

Future Directions

The paper opens several avenues for future research:

• Relaxing the technical assumption of the smoothness of the density and the manifold.
• Further improving the sampling complexity bounds, potentially reducing the dependency on $d$.
• Extending the methods to other types of generative models and exploring their behavior under manifold hypothesis.
• Investigating practical algorithms and optimization techniques to efficiently implement the proposed neural network-based score function approximators.

In conclusion, this work provides significant theoretical advancements in understanding the performance of diffusion models under the manifold hypothesis, potentially leading to more effective high-dimensional generative models. The findings not only contribute to the theoretical underpinnings of generative modeling but also have practical implications for designing efficient algorithms in high-dimensional data spaces.