Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models (2501.18863v1)

Abstract: Score-based generative models, which transform noise into data by learning to reverse a diffusion process, have become a cornerstone of modern generative AI. This paper contributes to establishing theoretical guarantees for the probability flow ODE, a widely used diffusion-based sampler known for its practical efficiency. While a number of prior works address its general convergence theory, it remains unclear whether the probability flow ODE sampler can adapt to the low-dimensional structures commonly present in natural image data. We demonstrate that, with accurate score function estimation, the probability flow ODE sampler achieves a convergence rate of $O(k/T)$ in total variation distance (ignoring logarithmic factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of iterations. This dimension-free convergence rate improves upon existing results that scale with the typically much larger ambient dimension, highlighting the ability of the probability flow ODE sampler to exploit intrinsic low-dimensional structures in the target distribution for faster sampling.

Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models

The paper authored by Jiaqi Tang and Yuling Yan provides a rigorous exploration into the theoretical underpinnings of score-based generative models (SGMs), with a specific focus on the adaptability and convergence of the probability flow ordinary differential equation (ODE) sampler. SGMs have gained prominence for their ability to simulate complex, high-dimensional data through diffusion processes, effectively reversing these processes to create realistic data samples. This work investigates whether the probability flow ODE sampler can adeptly harness low-dimensional data structures, often a characteristic of natural images and prevalent within various data distributions.

Central to this paper is the demonstration of the probability flow ODE sampler's convergence rate, denoted as $O(k/T)$, in total variation distance. Here, $k$ represents the intrinsic dimensionality of the target distribution, while $T$ denotes the number of iterations. This dimension-independent convergence rate significantly improves upon previous results which scale with the typically larger ambient dimension $d$, thereby affirming the sampler's capability to efficiently utilize intrinsic low-dimensional structures in the target distribution.

The paper elucidates the operational dynamics of SGMs, describing their foundation in dual processes: a forward process that gradually transforms data to noise, and a reverse process that reconstructs data from this noise. The authors focus particularly on probability flow ODEs, a deterministic alternative within DDIM-type samplers, which convert random samples back to realistic data by refining Gaussian noise.

Despite the prevailing use of SGMs, a critical challenge lies in the accurate estimation of stein score functions, approximated using neural networks trained via score-matching techniques. The authors emphasize the significance of achieving precision in these estimates, which directly influences the probability flow ODE sampler's convergence efficiency.

In light of empirical observations that suggest image data distributions are concentrated on manifolds or low-dimensional subspaces, this paper's findings are particularly salient. The authors propose a coefficient design enabling the probability flow ODE sampler to adaptively converge to such low-dimensional structures, thereby improving convergence speed without being shackled by the ambient space's high dimensionality.

Beyond offering a streamlined analysis framework, the paper sits within a broader dialogue of recent studies aiming to extrapolate SGMs' efficacy to broader data classes beyond those merely conforming to smooth or log-concave structures. However, the paper also acknowledges ongoing work, such as a contemporaneous paper by Liang et al., that explores similar adaptability challenges, highlighting the significance and timeliness of such investigations.

From a theoretical standpoint, this paper contributes to a more nuanced understanding of the operational mechanics and potential of SGMs and diffusion models. Practically, the findings could inform a new cadre of algorithms better capable of handling real-world data's inherent complexity and low-dimensional character. The implications for this are profound in the field of generative AI, enabling more efficient training and sampling processes.

Moving forward, this line of research might explore further optimizations and adaptations of deterministic sampler processes, potentially uncovering novel relationships or configurations that optimize data generation even further. The ideas presented in this work could also serve as a foundational tool for more expansive explorations into the implications of intrinsic dimensionality across different types of data applications within AI-driven environments.