Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models
This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of $O(k^{2}/\sqrt{T})$ (up to log factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.
Code (0)
등록된 구현이 없습니다.
Tasks
DenoisingMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data
Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of …
O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative task…
DenoisingDiffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This …
Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures
The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive a…
Low-dimensional adaptation of diffusion models: Convergence in total variation
This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the de…
Denoising