  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.
