This paper investigates the performance of score-based diffusion models, specifically the Denoising Diffusion Probabilistic Model (DDPM), when the target distribution is concentrated on or near low-dimensional manifolds within a higher-dimensional space. The authors identify a particular coefficient design that enables the DDPM sampler to adapt to unknown low-dimensional structures, achieving a dimension-free convergence rate of \( O(k^2 / \sqrt{T}) \), where \( k \) is the intrinsic dimension of the target distribution and \( T \) is the number of steps. This is the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures, highlighting the critical importance of coefficient design. The paper also provides a proof of this result, showing that the error bound is nearly dimension-free, with the ambient dimension appearing only in logarithmic terms. Additionally, the authors show that their choice of coefficients is unique in terms of achieving this dimension-free error bound, emphasizing the importance of careful coefficient design for the DDPM sampler.This paper investigates the performance of score-based diffusion models, specifically the Denoising Diffusion Probabilistic Model (DDPM), when the target distribution is concentrated on or near low-dimensional manifolds within a higher-dimensional space. The authors identify a particular coefficient design that enables the DDPM sampler to adapt to unknown low-dimensional structures, achieving a dimension-free convergence rate of \( O(k^2 / \sqrt{T}) \), where \( k \) is the intrinsic dimension of the target distribution and \( T \) is the number of steps. This is the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures, highlighting the critical importance of coefficient design. The paper also provides a proof of this result, showing that the error bound is nearly dimension-free, with the ambient dimension appearing only in logarithmic terms. Additionally, the authors show that their choice of coefficients is unique in terms of achieving this dimension-free error bound, emphasizing the importance of careful coefficient design for the DDPM sampler.