Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices

Diffusion models generate high-dimensional data by iteratively denoising Gaussian noise. While it is well-established that these models can exploit low-dimensional data structure to accelerate sampling, existing theoretical guarantees have largely been restricted to narrowly defined update coefficients tied to specific discretization schemes. This paper investigates whether this adaptation to low-dimensional structure is a robust property or if it is sensitive to the precise choice of update coefficients used in practice.

The authors develop a unified theoretical framework for a broad class of diffusion samplers. They define an auxiliary random process that tracks the sampling iterates and analyze the convergence of the sampler output in total variation (TV) distance. By defining a noise deviation sequence that measures the discrepancy between the effective noise level induced by the update coefficients and the canonical forward process, the authors prove that the discretization error depends on the intrinsic dimension of the data rather than the ambient dimension.

The study demonstrates that low-dimensional adaptation is a robust property of diffusion models. For a wide range of update coefficient choices, the number of iterations required to generate an epsilon-accurate sample is proportional to the intrinsic dimension k, up to logarithmic factors. This result provides a theoretical justification for why various practical samplers—such as the original DDPM, improved DDPM, and Analytic-DPM—exhibit similar empirical performance despite using different coefficient configurations. The authors further show that even when coefficient-induced noise deviations are non-zero, one can adjust the noise schedule to ensure the sampler output matches the target distribution's signal-to-noise ratio, maintaining the same low-dimensional convergence guarantees.