Sobolev Regularized Score Difference Estimation in Diffusion Models

Score-based diffusion models are increasingly considered in settings where the underlying data distribution may differ from the training distribution, yet existing theoretical guarantees largely focus on the no-shift setting. In this work, we study robust score estimation under Wasserstein perturbations of a reference distribution. For the Ornstein--Uhlenbeck diffusion, we show that robust estimation decomposes into two fundamental components: the statistical cost of learning the reference distribution and the intrinsic cost of distribution shift. The latter scales quadratically with the Wasserstein radius, and this dependence is minimax optimal. We construct an explicit finite-sample estimator achieving the resulting robust minimax rate without knowing the shift radius. When the reference distribution lies on an unknown low-dimensional subspace, the statistical term adapts to the intrinsic dimension while the shift cost remains unchanged. Finally, we show that the same decomposition governs positive-time reverse sampling and obtain matching minimax guarantees in KL divergence. Together, these results characterize how finite data, intrinsic dimension, and distribution shift affect the robustness of score-based diffusion models.

Generative modeling across related distributions raises a question: how can shared structure support accurate generation for a target distribution with limited observations? We develop a statistical framework for diffusion transfer that analyzes a shared pooled score together with a target-specific offset. Under Besov regularity of the pooled density and a stable hierarchical composition model for the offset, we derive convergence rates that separate the costs of learning these components and identify when a simpler offset yields a faster risk-bound rate than separate fitting. Under suitable boundary and regularity assumptions, we prove that the offset remains bounded and jointly regular as diffusion time approaches zero. These results allow the lower time cutoff to vanish polynomially and provide total-variation guarantees for diffusion models as density estimators. Synthetic simulations and image benchmarks provide a proof of concept for the framework.

Despite the remarkable empirical success of score-based diffusion models, their statistical guarantees remain underdeveloped. Existing analyses often provide pessimistic convergence rates that do not reflect the intrinsic low-dimensional structure common in real data, such as that arising in natural images. In this work, we study the statistical convergence of score-based diffusion models for learning an unknown distribution $μ$ from finitely many samples. Under mild regularity conditions on the forward diffusion process and the data distribution, we derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance. Unlike prior results, our guarantees hold for all $p \ge 1$ and require only a finite-moment assumption on $μ$, without compact-support, manifold, or smooth-density conditions. Specifically, given $n$ i.i.d.\ samples from $μ$ with finite $q$-th moment and appropriately chosen network architectures, hyperparameters, and discretization schemes, we show that the expected Wasserstein-$p$ error between the learned distribution $\hatμ$ and $μ$ scales as $\mathbb{E}\, \mathbb{W}_p(\hatμ,μ) = \widetilde{O}\!\left(n^{-1 / d^\ast_{p,q}(μ)}\right),$ where $d^\ast_{p,q}(μ)$ is the $(p,q)$-Wasserstein dimension of $μ$. Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data and mitigate the curse of dimensionality, since the convergence rate depends on $d^\ast_{p,q}(μ)$ rather than the ambient dimension. Moreover, our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport. The proposed $(p,q)$-Wasserstein dimension also extends the notion of classical Wasserstein dimension to distributions with unbounded support, which may be of independent theoretical interest.