Diffusion models are a new class of generative models that revolve around the estimation of the score function associated with a stochastic differential equation. Subsequent to its acquisition, the approximated score function is then harnessed to simulate the corresponding time-reversal process, ultimately enabling the generation of approximate data samples. Despite their evident practical significance these models carry, a notable challenge persists in the form of a lack of comprehensive quantitative results, especially in scenarios involving non-regular scores and estimators. In almost all reported bounds in Kullback Leibler (KL) divergence, it is assumed that either the score function or its approximation is Lipschitz uniformly in time. However, this condition is very restrictive in practice or appears to be difficult to establish. To circumvent this issue, previous works mainly focused on establishing convergence bounds in KL for an early stopped version of the diffusion model and a smoothed version of the data distribution, or assuming that the data distribution is supported on a compact manifold. These explorations have lead to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However, the question remains about the relevance of such early-stopping procedure or compactness conditions. In particular, if there exist a natural and mild condition ensuring explicit and sharp convergence bounds in KL. In this article, we tackle the aforementioned limitations by focusing on score diffusion models with fixed step size stemming from the Ornstein-Ulhenbeck semigroup and its kinetic counterpart. Our study provides a rigorous analysis, yielding simple, improved and sharp convergence bounds in KL applicable to any data distribution with finite Fisher information with respect to the standard Gaussian distribution.

翻译：扩散模型是一类新型生成模型，其核心在于估计与随机微分方程相关的分数函数。在获得近似分数函数后，通过模拟对应的时间反转过程，最终生成近似数据样本。尽管这些模型具有显著的实践价值，但一个关键挑战在于缺乏全面的定量结果，特别是在处理非正则分数和估计量的场景中。几乎所有基于Kullback-Leibler (KL)散度的已知理论界都假设分数函数或其近似在时间上一致Lipschitz连续，然而该条件在实际中极具限制性且难以验证。为规避此问题，先前研究主要致力于建立扩散模型提前停止版本与数据分布平滑版本之间的KL散度收敛界，或假设数据分布支撑于紧流形上。这些探索在Wasserstein或Fortet-Mourier度量下得到了有意义的界，但关于提前停止过程或紧致性条件的必要性仍存疑问——是否存在自然且温和的条件，能保证KL散度下显式且尖锐的收敛界？本文通过聚焦基于Ornstein-Uhlenbeck半群及其动力学对应形式的固定步长分数扩散模型，解决了上述局限性。我们的严谨分析给出了简洁、改进且尖锐的KL散度收敛界，该界适用于任何关于标准高斯分布具有有限Fisher信息的数据分布。