In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a \emph{reflected} diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, we establish minimax optimal rates of convergence in total variation, up to a polylogarithmic factor, under Sobolev smoothness assumptions. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

翻译：近年来，去噪扩散模型因其在快速发展的生成式人工智能领域的广泛应用而成为关键研究方向。尽管近期的统计学进展已为理想化去噪扩散模型在高维目标数据上的生成能力提供了理论解释，实际实现中常对生成过程引入阈值处理，以克服此类模型无界状态空间引发的问题。扩散模型理论设计与实际实现之间的这种不匹配，在经验上已通过采用反射扩散过程作为噪声驱动机制得到解决。本文系统研究这类反射去噪扩散模型的统计保证。具体而言，我们在Sobolev平滑性假设下，建立了总变差距离中达到多对数因子最优的极小极大收敛速率。主要贡献包括：对此新型反射去噪扩散模型类别的统计分析，以及结合谱分解与严格神经网络分析、在时间与空间维度同步优化的精细化分数逼近方法。