While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

翻译：尽管基于分数的生成模型在无约束欧几里得空间中的数学基础日益清晰，但许多实际应用涉及限制在有界区域内的数据。本文针对超立方体 $[0,1]^D$ 上的反射扩散模型进行了统计分析，目标分布支持在 $d$ 维线性子空间上。该设定下的主要挑战在于缺乏高斯转移核——而高斯转移核在 $\mathbb{R}^D$ 的标准理论中处于核心地位。通过采用易于实现的转移密度无穷级数展开，我们开发了分析工具来限制得分函数及其通过稀疏ReLU网络的近似。对于具有Sobolev光滑性 $α$ 的目标密度，我们建立了 $1$-Wasserstein 距离下的收敛率为 $n^{-\frac{α+1-δ}{2α+d}}$（对任意小 $δ> 0$），表明该生成算法完全适应于内在维度 $d$。这些结果证实，反射边界的存在不会降低扩散范式的基本统计效率，达到了与无约束设定中已知的近乎最优速率相匹配。