Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel $δ$-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.

翻译：许多空间模型展现出局部性结构，这些结构能有效降低其内在维度，从而实现高维分布的高效近似与采样。然而，现有近似技术主要关注联合分布，未能为实际应用中备受关注的低维边缘分布提供精确的精度控制。通过利用局部性结构，我们为近似分布的边缘建立了与维度无关的均匀误差界。受Stein方法的启发，我们提出了一种新的$δ$-局部性条件来量化分布中的局部性特征，并将其与稀疏图模型等结构假设相关联。该理论保证推动了现有采样方法的局部化改进，我们通过局部化似然感知子空间方法与局部化得分匹配方法进行了具体阐释。研究表明，通过利用局部性结构，这些方法借助局部化并行实现显著降低了样本复杂度与计算成本。