l_inf-approximation of localized distributions

Distributions in spatial model often exhibit localized features. Intuitively, this locality implies a low intrinsic dimensionality, which can be exploited for efficient approximation and computation of complex distributions. However, existing approximation theory mainly considers the joint distributions, which does not guarantee that the marginal errors are small. In this work, we establish a dimension independent error bound for the marginals of approximate distributions. This $\ell_\infty$-approximation error is obtained using Stein's method, and we propose a $\delta$-locality condition that quantifies the degree of localization in a distribution. We also show how $\delta$-locality can be derived from different conditions that characterize the distribution's locality. Our $\ell_\infty$ bound motivates the localization of existing approximation methods to respect the locality. As examples, we show how to use localized likelihood-informed subspace method and localized score matching, which not only avoid dimension dependence in the approximation error, but also significantly reduce the computational cost due to the local and parallel implementation based on the localized structure.