What is the optimal way to approximate a high-dimensional diffusion process by one in which the coordinates are independent? This paper presents a construction, called the \emph{independent projection}, which is optimal for two natural criteria. First, when the original diffusion is reversible with invariant measure $\rho_*$, the independent projection serves as the Wasserstein gradient flow for the relative entropy $H(\cdot\,|\,\rho_*)$ constrained to the space of product measures. This is related to recent Langevin-based sampling schemes proposed in the statistical literature on mean field variational inference. In addition, we provide both qualitative and quantitative results on the long-time convergence of the independent projection, with quantitative results in the log-concave case derived via a new variant of the logarithmic Sobolev inequality. Second, among all processes with independent coordinates, the independent projection is shown to exhibit the slowest growth rate of path-space entropy relative to the original diffusion. This sheds new light on the classical McKean-Vlasov equation and recent variants proposed for non-exchangeable systems, which can be viewed as special cases of the independent projection.

翻译：如何最优地用一个坐标独立的过程来近似高维扩散过程？本文提出了一种称为“独立投影”的构造，该构造在两种自然准则下均为最优。首先，当原始扩散过程具有可逆性且不变测度为$\rho_*$时，独立投影可作为相对熵$H(\cdot\,|\,\rho_*)$在乘积测度空间约束下的Wasserstein梯度流。这一定理与统计文献中关于均场变分推断的近期朗之万采样方案相关。此外，我们提供了独立投影长期收敛性的定性与定量结果，其中对数凹情形下的定量结果通过一种新的对数索博列夫不等式变体推导得出。其次，在所有坐标独立的过程中，独立投影被证明具有相对于原始扩散过程最慢的路径空间熵增长率。这为经典的McKean-Vlasov方程及其针对非可交换系统提出的近期变体（均可视为独立投影的特例）提供了新的视角。