  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.
