  We propose a projected Wasserstein gradient descent method (pWGD) for
high-dimensional Bayesian inference problems. The underlying density function
of a particle system of WGD is approximated by kernel density estimation (KDE),
which faces the long-standing curse of dimensionality. We overcome this
challenge by exploiting the intrinsic low-rank structure in the difference
between the posterior and prior distributions. The parameters are projected
into a low-dimensional subspace to alleviate the approximation error of KDE in
high dimensions. We formulate a projected Wasserstein gradient flow and analyze
its convergence property under mild assumptions. Several numerical experiments
illustrate the accuracy, convergence, and complexity scalability of pWGD with
respect to parameter dimension, sample size, and processor cores.
