Projected Wasserstein gradient descent for high-dimensional Bayesian inference
This addresses computational challenges in Bayesian inference for high-dimensional problems, though it appears incremental as it builds on existing Wasserstein gradient methods with a projection technique.
The authors tackled the curse of dimensionality in high-dimensional Bayesian inference by proposing a projected Wasserstein gradient descent method that exploits low-rank structure between posterior and prior distributions, achieving scalable accuracy and convergence in numerical experiments.
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.