  We consider the problem of sampling from a distribution governed by a
potential function. This work proposes an explicit score based MCMC method that
is deterministic, resulting in a deterministic evolution for particles rather
than a stochastic differential equation evolution. The score term is given in
closed form by a regularized Wasserstein proximal, using a kernel convolution
that is approximated by sampling. We demonstrate fast convergence on various
problems and show improved dimensional dependence of mixing time bounds for the
case of Gaussian distributions compared to the unadjusted Langevin algorithm
(ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally
derive closed form expressions for the distributions at each iterate for
quadratic potential functions, characterizing the variance reduction. Empirical
results demonstrate that the particles behave in an organized manner, lying on
level set contours of the potential. Moreover, the posterior mean estimator of
the proposed method is shown to be closer to the maximum a-posteriori estimator
compared to ULA and MALA in the context of Bayesian logistic regression.
Additional examples demonstrate competitive performance for Bayesian neural
network training.
