  The sliced Wasserstein (SW) distances between two probability measures are
defined as the expectation of the Wasserstein distance between two
one-dimensional projections of the two measures. The randomness comes from a
projecting direction that is used to project the two input measures to one
dimension. Due to the intractability of the expectation, Monte Carlo
integration is performed to estimate the value of the SW distance. Despite
having various variants, there has been no prior work that improves the Monte
Carlo estimation scheme for the SW distance in terms of controlling its
variance. To bridge the literature on variance reduction and the literature on
the SW distance, we propose computationally efficient control variates to
reduce the variance of the empirical estimation of the SW distance. The key
idea is to first find Gaussian approximations of projected one-dimensional
measures, then we utilize the closed-form of the Wasserstein-2 distance between
two Gaussian distributions to design the control variates. In particular, we
propose using a lower bound and an upper bound of the Wasserstein-2 distance
between two fitted Gaussians as two computationally efficient control variates.
We empirically show that the proposed control variate estimators can help to
reduce the variance considerably when comparing measures over images and
point-clouds. Finally, we demonstrate the favorable performance of the proposed
control variate estimators in gradient flows to interpolate between two
point-clouds and in deep generative modeling on standard image datasets, such
as CIFAR10 and CelebA.
