  The sliced Wasserstein (SW) distance has been widely recognized as a
statistically effective and computationally efficient metric between two
probability measures. A key component of the SW distance is the slicing
distribution. There are two existing approaches for choosing this distribution.
The first approach is using a fixed prior distribution. The second approach is
optimizing for the best distribution which belongs to a parametric family of
distributions and can maximize the expected distance. However, both approaches
have their limitations. A fixed prior distribution is non-informative in terms
of highlighting projecting directions that can discriminate two general
probability measures. Doing optimization for the best distribution is often
expensive and unstable. Moreover, designing the parametric family of the
candidate distribution could be easily misspecified. To address the issues, we
propose to design the slicing distribution as an energy-based distribution that
is parameter-free and has the density proportional to an energy function of the
projected one-dimensional Wasserstein distance. We then derive a novel sliced
Wasserstein metric, energy-based sliced Waserstein (EBSW) distance, and
investigate its topological, statistical, and computational properties via
importance sampling, sampling importance resampling, and Markov Chain methods.
Finally, we conduct experiments on point-cloud gradient flow, color transfer,
and point-cloud reconstruction to show the favorable performance of the EBSW.
