  The Sliced Wasserstein (SW) distance has become a popular alternative to the
Wasserstein distance for comparing probability measures. Widespread
applications include image processing, domain adaptation and generative
modelling, where it is common to optimise some parameters in order to minimise
SW, which serves as a loss function between discrete probability measures
(since measures admitting densities are numerically unattainable). All these
optimisation problems bear the same sub-problem, which is minimising the Sliced
Wasserstein energy. In this paper we study the properties of $\mathcal{E}: Y
\longmapsto \mathrm{SW}_2^2(\gamma_Y, \gamma_Z)$, i.e. the SW distance between
two uniform discrete measures with the same amount of points as a function of
the support $Y \in \mathbb{R}^{n \times d}$ of one of the measures. We
investigate the regularity and optimisation properties of this energy, as well
as its Monte-Carlo approximation $\mathcal{E}_p$ (estimating the expectation in
SW using only $p$ samples) and show convergence results on the critical points
of $\mathcal{E}_p$ to those of $\mathcal{E}$, as well as an almost-sure uniform
convergence and a uniform Central Limit result on the process
$\mathcal{E}_p(Y)$. Finally, we show that in a certain sense, Stochastic
Gradient Descent methods minimising $\mathcal{E}$ and $\mathcal{E}_p$ converge
towards (Clarke) critical points of these energies.
