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. 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.

翻译：切片瓦瑟斯坦（Sliced Wasserstein, SW）距离已成为比较概率测度时瓦瑟斯坦距离的流行替代方案。其广泛应用包括图像处理、领域自适应和生成建模，在这些场景中，通常优化某些参数以最小化SW，该距离作为离散概率测度之间的损失函数（因为具有密度的测度在数值上难以实现）。所有这些优化问题都蕴含同一子问题，即最小化切片瓦瑟斯坦能量。本文研究$\mathcal{E}: Y \longmapsto \mathrm{SW}_2^2(\gamma_Y, \gamma_Z)$的性质，即两个具有相同点数的均匀离散测度之间的SW距离，将其中一个测度的支撑集$Y \in \mathbb{R}^{n \times d}$视为函数。我们探讨该能量的正则性与优化性质，及其蒙特卡洛近似$\mathcal{E}_p$（仅用$p$个样本估计SW中的期望），证明了$\mathcal{E}_p$的临界点收敛到$\mathcal{E}$的临界点的结果，以及几乎必然一致收敛性。最后，我们证明在某种意义上，最小化$\mathcal{E}$和$\mathcal{E}_p$的随机梯度下降方法收敛到这些能量的（Clarke）临界点。