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Wasserstein（Sliced Wasserstein, SW）距离已日益成为Wasserstein距离在概率测度比较中的流行替代方案。其广泛应用涵盖图像处理、域自适应及生成建模等领域，在这些场景中，常见做法是优化某些参数以最小化SW，将其作为离散概率测度间的损失函数（因为具有密度的测度在数值上难以实现）。这些优化问题均涉及同一子问题，即最小化切片Wasserstein能量。本文研究$\mathcal{E}: Y \longmapsto \mathrm{SW}_2^2(\gamma_Y, \gamma_Z)$的性质，其中$\mathcal{E}$表示具有相同点数的两个均匀离散测度之间的SW距离，作为其中一个测度的支撑集$Y \in \mathbb{R}^{n \times d}$的函数。我们探究该能量的正则性与优化性质，以及其蒙特卡洛近似$\mathcal{E}_p$（仅使用$p$个样本估计SW中的期望），并证明$\mathcal{E}_p$的临界点收敛至$\mathcal{E}$的临界点的结果，同时给出几乎必然一致收敛性。最后，我们表明，在特定意义下，最小化$\mathcal{E}$与$\mathcal{E}_p$的随机梯度下降方法收敛至这些能量的（Clarke）临界点。