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.

翻译：切片Wasserstein（SW）距离定义为两个概率测度的一维投影之间Wasserstein距离的期望值。其随机性来源于用于将两个输入测度投影到一维的投影方向。由于期望值的不可计算性，通常采用蒙特卡洛积分来估计SW距离。尽管已有多种变体，但尚无先前工作在控制估计方差方面改进SW距离的蒙特卡洛估计方案。为桥接方差缩减与SW距离的文献，我们提出计算高效的**控制变量**方法以降低SW距离经验估计的方差。核心思想为：首先寻找投影后一维测度的高斯近似，继而利用两个高斯分布之间Wasserstein-2距离的闭式解设计控制变量。具体而言，我们提出将两个拟合高斯分布之间Wasserstein-2距离的下界与上界作为两种计算高效的控制变量。实验表明，所提出的控制变量估计器在图像与点云测度比较中能显著降低方差。最后，我们展示了该控制变量估计器在梯度流插值点云以及标准图像数据集（如CIFAR10与CelebA）上深度生成建模中的优异性能。