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经一维投影后所得Wasserstein距离的期望值，其随机性源于用于将两个输入测度投影至一维的投影方向。由于期望值难以直接计算，通常采用蒙特卡罗积分来估计SW距离。尽管已有多种变体，但尚无先例研究通过控制方差来改进SW距离的蒙特卡罗估计方案。为弥合方差缩减理论与SW距离研究之间的空白，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。其核心思路是：首先对投影后的一维测度进行高斯近似，进而利用两个高斯分布之间Wasserstein-2距离的闭式解来设计控制变量。具体而言，我们提出将两个拟合高斯分布Wasserstein-2距离的下界和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和点云测度比较中能显著降低方差。最后，我们在点云插值的梯度流以及CIFAR10和CelebA等标准图像数据集的深度生成模型中，展示了所提控制变量估计器的优异性能。