Sliced Wasserstein (SW) distance has been widely used in different application scenarios since it can be scaled to a large number of supports without suffering from the curse of dimensionality. The value of sliced Wasserstein distance is the average of transportation cost between one-dimensional representations (projections) of original measures that are obtained by Radon Transform (RT). Despite its efficiency in the number of supports, estimating the sliced Wasserstein requires a relatively large number of projections in high-dimensional settings. Therefore, for applications where the number of supports is relatively small compared with the dimension, e.g., several deep learning applications where the mini-batch approaches are utilized, the complexities from matrix multiplication of Radon Transform become the main computational bottleneck. To address this issue, we propose to derive projections by linearly and randomly combining a smaller number of projections which are named bottleneck projections. We explain the usage of these projections by introducing Hierarchical Radon Transform (HRT) which is constructed by applying Radon Transform variants recursively. We then formulate the approach into a new metric between measures, named Hierarchical Sliced Wasserstein (HSW) distance. By proving the injectivity of HRT, we derive the metricity of HSW. Moreover, we investigate the theoretical properties of HSW including its connection to SW variants and its computational and sample complexities. Finally, we compare the computational cost and generative quality of HSW with the conventional SW on the task of deep generative modeling using various benchmark datasets including CIFAR10, CelebA, and Tiny ImageNet.
翻译:Sliced Wasserstein (SW) 距离在不同的常规应用情景中被广泛使用,因为它可以缩放成大量支持,而不受维度诅咒的影响。切片瓦瑟斯坦距离的价值是Radon 变换公司(RT)获得的原始测量的一维表现(预测)之间的平均运输成本。尽管其支持数量效率很高,但估计被切片瓦瑟斯坦(Swest)需要在高维环境中进行数量较大的预测。因此,对于支持数量相对维度而言相对较小的应用应用,例如,在使用小型质量方法的一些深层次学习应用中,来自拉登变异的矩阵变异的复杂性成为主要的计算瓶颈。为了解决这一问题,我们提议通过直线和随机结合较少数量的预测来作出预测,这些预测的用途是高度变异异性模型(HRT),该变异性是应用Radon 变异性(HRT) 。我们随后将这一方法编成一个新的测量尺度,名为Red-batch roal 变异性 变异性 变异性 变式的Ral 变异性 变异性 变式 变式 变式 变式 变式 变式 变式 变式 变式 变式, 变式 变式 变式 变式 变式 变式 变式