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：切片沃瑟斯坦（SW）距离已被广泛应用于不同场景，因其可扩展至大量支持点且不受维数灾难影响。该距离的本质是原始测度经拉东变换（RT）得到的一维表示（投影）之间运输代价的平均值。尽管SW距离在支持点数量上具有高效性，但在高维场景下仍需大量投影才能准确估计。因此，当支持点数量相对于维度较小时（例如采用小批量方法的深度学习应用场景），拉东变换的矩阵乘法复杂度将成为主要计算瓶颈。为解决该问题，我们提出通过线性随机组合少量投影（称为瓶颈投影）来生成投影。通过引入层级拉东变换（HRT）——即递归应用拉东变换变体构建的变换——我们阐述了此类投影的用法，并将其形式化为测度间的新型度量——层级切片沃瑟斯坦（HSW）距离。通过证明HRT的单射性，推导了HSW的度量性质。此外，我们探究了HSW的理论特性，包括其与SW变体的关联性及其计算复杂度与样本复杂度。最后，我们在CIFAR10、CelebA和Tiny ImageNet等基准数据集上，基于深度生成建模任务对比了HSW与传统SW的计算成本与生成质量。