  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.
