The ability to implement effective autonomous hypersonic strikes against an adversary can potentially change the balance of power for our forces located in proximity to a powerful adversary, significantly extending the depth of our defense. Hypersonic weapons have an advantage over slower missiles in that they both reduce the response time of a strike and are more difficult to intercept. The fastest response time for weapons deployed at large distances from the theater of operations would be achieved by boost-glide hypersonic strike weapons (HSW) or maneuverable ballistic re-entry vehicles (MBRV) launched on a depressed trajectory. Both approaches could be made more economical by launching multiple HSW or MBRV from a single rocket, similar to our approach with our nuclear deterrent. Hypersonic cruise missiles launched from strategic bombers would have slower response times, but may be more economical.

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. Code for experiments in the paper is published at the following link https://github.com/ Despite the increasing importance of Wasserstein distance in applications, prior works have alluded to the concerns surrounding the high computational complexity of that distance. Additionally, it suffers from the curse of dimensionality, i.e., its sample complexity (the bounding gap of the distance between a probability measure and the empirical measures from its random samples) is of the order of O(n Over the years, numerous attempts have been made to improve the computational and sample complexities of the Wasserstein distance.