Sliced Wasserstein distances preserve properties of classic Wasserstein
distances while being more scalable for computation and estimation in high
dimensions. The goal of this work is to quantify this scalability from three
key aspects: (i) empirical convergence rates; (ii) robustness to data
contamination; and (iii) efficient computational methods. For empirical
convergence, we derive fast rates with explicit dependence of constants on
dimension, subject to log-concavity of the population distributions. For
robustness, we characterize minimax optimal, dimension-free robust estimation
risks, and show an equivalence between robust sliced 1-Wasserstein estimation
and robust mean estimation. This enables lifting statistical and algorithmic
guarantees available for the latter to the sliced 1-Wasserstein setting. Moving
on to computational aspects, we analyze the Monte Carlo estimator for the
average-sliced distance, demonstrating that larger dimension can result in
faster convergence of the numerical integration error. For the max-sliced
distance, we focus on a subgradient-based local optimization algorithm that is
frequently used in practice, albeit without formal guarantees, and establish an
$O(\epsilon^{-4})$ computational complexity bound for it. Our theory is
validated by numerical experiments, which altogether provide a comprehensive
quantitative account of the scalability question.

本文研究了切片瓦瑟斯坦距离在不同方面的可扩展性，包括实证收敛性、数据污染下的鲁棒性、以及高效的计算方法，并提出了用于切片瓦瑟斯坦距离常数维度的快速率。同时，本文研究了蒙特卡洛估计器和局部优化算法等方面，验证了理论研究结果。

基于切片 Wasserstein 距离的统计、稳健性、和计算保证

Statistical, Robustness, and Computational Guarantees for Sliced  Wasserstein Distances

Computing optimal transport (OT) between measures in high dimensions is
doomed by the curse of dimensionality. A popular approach to avoid this curse
is to project input measures on lower-dimensional subspaces (1D lines in the
case of sliced Wasserstein distances), solve the OT problem between these
reduced measures, and settle for the Wasserstein distance between these
reductions, rather than that between the original measures. This approach is
however difficult to extend to the case in which one wants to compute an OT map
(a Monge map) between the original measures. Since computations are carried out
on lower-dimensional projections, classical map estimation techniques can only
produce maps operating in these reduced dimensions. We propose in this work two
methods to extrapolate, from an transport map that is optimal on a subspace,
one that is nearly optimal in the entire space. We prove that the best optimal
transport plan that takes such "subspace detours" is a generalization of the
Knothe-Rosenblatt transport. We show that these plans can be explicitly
formulated when comparing Gaussian measures (between which the Wasserstein
distance is commonly referred to as the Bures or Fr\'echet distance). We
provide an algorithm to select optimal subspaces given pairs of Gaussian
measures, and study scenarios in which that mediating subspace can be selected
using prior information. We consider applications to semantic mediation between
elliptic word embeddings and domain adaptation with Gaussian mixture models.

本文提出了两种方法，从子空间中的最优传输中推断出在整个空间中的近似最优传输， 进而研究了高斯混合模型的领域适应方案并应用于椭圆词嵌入的语义中介。

子空间绕行：构建在子空间投影上最优的传输计划

Subspace Detours: Building Transport Plans that are Optimal on Subspace  Projections

Wasserstein distances are increasingly used in a wide variety of applications
in machine learning. Sliced Wasserstein distances form an important subclass
which may be estimated efficiently through one-dimensional sorting operations.
In this paper, we propose a new variant of sliced Wasserstein distance, study
the use of orthogonal coupling in Monte Carlo estimation of Wasserstein
distances and draw connections with stratified sampling, and evaluate our
approaches experimentally in a range of large-scale experiments in generative
modelling and reinforcement learning.

本文介绍了切片 Wasserstein 距离的一种新变体，并研究了在 Wasserstein 距离的蒙特卡罗估计中使用正交耦合的方法，并与分层抽样建立了联系，并在生成建模和强化学习的大规模实验中进行了性能评估。