Wasserstein distances are metrics on probability distributions inspired by
the problem of optimal mass transportation. Roughly speaking, they measure the
minimal effort required to reconfigure the probability mass of one distribution
in order to recover the other distribution. They are ubiquitous in mathematics,
with a long history that has seen them catalyse core developments in analysis,
optimization, and probability. Beyond their intrinsic mathematical richness,
they possess attractive features that make them a versatile tool for the
statistician: they can be used to derive weak convergence and convergence of
moments, and can be easily bounded; they are well-adapted to quantify a natural
notion of perturbation of a probability distribution; and they seamlessly
incorporate the geometry of the domain of the distributions in question, thus
being useful for contrasting complex objects. Consequently, they frequently
appear in the development of statistical theory and inferential methodology,
and have recently become an object of inference in themselves. In this review,
we provide a snapshot of the main concepts involved in Wasserstein distances
and optimal transportation, and a succinct overview of some of their many
statistical aspects.

本文介绍了 Wasserstein 距离作为概率分布度量的数学概念，以及其在统计学中的重要应用，包括了弱收敛、矩收敛、概率分布扰动等内容。

Wasserstein 距离的统计学特征

Statistical Aspects of Wasserstein Distances

Deep learning is the mainstream technique for many machine learning tasks,
including image recognition, machine translation, speech recognition, and so
on. It has outperformed conventional methods in various fields and achieved
great successes. Unfortunately, the understanding on how it works remains
unclear. It has the central importance to lay down the theoretic foundation for
deep learning.
In this work, we give a geometric view to understand deep learning: we show
that the fundamental principle attributing to the success is the manifold
structure in data, namely natural high dimensional data concentrates close to a
low-dimensional manifold, deep learning learns the manifold and the probability
distribution on it.
We further introduce the concepts of rectified linear complexity for deep
neural network measuring its learning capability, rectified linear complexity
of an embedding manifold describing the difficulty to be learned. Then we show
for any deep neural network with fixed architecture, there exists a manifold
that cannot be learned by the network. Finally, we propose to apply optimal
mass transportation theory to control the probability distribution in the
latent space.

该研究通过几何视角揭示深度学习的关键在于学习数据的流形结构和概率分布，并提出一种控制潜在空间中概率分布的最优质量传递理论。