Comparing probability distributions is a fundamental problem in data
sciences. Simple norms and divergences such as the total variation and the
relative entropy only compare densities in a point-wise manner and fail to
capture the geometric nature of the problem. In sharp contrast, Maximum Mean
Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of
distances between measures that take into account the geometry of the
underlying space and metrize the convergence in law.
This paper studies the Sinkhorn divergences, a family of geometric
divergences that interpolates between MMD and OT. Relying on a new notion of
geometric entropy, we provide theoretical guarantees for these divergences:
positivity, convexity and metrization of the convergence in law. On the
practical side, we detail a numerical scheme that enables the large scale
application of these divergences for machine learning: on the GPU, gradients of
the Sinkhorn loss can be computed for batches of a million samples.

本文研究 Sinkhorn 差异，一种几何差异的系列，介于 MMD 和 OT 之间，提供了这些差异的理论保证和实际应用的数值方案。

使用 Sinkhorn 散度在最优传输和 MMD 之间插值

Interpolating between Optimal Transport and MMD using Sinkhorn  Divergences

Optimal transport (OT) theory can be informally described using the words of
the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in
hand has to move a large pile of sand lying on a construction site. The goal of
the worker is to erect with all that sand a target pile with a prescribed shape
(for example, that of a giant sand castle). Naturally, the worker wishes to
minimize her total effort, quantified for instance as the total distance or
time spent carrying shovelfuls of sand. Mathematicians interested in OT cast
that problem as that of comparing two probability distributions, two different
piles of sand of the same volume. They consider all of the many possible ways
to morph, transport or reshape the first pile into the second, and associate a
"global" cost to every such transport, using the "local" consideration of how
much it costs to move a grain of sand from one place to another. Recent years
have witnessed the spread of OT in several fields, thanks to the emergence of
approximate solvers that can scale to sizes and dimensions that are relevant to
data sciences. Thanks to this newfound scalability, OT is being increasingly
used to unlock various problems in imaging sciences (such as color or texture
processing), computer vision and graphics (for shape manipulation) or machine
learning (for regression, classification and density fitting). This short book
reviews OT with a bias toward numerical methods and their applications in data
sciences, and sheds lights on the theoretical properties of OT that make it
particularly useful for some of these applications.

本短篇论文着重回顾了优化输运相关理论（Optimal transport theory）及其在数据科学中的应用，重点在于阐述其针对分类、回归、密度拟合等机器学习等领域的优势，介绍了它的数值方法，并介绍了一些学术性质。