Many problems in machine learning involve calculating correspondences between
sets of objects, such as point clouds or images. Discrete optimal transport
provides a natural and successful approach to such tasks whenever the two sets
of objects can be represented in the same space, or at least distances between
them can be directly evaluated. Unfortunately neither requirement is likely to
hold when object representations are learned from data. Indeed, automatically
derived representations such as word embeddings are typically fixed only up to
some global transformations, for example, reflection or rotation. As a result,
pairwise distances across two such instances are ill-defined without specifying
their relative transformation. In this work, we propose a general framework for
optimal transport in the presence of latent global transformations. We cast the
problem as a joint optimization over transport couplings and transformations
chosen from a flexible class of invariances, propose algorithms to solve it,
and show promising results in various tasks, including a popular unsupervised
word translation benchmark.

该论文提出了一种在潜在的全局转换情况下进行离散最优传输的通用框架，并通过采用灵活类的不变性来选择转换进行联合最优化求解，成功解决了包括无监督词汇翻译基准在内的各种任务。

全局不变性下的最优输运

Towards Optimal Transport with Global Invariances

This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification.

本文提出了一种针对离散最优输运问题的平滑凸正则化统一框架，并基于 Bregman 差异将正则化最优输运等效于矩阵相似问题，其中的算法包括基于 Sinkhorn-Knopp 以及 Dykstra 的交替投影算法，以及基于牛顿 - 拉夫逊法的扩展算法。此外，还将该框架应用到了机器学习和信息几何等领域，并通过实验进行了验证。

正则化最优输运及 Rot Mover's 距离

Regularized Optimal Transport and the Rot Mover's Distance

This article introduces a generalization of the discrete optimal transport,
with applications to color image manipulations. This new formulation includes a
relaxation of the mass conservation constraint and a regularization term. These
two features are crucial for image processing tasks, which necessitate to take
into account families of multimodal histograms, with large mass variation
across modes.
The corresponding relaxed and regularized transportation problem is the
solution of a convex optimization problem. Depending on the regularization
used, this minimization can be solved using standard linear programming methods
or first order proximal splitting schemes.
The resulting transportation plan can be used as a color transfer map, which
is robust to mass variation across images color palettes. Furthermore, the
regularization of the transport plan helps to remove colorization artifacts due
to noise amplification.
We also extend this framework to the computation of barycenters of
distributions. The barycenter is the solution of an optimization problem, which
is separately convex with respect to the barycenter and the transportation
plans, but not jointly convex. A block coordinate descent scheme converges to a
stationary point of the energy. We show that the resulting algorithm can be
used for color normalization across several images. The relaxed and regularized
barycenter defines a common color palette for those images. Applying color
transfer toward this average palette performs a color normalization of the
input images.

本文介绍了一种离散最优传输的推广，应用于彩色图像处理，并将其扩展到 distribution 的 Barycenters 计算，它们的混合体是通过图像彩色调整用于颜色归一化。

正则离散最优输运

Regularized Discrete Optimal Transport

In this paper, we develop several related finite dimensional variational
principles for discrete optimal transport (DOT), Minkowski type problems for
convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the
discrete optimal transport, discrete Monge-Ampere equation and the power
diagram in computational geometry is established.

本文研究了离散优化输运、凸多面体的黎曼度量问题以及离散 Monge-Ampere 方程，建立了这些问题与计算几何中的功率图之间的联系。