We commonly encounter the problem of identifying an optimally weight adjusted
version of the empirical distribution of observed data, adhering to predefined
constraints on the weights. Such constraints often manifest as restrictions on
the moments, tail behaviour, shapes, number of modes, etc., of the resulting
weight adjusted empirical distribution. In this article, we substantially
enhance the flexibility of such methodology by introducing a nonparametrically
imbued distributional constraints on the weights, and developing a general
framework leveraging the maximum entropy principle and tools from optimal
transport. The key idea is to ensure that the maximum entropy weight adjusted
empirical distribution of the observed data is close to a pre-specified
probability distribution in terms of the optimal transport metric while
allowing for subtle departures. The versatility of the framework is
demonstrated in the context of three disparate applications where data
re-weighting is warranted to satisfy side constraints on the optimization
problem at the heart of the statistical task: namely, portfolio allocation,
semi-parametric inference for complex surveys, and ensuring algorithmic
fairness in machine learning algorithms.

介绍了一种通过最大熵原理和最优传输工具，引入非参数分布约束的权重调整经验分布方法，可以在满足预定义约束的情况下对观测数据的最大熵权重调整经验分布进行灵活性增强，展示了在投资组合分配、复杂调查的半参数推断以及机器学习算法中实现数据重新加权的多功能性。

约束分布的重新加权：一种最优输运方法

Constrained Reweighting of Distributions: an Optimal Transport Approach

Datasets often contain biases which unfairly disadvantage certain groups, and
classifiers trained on such datasets can inherit these biases. In this paper,
we provide a mathematical formulation of how this bias can arise. We do so by
assuming the existence of underlying, unknown, and unbiased labels which are
overwritten by an agent who intends to provide accurate labels but may have
biases against certain groups. Despite the fact that we only observe the biased
labels, we are able to show that the bias may nevertheless be corrected by
re-weighting the data points without changing the labels. We show, with
theoretical guarantees, that training on the re-weighted dataset corresponds to
training on the unobserved but unbiased labels, thus leading to an unbiased
machine learning classifier. Our procedure is fast and robust and can be used
with virtually any learning algorithm. We evaluate on a number of standard
machine learning fairness datasets and a variety of fairness notions, finding
that our method outperforms standard approaches in achieving fair
classification.

本文提出了一种数学公式来证明数据集中的偏见对分类器的影响，通过无需改变标签来重新计算数据点的值，从而得到一个无偏分类器，这种方法可以在各种机器学习算法中使用，并实现了公平分类的目标。