This paper defines fair principal component analysis (PCA) as minimizing the
maximum mean discrepancy (MMD) between dimensionality-reduced conditional
distributions of different protected classes. The incorporation of MMD
naturally leads to an exact and tractable mathematical formulation of fairness
with good statistical properties. We formulate the problem of fair PCA subject
to MMD constraints as a non-convex optimization over the Stiefel manifold and
solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu
and Boumal, 2019). Importantly, we provide local optimality guarantees and
explicitly show the theoretical effect of each hyperparameter in practical
settings, extending previous results. Experimental comparisons based on
synthetic and UCI datasets show that our approach outperforms prior work in
explained variance, fairness, and runtime.

本文提出了一种公平的主成分分析方法，该方法使用最大均值差异法对不同保护类别的条件分布进行降维，并将多目标策略与数据降维技术相结合，通过在 Stiefel 流形上的非凸优化过程，优化约束条件下的主成分分析，得到较高的计算效率和分类精确性。使用合成数据和 UCI 数据集进行的实验结果表明该方法在自变量解释度，公平性和运行时间方面均优于现有工作。

基于 MMD 的 Stiefel 流形优化的快速高效公平 PCA 算法

Fast and Efficient MMD-based Fair PCA via Optimization over Stiefel  Manifold

Clustering is a fundamental tool in data mining. It partitions points into
groups (clusters) and may be used to make decisions for each point based on its
group. However, this process may harm protected (minority) classes if the
clustering algorithm does not adequately represent them in desirable clusters
-- especially if the data is already biased.
At NIPS 2017, Chierichetti et al. proposed a model for fair clustering
requiring the representation in each cluster to (approximately) preserve the
global fraction of each protected class. Restricting to two protected classes,
they developed both a 4-approximation for the fair $k$-center problem and a
$O(t)$-approximation for the fair $k$-median problem, where $t$ is a parameter
for the fairness model. For multiple protected classes, the best known result
is a 14-approximation for fair $k$-center.
We extend and improve the known results. Firstly, we give a 5-approximation
for the fair $k$-center problem with multiple protected classes. Secondly, we
propose a relaxed fairness notion under which we can give bicriteria
constant-factor approximations for all of the classical clustering objectives
$k$-center, $k$-supplier, $k$-median, $k$-means and facility location. The
latter approximations are achieved by a framework that takes an arbitrary
existing unfair (integral) solution and a fair (fractional) LP solution and
combines them into an essentially fair clustering with a weakly supervised
rounding scheme. In this way, a fair clustering can be established belatedly,
in a situation where the centers are already fixed.

本文提出了针对多个保护类的公平聚类方法，并且提出了一种松散的公平概念，在这种概念下，可以对所有经典聚类目标进行双标准常数因子近似，这是通过将任意现有不公平的（整数）解和公平的（分数）线性规划解结合起来实现的。

论基本公平聚类的代价

On the cost of essentially fair clusterings

We study the question of fair clustering under the {\em disparate impact}
doctrine, where each protected class must have approximately equal
representation in every cluster. We formulate the fair clustering problem under
both the $k$-center and the $k$-median objectives, and show that even with two
protected classes the problem is challenging, as the optimum solution can
violate common conventions---for instance a point may no longer be assigned to
its nearest cluster center! En route we introduce the concept of fairlets,
which are minimal sets that satisfy fair representation while approximately
preserving the clustering objective. We show that any fair clustering problem
can be decomposed into first finding good fairlets, and then using existing
machinery for traditional clustering algorithms. While finding good fairlets
can be NP-hard, we proceed to obtain efficient approximation algorithms based
on minimum cost flow. We empirically quantify the value of fair clustering on
real-world datasets with sensitive attributes.

本文研究了如何在保护隐私的前提下实现公平聚类，提出了公平小集（fairlets）的概念，并基于最小代价流设计出高效的近似算法，在实际数据集中验证了公平聚类的价值。

公平分簇

Fair Clustering Through Fairlets

Machine learnt systems inherit biases against protected classes, historically
disparaged groups, from training data. Usually, these biases are not explicit,
they rely on subtle correlations discovered by training algorithms, and are
therefore difficult to detect. We formalize proxy discrimination in data-driven
systems, a class of properties indicative of bias, as the presence of protected
class correlates that have causal influence on the system's output. We evaluate
an implementation on a corpus of social datasets, demonstrating how to validate
systems against these properties and to repair violations where they occur.

该研究旨在解决机器学习算法在训练中自然而然会对历史上被歧视的群体产生偏差的问题，并通过实验证明了可以通过验证和修复违规来解决这些问题。