Fair clustering aims to divide data into distinct clusters while preventing sensitive attributes (\textit{e.g.}, gender, race, RNA sequencing technique) from dominating the clustering. Although a number of works have been conducted and achieved huge success recently, most of them are heuristical, and there lacks a unified theory for algorithm design. In this work, we fill this blank by developing a mutual information theory for deep fair clustering and accordingly designing a novel algorithm, dubbed FCMI. In brief, through maximizing and minimizing mutual information, FCMI is designed to achieve four characteristics highly expected by deep fair clustering, \textit{i.e.}, compact, balanced, and fair clusters, as well as informative features. Besides the contributions to theory and algorithm, another contribution of this work is proposing a novel fair clustering metric built upon information theory as well. Unlike existing evaluation metrics, our metric measures the clustering quality and fairness as a whole instead of separate manner. To verify the effectiveness of the proposed FCMI, we conduct experiments on six benchmarks including a single-cell RNA-seq atlas compared with 11 state-of-the-art methods in terms of five metrics. The code could be accessed from \url{ https://pengxi.me}.

翻译：公平聚类旨在将数据划分为不同的簇，同时防止敏感属性（如性别、种族、RNA测序技术）主导聚类过程。尽管近年来已有大量研究并取得了显著成功，但大多数方法仍基于启发式策略，且缺乏统一的算法设计理论。在本工作中，我们通过发展深度公平聚类的互信息理论，并据此设计了一种名为FCMI的新颖算法，填补了这一空白。简言之，FCMI通过最大化与最小化互信息，实现了深度公平聚类高度期望的四个特性，即紧凑性、平衡性、公平性簇以及信息丰富的特征。除了在理论与算法上的贡献外，本工作的另一贡献是基于信息论提出了一种新颖的公平聚类度量。与现有评估度量不同，我们的度量将聚类质量与公平性作为一个整体而非分离的方式进行衡量。为验证所提FCMI的有效性，我们在六个基准数据集（包括一个单细胞RNA-seq图谱）上进行了实验，并与11种最先进方法在五项指标上进行了比较。代码可从\url{https://pengxi.me}获取。