  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}.
