Spectral clustering with scale fairness constraints

Spectral clustering is one of the most common unsupervised learning algorithms in machine learning and plays an important role in data science. Fair spectral clustering has also become a hot topic with the extensive research on fair machine learning in recent years. Current iterations of fair spectral clustering methods are based on the concepts of group and individual fairness. These concepts act as mechanisms to mitigate decision bias, particularly for individuals with analogous characteristics and groups that are considered to be sensitive. Existing algorithms in fair spectral clustering have made progress in redistributing resources during clustering to mitigate inequities for certain individuals or subgroups. However, these algorithms still suffer from an unresolved problem at the global level: the resulting clusters tend to be oversized and undersized. To this end, the first original research on scale fairness is presented, aiming to explore how to enhance scale fairness in spectral clustering. We define it as a cluster attribution problem for uncertain data points and introduce entropy to enhance scale fairness. We measure the scale fairness of clustering by designing two statistical metrics. In addition, two scale fair spectral clustering algorithms are proposed, the entropy weighted spectral clustering (EWSC) and the scale fair spectral clustering (SFSC). We have experimentally verified on several publicly available real datasets of different sizes that EWSC and SFSC have excellent scale fairness performance, along with comparable clustering effects.