Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

Abstract: Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.