Non-Euclidean Geometries and Fairness Constraints in Advanced Clustering

Abstract

A fundamental challenge in modern unsupervised learning is adapting classical clustering algorithms to handle complex, real-world data constraints. Traditional models often assume data resides in a flat, Euclidean space and optimize strictly for cluster cohesion, thereby failing to capture intrinsic hierarchical structures and ignoring sociotechnical demographic biases. This thesis addresses these critical limitations by extending generalized mean-shift dynamics into two novel clustering frameworks. First, to natively accommodate data with tree-like structures (e.g., taxonomies and social networks), we propose Hyperbolic Gaussian Blurring Mean Shift (HypeGBMS). By projecting data into the Poincar´e ball model and utilizing M¨obius vector space operations, HypeGBMS successfully generalizes density-based clustering to non-Euclidean manifolds. Second, to tackle algorithmic bias in noisy datasets, we introduce Fair Possibilistic C-Means (F-PCM). By embedding a group-fairness Kullback-Leibler divergence penalty into the possibilistic objective function, F-PCM explicitly enforces demographic parity without sacrificing the outlier-robust nature of possibilistic typicalities. We provide rigorous theoretical proofs for both methodologies, including convergence guarantees, statistical consistency, and optimization bounds via Majorization-Minimization. Extensive experiments on complex real-world datasets demonstrate that HypeGBMS dramatically improves cluster quality on hierarchical data, while F-PCM maintains strict fairness criteria while matching the computational efficiency of traditional baselines.