Thesis title: Enhancing Graph-based Learning through Sheaf Theory
This thesis addresses a fundamental limitation in contemporary network modeling: the inability of traditional graph-based approaches to effectively capture complex, multidimensional relationships that characterize real-world systems. While hypergraphs naturally represent higher-order interactions, current hypergraph neural networks suffer from over-smoothing, where node representations become too uniform during propagation, limiting their effectiveness.
This thesis introduces cellular sheaf theory as a powerful mathematical framework to enhance network expressivity. By associating vector spaces (stalks) with network elements and defining restriction maps between them, sheaves provide a principled approach for modeling context-dependent transformations rather than simple averaging. This better reflects how information propagates in complex systems -- individuals can express different opinions in different contexts while maintaining apparent consensus.
The thesis makes three primary contributions: First, it develops the formal mathematical construction of cellular sheaves for hypergraphs, significantly enhancing expressivity while preserving higher-order connectivity. Second, it introduces both linear and non-linear formulations of sheaf hypergraph Laplacians with rigorous theoretical characterization of their improved inductive biases and over-smoothing resistance. Third, it demonstrates the framework's versatility through novel architectures like SheafHyperGNN, SheafHyperGCN, and Sheaf4Rec, a pioneering application to recommendation systems.
Empirical evaluations demonstrate consistent and substantial improvements over state-of-the-art methods across diverse tasks. For recommendation systems, Sheaf4Rec significantly outperforms existing approaches on standard ranking metrics while simultaneously reducing computational demands. These results confirm that sheaf theory not only provides an elegant mathematical framework but also offers practical solutions to fundamental challenges in modeling complex networked systems.
Through a comprehensive analysis of energy functions, diffusion processes, and computational complexity, this thesis establishes sheaf theory as a powerful paradigm for understanding and leveraging the inherently relational nature of complex systems across diverse domains.