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This paper extends the theory of graph fibrations to include weighted graphs and those labeled with various algebraic structures, creating a comprehensive framework for analyzing fibrations on graphs. The authors demonstrate the applicability of this framework to neural network compression, providing a robust theoretical foundation for recent advancements in geometric deep learning. Key results indicate that this approach not only enhances understanding of neural network structures but also facilitates more effective compression techniques.
Compressing neural networks just got a theoretical boost with a new framework that integrates weighted graphs and algebraic structures.
The purpose of this paper is to provide a general, comprehensive, theoretical framework that allows one to deal with fibrations on graphs labelled on a commutative monoid. This is a genuine extension of the theory of graph fibrations (as introduced in"Fibrations of Graphs"[Discrete Math., vol. 243, pp. 21-66, 2002]), that makes it possible to deal with weighted graphs, and also graphs labelled with other algebraic structures. The derived theory also lends itself naturally to consider approximate fibrations. As an example, we show how this framework can be applied to the compression of arbitrary neural networks (including CNNs), providing a strong theoretical underpinning to the recent results in"The role of fibration symmetries in geometric deep learning"[Proc. Natl. Acad. Sci. USA, vol. 123, no. 4, p. e2416552123, 2026]
