On the Sheafification of Higher-Order Message Passing
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915517403496448 |
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| author | Hume, Jacob Liò, Pietro |
| author_facet | Hume, Jacob Liò, Pietro |
| contents | Recent work in Topological Deep Learning (TDL) seeks to generalize graph learning's preeminent $message \ passing$ paradigm to more complex relational structures: simplicial complexes, cell complexes, hypergraphs, and combinations thereof. Many approaches to such ${higher\text{-}order \ message \ passing}$ (HOMP) admit formulation in terms of nonlinear diffusion with the Hodge (combinatorial) Laplacian, a graded operator which carries an inductive bias that dimension-$k$ data features correlate with dimension-$k$ topological features encoded in the (singular) cohomology of the underlying domain. For $k=0$ this recovers the graph Laplacian and its well-studied homophily bias. In higher gradings, however, the Hodge Laplacian's bias is more opaque and potentially even degenerate. In this essay, we position sheaf theory as a natural and principled formalism for modifying the Hodge Laplacian's diffusion-mediated interface between local and global descriptors toward more expressive message passing. The sheaf Laplacian's inductive bias correlates dimension-$k$ data features with dimension-$k$ $sheaf$ cohomology, a data-aware generalization of singular cohomology. We will contextualize and novelly extend prior theory on sheaf diffusion in graph learning ($k=0$) in such a light -- and explore how it fails to generalize to $k>0$ -- before developing novel theory and practice for the higher-order setting. Our exposition is accompanied by a self-contained introduction shepherding sheaves from the abstract to the applied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Sheafification of Higher-Order Message Passing Hume, Jacob Liò, Pietro Machine Learning Algebraic Topology Recent work in Topological Deep Learning (TDL) seeks to generalize graph learning's preeminent $message \ passing$ paradigm to more complex relational structures: simplicial complexes, cell complexes, hypergraphs, and combinations thereof. Many approaches to such ${higher\text{-}order \ message \ passing}$ (HOMP) admit formulation in terms of nonlinear diffusion with the Hodge (combinatorial) Laplacian, a graded operator which carries an inductive bias that dimension-$k$ data features correlate with dimension-$k$ topological features encoded in the (singular) cohomology of the underlying domain. For $k=0$ this recovers the graph Laplacian and its well-studied homophily bias. In higher gradings, however, the Hodge Laplacian's bias is more opaque and potentially even degenerate. In this essay, we position sheaf theory as a natural and principled formalism for modifying the Hodge Laplacian's diffusion-mediated interface between local and global descriptors toward more expressive message passing. The sheaf Laplacian's inductive bias correlates dimension-$k$ data features with dimension-$k$ $sheaf$ cohomology, a data-aware generalization of singular cohomology. We will contextualize and novelly extend prior theory on sheaf diffusion in graph learning ($k=0$) in such a light -- and explore how it fails to generalize to $k>0$ -- before developing novel theory and practice for the higher-order setting. Our exposition is accompanied by a self-contained introduction shepherding sheaves from the abstract to the applied. |
| title | On the Sheafification of Higher-Order Message Passing |
| topic | Machine Learning Algebraic Topology |
| url | https://arxiv.org/abs/2509.23020 |