{"ID":2864508,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.23020","arxiv_id":"2509.23020","title":"On the Sheafification of Higher-Order Message Passing","abstract":"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 $k\u003e0$ -- 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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2509.23020","url_pdf":"https://arxiv.org/pdf/2509.23020v1","authors":"[\"Jacob Hume\",\"Pietro Liò\"]","published":"2025-09-27T00:33:29Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.AT\"]","methods":"[\"Diffusion Model\"]","has_code":false}