[Submitted on 12 Aug 2026]
Title:Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation
View a PDF of the paper titled Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation, by R\'emi Bourgerie and 2 other authors
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Abstract:Implicit Graph Neural Networks (IGNNs) define node representations as fixed points of message-passing operators, enabling effectively infinite-depth propagation, iteration-independent parameterization, and flexible test-time computation. Yet these benefits depend on the equilibrium being unique and attainable by fixed-point iteration. Existing constructions often impose constraints on recurrent updates to obtain these guarantees, limiting the transformations available at equilibrium. This raises a central question: can IGNNs gain expressiveness through richer, edge-dependent transformations while retaining the inherent strengths of their equilibrium formulation? We introduce SheafDEQ, a subhomogeneous deep-equilibrium architecture with adaptive neural-sheaf propagation. Its learned, matrix-valued sheaf restriction maps can align, mix, or reverse neighbouring representations. Under mild regularity conditions, we prove that SheafDEQ admits a unique equilibrium reached globally by fixed-point iteration from any positive initialization. Contractivity further guarantees convergence under bounded communication staleness. We evaluate SheafDEQ on distributed-inference tasks requiring repeated nonlocal aggregation and on community detection whose rewiring increasingly favours cross-community interactions. SheafDEQ improves over fixed-propagation implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic. Continued-iteration diagnostics show decreasing residuals and low prediction sensitivity after 100 iterations for initialization scales from $0.001$ to $10$, while delayed-update experiments show low sensitivity to bounded communication staleness.
Comments: Submitted to the Learning on Graphs Conference 2026
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2609.30277 [cs.LG]
(or arXiv:2609.30277v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.30277
arXiv-issued DOI via DataCite
Submission history
From: Rémi Bourgerie [view email] [v1] Wed, 12 Aug 2026 21:48:07 UTC (1,001 KB)
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