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[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 View PDF HTML (experimental) 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) Full-text links: Access Paper: 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 View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)