Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation

2026-08-12

AI Summary

This study addresses the challenge that implicit graph neural networks face in guaranteeing equilibrium uniqueness and convergence when enhancing expressive power. To this end, we propose SheafDEQ, an architecture incorporating sub-homogeneous deep equilibrium design and matrix-valued restriction mappings to enable rich feature transformations via adaptive neural layer propagation. We theoretically prove that, from any positive initialization, fixed-point iterations globally converge to a unique equilibrium. Experimental results demonstrate that SheafDEQ significantly outperforms existing baselines across multi-task learning benchmarks while maintaining high test accuracy. Furthermore, it exhibits low sensitivity to communication latency, offering an efficient and reliable solution for distributed inference.

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.

Problem

• Implicit Graph Neural Networks
• Deep Equilibrium Models
• Neural Sheaves
• Fixed-Point Iteration
• Expressiveness

Innovation

• Implicit Graph Neural Networks
• Deep Equilibrium Models
• Neural Sheaves
• Fixed-Point Iteration
• Heterophily