无固定扩散的固定点：用于收敛测试时计算的隐式神经层丛
Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation
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中文总结 AI 辅助
提出SheafDEQ，一种具有自适应神经层丛传播的隐式图神经网络架构，通过次齐次深度平衡公式实现唯一平衡点，在分布式推理和社区检测任务上优于固定传播基线。
中文摘要 AI 辅助
隐式图神经网络（IGNNs）将节点表示定义为消息传递算子的固定点，从而实现有效无限深度传播、迭代无关参数化和灵活的测试时计算。然而，这些优势依赖于平衡点的唯一性以及通过固定点迭代可达性。现有构造通常对循环更新施加约束以获得这些保证，从而限制了平衡点处可用的变换。这引发了一个核心问题：IGNNs能否通过更丰富的、依赖于边的变换获得表达能力，同时保留其平衡公式的固有优势？我们提出了SheafDEQ，一种具有自适应神经层丛传播的次齐次深度平衡架构。其学习到的矩阵值层丛限制映射可以对齐、混合或反转相邻表示。在温和的正则条件下，我们证明了SheafDEQ具有唯一平衡点，该平衡点可从任何正初始化通过固定点迭代全局达到。收缩性进一步保证了在有界通信停滞下的收敛性。我们在需要重复非局部聚合的分布式推理任务以及重连越来越有利于跨社区交互的社区检测任务上评估了SheafDEQ。在Sums、MNIST Terrain和Coordinates上，SheafDEQ优于固定传播隐式基线，并且在连接性越来越异质时，在社区检测上也优于基线。持续迭代诊断显示，在初始化尺度从0.001到10的情况下，经过100次迭代后残差递减且预测敏感性低，而延迟更新实验显示对有界通信停滞的敏感性低。
英文摘要
Implicit Graph Neural Networks (IGNNs) define node representations as fixed points of graph neural operators, enabling effectively infinite-depth propagation, iteration-independent parameterization, and flexible test-time computation. Yet these benefits depend on the equilibrium being unique and reached 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 initial state. 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 implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic. SheafDEQ remains competitive with recurrent models without equilibrium guarantees. Continued-iteration diagnostics show decreasing residuals and retained test accuracy through 100 iterations, unlike the finite-horizon control, while delayed-update experiments show low sensitivity to bounded communication staleness.
发表机构
- KTH Royal Institute of Technology（瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
