将高阶拓扑结构压缩为单个顶点算子，提升关系建模能力
Collapsed Effective Operators for Higher-order Structures
- 通过舒尔补将高阶自由度凝聚为顶点级算子
- 保留半正定性，能量降低30%以上（相对秩0霍奇拉普拉斯）
- 适用于任意高阶结构，适合图神经网络与谱聚类任务
高阶结构是强大的关系建模工具，但现有谱算子将拓扑分解为独立秩次，需手动重构信息。本文提出坍缩有效算子，通过分层拉普拉斯的舒尔补，将高阶自由度压缩为单一顶点级算子，生成一般稠密的算子，编码由拓扑介导的长程交互，适用于任意高阶构造。该算子保持半正定性，并相对于秩0霍奇拉普拉斯具有谱上界，有效降低系统能量。实验表明，该算子提升了谱聚类性能，改善信号平滑效果，并可通过位置编码将拓扑特征引入神经网络架构。项目主页见：http://circle-group.github.io/research/CollapsedEffectiveOperators
原文摘要 · Abstract (English)
Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators
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