Collapsed Effective Operators for Higher-order Structures
2026-06-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors propose a new mathematical tool called Collapsed Effective Operators to better capture complex relationships in higher-order structures, like networks with more than simple connections. Unlike older methods that separate these relationships into different layers, their approach merges all the information into one operator focused on individual points (vertices). This operator maintains important mathematical properties and helps improve tasks such as grouping similar data points, smoothing signals, and adding topology-based information to neural networks. Their method works for many complex network types and helps represent long-range interactions more effectively.
higher-order structuresspectral operatorsSchur complementationgraded Laplacianpositive semi-definitenessHodge Laplacianspectral clusteringsignal smoothingpositional encodingtopological features
Authors
Maximilian Krahn, Lennart Bastian, Vikas Garg, Björn Schuller, Tolga Birdal
Abstract
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