Collapsed Effective Operators for Higher-order Structures
Maximilian Krahn, Lennart Bastian, Vikas Garg, Björn Schuller, Tolga Birdal
cs.LG, stat.ML
Submitted: 2026-06-22
Comments: Accepted at ICML 2026
Project page: http://circle-group.github.io/research/CollapsedEffectiveOperators
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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
Terminology
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
Sources
- Topological Neural Operators
- Attending to Topological Spaces: The Cellular Transformer
- Continuous Simplicial Neural Networks
- Topological Deep Learning: Going Beyond Graph Data
- Topology Adaptive Graph Convolutional Networks
- Convolutional Learning on Simplicial Complexes
- Root-to-Leaf Path Random Walks, Normalized Hodge Laplacians, and Cheeger Inequalities on Simplicial Complexes
- Topotein: Topological Deep Learning for Protein Representation Learning
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