Collapsed effective operators for higher-order structures
File(s)
Accepted version
Author(s)
Krahn, Max
Bastian, Lennart
Garg, Vikas
Schuller, Bjoern
Birdal, Tolga
Type
Conference Paper
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 here.
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 here.
Date Acceptance
2026-04-30
Citation
306
Publisher
The Proceedings of Machine Learning Research
Volume
306
Copyright Statement
Copyright This paper is embargoed until publication. Once published the Version of Record (VoR) will be available on immediate open access.
Source
Forty-Third International Conference on Machine Learning
Publication Status
Accepted
Start Date
2026-07-06
Finish Date
2026-07-11
Coverage Spatial
Seoul, South Korea