Learning on Higher-Order Structures with Effective Operators
Maximilian Krahn, Lennart Bastian, Tolga Birdal, Björn Schuller, Vikas Garg
Abstract
Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices through ad-hoc choices. We introduce _Collapsed Effective Operators_, which marginalize higher-order structures into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a 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 strict spectral upper bound relative to the rank-0 Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator significantly improves spectral clustering, enables diffusion over topological structures, and accelerates the processing of higher-order structures with neural networks.
BibTeX
@inproceedings{
krahn2026collapsed,
title={Collapsed Effective Operators for Higher-order Structures},
author={Maximilian Krahn and Lennart Bastian and Vikas K Garg and Bj{\"o}rn Schuller and Tolga Birdal},
booktitle={Forty-third International Conference on Machine Learning},
year={2026},
url={https://openreview.net/forum?id=arc2pWtZLN}
}