ArXiv

Collapsed Effective Operators for Higher-order Structures

2026-06-22 · 16:04 UTC ·Maximilian Krahn, Lennart Bastian, Vikas Garg... ·1 min read
Authors
Maximilian Krahn, Lennart Bastian, Vikas Garg...
Categories
cs.LG, stat.ML
arXiv
https://arxiv.org/abs/2606.23517v1
PDF
https://arxiv.org/pdf/2606.23517v1

Brief

Collapsed Effective Operators (CEOs) address the fusion problem for higher-order spectral operators by Schur-complementing a graded Laplacian to produce a single vertex-level operator that encodes long-range, topology-mediated interactions. CEOs preserve PSD, give a spectral upper bound versus the rank-0 Hodge Laplacian, and empirically improve spectral clustering, signal smoothing, and positional encodings in neural networks. Full paper on arXiv and ICML 2026 acceptance.

Why it matters

Introduces Collapsed Effective Operators (CEOs) that condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian; the resulting (generally dense) operator preserves positive semi-definiteness and satisfies a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity.

Key details

  • CEOs are applicable to arbitrary higher-order constructs and empirically improve spectral clustering and signal smoothing, while enabling topological features as positional encodings in neural architectures; paper by Maximilian Krahn, Lennart Bastian, Vikas Garg, Björn Schuller, and Tolga Birdal (arXiv:2606.23517v1), accepted at ICML 2026 and published 2026-06-22 (project: http://circle-group.github.io/research/CollapsedEffectiveOperators).
Source evidence

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

Comment: Accepted at ICML 2026