Collapsed Effective Operators for Higher-order Structures

2026-06-22

AI Summary

Existing spectral operators require manual integration of higher-order structural information to obtain vertex-level representations, lacking a unified and effective mechanism. This work proposes "Collapsed Effective Operators," which automatically compress arbitrary higher-order topological information into a single vertex-level operator via the Schur complement of hierarchical Laplacian matrices. The method preserves positive semi-definiteness and, for the first time, achieves an effective collapse of higher-order structures onto the vertex level, accompanied by a spectral upper bound relative to the 0-th order Hodge Laplacian. By naturally encoding topology-mediated long-range interactions, the proposed operators significantly enhance performance in spectral clustering and signal smoothing tasks and can be readily incorporated as positional encodings within neural network architectures.

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

Problem

• higher-order structures
• spectral operators
• topological modeling
• vertex-level representation
• Hodge Laplacian

Innovation

• Collapsed Effective Operators
• Higher-order Structures
• Schur Complementation
• Spectral Clustering
• Topological Signal Processing