Related papers: Collapsed Effective Operators for Higher-order Structures

Collapsed Effective Operators for Higher-order Structures

URL: http://arxiv.org/abs/2606.23517v1
Date: Mon, 22 Jun 2026 16:04:42 GMT
Title: Collapsed Effective Operators for Higher-order Structures
Abstract summary: We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single-level operator.<n>This yields a (generally dense) operator that encodes long-range interactions mediated by topology.<n>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.
Score: 37.68493885832665
License: http://creativecommons.org/licenses/by/4.0/
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

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