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Collapsed Effective Operators for Higher-Order Structures

MCML Authors

Lennart Bastian

→ Group Nassir Navab
Computer Aided Medical Procedures & Augmented Reality

Björn Schuller

Prof. Dr.

Core PI

Health Informatics

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 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 strict 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.

inproceedings KBG+26