The authors introduce Collapsed Effective Operators, a method that condenses higher-order degrees of freedom into a single vertex-level operator using Schur complementation of a graded Laplacian. This approach yields a dense operator encoding long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs.
- Preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian.
- Effectively lowers system energy under higher-order connectivity.
- Improves performance in spectral clustering and signal smoothing tasks.
- Enables the inclusion of topological features in neural network architectures via positional encoding.
This method addresses the limitation of existing spectral operators that decompose topology into separate ranks, providing a unified way to incorporate higher-order structural information.