Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes
arXiv:2607.20910
2026
Architecture
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper gives an explicit spectral certificate for incidence-based operators on simplicial complexes: the eigenvalue vector of the up-Laplacian is majorized by the conjugate degree sequence of codimension-one faces. This converts local combinatorial degree information into bounds on every Ky Fan partial sum of a global propagation operator, which is directly useful for controlling oversmoothing and exploding activations in simplicial or hypergraph neural networks. The most practical transfer is to use the degree sequence as a cheap spectral budget, either as a regularizer on the top eigenvalues of a learnable incidence operator or as a per-layer residual-step-size ceiling. The Brouwer-type bound additionally gives a topology-size-only fallback when individual face degrees are unavailable.
Ideas from this paper
Unverified
2026
Set the residual propagation coefficient of a simplicial neural layer from a cheap upper bound on the operator spectrum instead of tuning it blindly. The degree-majorization theorem supplies a bound on the largest eigenvalue, while the Brouwer-type inequality supplies a topology-count-based bound on sums of the top eigenvalues.
Useful6/10
Difficulty3/10
Novelty6/10
Unverified
2026
Use the conjugate degree sequence of codimension-one faces as a mathematically justified upper envelope for the spectrum of a simplicial up-Laplacian. Penalize violations of the corresponding top-k eigenvalue budgets in a simplicial message-passing layer, discouraging a few dominant propagation modes that cause oversmoothing or unstable amplification.
Useful6/10
Difficulty6/10
Novelty7/10