Eigenvalue growth of the discrete Hodge Laplacian across dimensions

What the math gives to ML

The paper proves a dimension-monotonicity principle for the largest eigenvalue of the up-Laplacian on simplicial complexes: the maximum eigenvalue at order k is no larger than that at order k-1. This gives a concrete cross-order stability certificate for neural operators that propagate signals between vertices, edges, triangles, and higher simplices. A practical transfer is to use the lower-dimensional spectral radius to normalize or constrain higher-order message-passing blocks, avoiding expensive eigenvalue estimation separately at every simplex order. The payoff is most relevant for simplicial or Hodge neural networks, where explicit diffusion steps otherwise become unstable as the order increases.