Spectral Bounds and Shifted Complexes: Eigenvalues of the Up-Laplacian via Face Degrees

What the math gives to ML

The paper gives an exact combinatorial certificate for when a pure simplicial complex has an extremal up-Laplacian spectrum: the difference between the squared conjugate vertex-degree partition and the squared nonzero spectrum equals twice the number of failed elementary facet shifts. This turns shiftedness into a computable spectral defect rather than an existence statement or a loose bound. A transferable application is to regularize learned hypergraph connectivity or sparse routing structures toward ordered, spectrally concentrated incidence patterns while preserving task loss. Because the theorem is exact only for hard unweighted complexes, implementation should use periodic hard top-k rewiring with a straight-through or score-based surrogate, and evaluate the exact defect on the resulting discrete structure.