Bottom of the Spectrum of Complete Kähler Metrics from Finite-Mass Plurisubharmonic Exhaustions
arXiv:2607.03036
2026
Optimization
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper constructs a complete Kähler metric from the logarithmic exhaustion g=-log(-rho) and proves that its Laplacian has exact spectral bottom n^2 under a finite weighted Monge–Ampère mass condition. The transferable mechanism is a logarithmic barrier metric: it makes approach to the boundary infinitely long while keeping the exhaustion gradient bounded through |partial g|_omega^2 <= 1. In neural optimization, this suggests replacing Euclidean updates on bounded parameters or normalized representations with Riemannian updates using the Hessian of a logarithmic barrier. The exact n^2 theorem is not automatically an optimizer guarantee, but the construction gives a concrete, testable way to suppress boundary-induced instability.
Ideas from this paper
Unverified
2026
Constrain a neural parameter block to a bounded open domain and replace its Euclidean optimizer with a Riemannian gradient induced by the Hessian of the logarithmic barrier g=-log(-rho). The metric diverges near the boundary, so updates automatically become small when parameters approach saturation or an invalid region, while the logarithmic exhaustion has bounded intrinsic gradient.
Useful6/10
Difficulty5/10
Novelty6/10
Unverified
2026
Use the logarithmic exhaustion as a geometry for bounded hidden representations rather than only as a parameter constraint. A representation approaching the boundary receives an increasingly large metric, making ordinary Euclidean motion expensive and discouraging brittle saturation while preserving a bounded intrinsic gradient for the boundary coordinate.
Useful5/10
Difficulty6/10
Novelty7/10