arXiv:2608.04903 2026 Regularization 1 ideas extracted · analyzed Aug 31, 2026
The paper provides a structural alternative to eigenvalue-gap regularization for learned spectral models: infer symmetry sectors from the linear commutant of an observed operator family, rather than clustering nearly equal eigenvalues. The transferable asset is a discrete algebraic gate that replaces ill-defined per-eigenvector targets by projector or block-trace targets on symmetry-forced multiplets. This removes the genuine inverse-gap singularity at protected crossings while preserving per-level supervision where levels are not symmetry-forced. A neural implementation can place the gate in the spectral loss for a network predicting parameterized Hermitian matrices, with the commutant and block projectors computed from noisy reference operators and refreshed periodically.
Replace per-eigenvector spectral supervision on symmetry-forced multiplets by a projector-trace target determined from the operator family's commutant. Use individual eigenvalue or eigenvector targets only outside detected forced blocks, avoiding arbitrary basis choices and exploding gradients at protected crossings.