On a family of one-dimensional oscillation inequalities
arXiv:2608.04639
2026
Regularization
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper gives sharp lower bounds linking the number of sign changes of a mean-zero one-dimensional signal to its negative Sobolev energy and several norm ratios. The transferable asset is a computable oscillation certificate: a signal with substantial L1/Lp mass and small low-frequency inverse-derivative energy must contain many sign changes. This can become a regularizer for logits, hidden states, or temporal trajectories that suppresses spurious rapid alternation without simply penalizing amplitude. The most practical first test is on ordered neural outputs, using FFT-based H^{-1} estimation and a differentiable soft count of sign transitions.
Ideas from this paper
Unverified
2026
Regularize a neural signal defined along an ordered axis so that it does not achieve large norm mass while simultaneously having very small negative-Sobolev energy, a combination that mathematically forces many sign changes. Apply the penalty to logits along time, spatial scanlines, token positions, or latent interpolation paths, preserving task-relevant amplitude through normalization and only discouraging unexplained rapid alternation.
Useful6/10
Difficulty4/10
Novelty8/10