arXiv:2606.31647 2026 Regularization 2 ideas extracted · analyzed Aug 29, 2026
The paper provides a constructive way to turn order-parameter inequalities into positivity of Hankel matrices by representing normalized quantities as moments of a positive measure. The transferable asset is not any particular special function, but the guarantee that positive mixtures of exponentials or Mellin powers generate log-convexity and an entire hierarchy of nonnegative Hankel minors, rather than only a two-point inequality. In neural networks this can be used either as a structural parameterization for order-dependent gates or as a regularizer that detects whether a learned response curve is a valid moment sequence. The strongest initial experiments should target monotone or scale-indexed components such as adaptive loss weights, diffusion schedules, attention-temperature gates, or neural scaling-law predictors, where smoothness across an ordered parameter is desirable.
Replace a free-form order-dependent gate with a positive mixture of Mellin powers $(1+s)^{-a}$. This gives a small, interpretable module whose response across the order variable is automatically generated by a positive measure and therefore inherits complete monotonicity, log-convexity, and Hankel-moment structure.
Regularize a neural network's response along an ordered variable by requiring its sampled values to form a positive Hankel moment sequence. This upgrades ordinary pairwise monotonicity or log-convexity penalties into simultaneous constraints on several higher-order interactions, while remaining differentiable and inexpensive for small Hankel order.