arXiv:2607.03450 2026 Regularization 2 ideas extracted · analyzed Aug 30, 2026
The paper provides a constructive way to reduce oscillation in scalar functions without changing their value distribution: sorting a function into its non-increasing rearrangement cannot increase its Riesz fractional variation. This is useful for one-dimensional signals, neural fields, sequence channels, and monotone bottlenecks where preserving the activation histogram while suppressing spatial or temporal oscillations is desirable. The most direct transfer is a histogram-preserving rearrangement layer or regularizer, with the theorem supplying a falsifiable guarantee for the hard-sorting version and the fractional variation supplying a tunable smoothness penalty for differentiable approximations.
Add a fractional oscillation penalty to scalar functions produced by a neural network on an ordered grid. Unlike a derivative penalty, this remains meaningful for nonsmooth or nowhere-differentiable outputs and interpolates between total-variation-like behavior and Sobolev-like smoothness.
Insert a rearrangement operation on scalar feature maps sampled along an ordered coordinate such as time, spatial position, or a neural-field input grid. The operation sorts values into non-increasing order, preserving the empirical histogram exactly while provably not increasing the Riesz fractional variation in the ideal one-dimensional continuous setting.