On optimal endpoints for integral kernel operators

What the math gives to ML

The paper develops pointwise upper bounds for the decreasing rearrangement of integral transforms, meaning that the entire output-value distribution can be controlled by an integral transform of the input quantile function. The concrete Laplace-transform inequality is directly transferable to positive learned integral layers: it provides a distributional envelope that is stronger than controlling only an average norm or spectral norm. A practical adaptation is to add a quantile-envelope constraint to kernel layers or attention-like continuous operators, penalizing outputs whose upper tail exceeds the rearrangement bound predicted from the input. This should primarily improve activation stability and robustness to outlier features, although the bound may be loose for general learned kernels.