Reversed inequality of the Herbst-type and the related Euler-Lagrange system

What the math gives to ML

The paper gives a reversed weighted Hardy-Littlewood-Sobolev inequality in the sublinear regime, where a nonlocal interaction is bounded below by the product of two quasi-norms. This can be transferred into a representation regularizer for nonnegative feature maps: it rewards distributed cross-view or cross-branch interaction and may reduce spatial feature collapse. The most practical implementation is a low-resolution kernel computation on feature grids, using a scale-normalized interaction ratio. The Euler-Lagrange system further suggests coupled fractional-potential modules, but the inequality itself offers the clearest first experiment.