Some Reverse Hardy-Littlewood-Sobolev Type Inequalities

What the math gives to ML

The paper supplies a sharp reverse inequality: a Riesz potential cannot have arbitrarily small output norm relative to the input mass and sub-unit L^q quasi-norm. This is useful as an anti-collapse certificate for nonnegative feature distributions, routing probabilities, or learned particle sets, because it penalizes representations whose potential becomes degenerate. The transferable asset is the explicit interpolation exponent and the threshold q>n/alpha, together with the half-space analogue for boundary-supported data. A practical adaptation is a differentiable hinge loss enforcing the reverse-HLS lower bound on minibatch feature densities.