Sharp $L^2$-Caffarelli--Kohn--Nirenberg and weighted Poincaré inequalities on half-spaces and orthants and their stability

What the math gives to ML

The paper derives sharp weighted Caffarelli–Kohn–Nirenberg inequalities and stability deficits on half-spaces and orthants. Their transferable asset is a scale-aware relation between weighted feature magnitude and weighted input sensitivity, with explicit extremal structure and deficit measurements. A neural-network adaptation is to regularize intermediate features whose responses are excessively concentrated in input space or require large Jacobians. The resulting penalty is distinct from ordinary weight decay because it controls the joint geometry of activations and input derivatives.