Sharp homogeneous Gagliardo--Nirenberg inequalities with applications to normalized solutions for a generalized MMT-type equation

What the math gives to ML

The paper supplies an explicit scale-consistent coupling between low-order fractional variation, an Lp amplitude norm, and high-order fractional energy through a sharp homogeneous Gagliardo–Nirenberg inequality. The transferable asset is not merely a Sobolev penalty, but a dimension- and derivative-order-aware ratio whose scaling is neutralized by the exponent determined from the interpolation relation. A neural-network version can regularize spatial feature maps or continuous neural fields by penalizing violations of this interpolation envelope, rather than independently tuning amplitude, smoothness, and high-frequency penalties. The optimizer-existence result suggests that the ratio has well-behaved normalized minimization targets, although the sharp constant must be estimated empirically on the chosen tensor domain.