The Cone Generated by Positive Semidefinite Mean Polynomials

What the math gives to ML

The paper supplies a constructive cone of globally nonnegative even-degree polynomials built from weighted power-mean differences, unifying SOS and circuit-polynomial certificates while strictly extending SOSONC. The transferable asset is a differentiable, compositional positivity certificate: each mean-polynomial atom is nonnegative by Jensen's inequality, and sums remain nonnegative without requiring a positive-semidefinite Gram matrix. This can parameterize globally nonnegative neural penalties, energy functions, or barrier losses with richer expressivity than SOS or SONC at comparable shallow depth. The most practical first test is to replace an SOS-based polynomial regularizer or energy head with a small learned sum of mean-polynomial atoms and compare optimization, expressivity, and constraint-violation rates.