Weak Limits of Wiener Chaos: Primitive-Fock Classification and Hilbert-Stein Extraction

What the math gives to ML

The paper gives a constructive way to represent weak limits of fixed-order Gaussian polynomials when the underlying Gaussian feature space changes: each physical weight decomposes into a decomposable span generated by products and a primitive orthogonal complement. The resulting weighted-Fock sectors separate interactions explainable by lower-order products from genuinely new Gaussian factors, while the primitive component is canonical up to orthogonal changes of coordinates. This suggests a neural polynomial-feature block that explicitly projects learned high-order interactions into decomposable and primitive channels, with an orthogonality penalty preventing redundant product features. The most realistic first test is a small q=2 or q=3 tensorized MLP on synthetic compositional data, comparing parameter efficiency and generalization against an unconstrained polynomial layer.