Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods

What the math gives to ML

The paper gives a finite, constructive way to distinguish multivariate polynomials up to orthogonal change of coordinates using joint moments of correlated Gaussian replicas, whereas the marginal output distribution loses important structure. The transferable asset is that these probe statistics are polynomial functions of covariance parameters, so a finite rational covariance grid and interpolation can recover Wick-contraction invariants without solving a nonconvex alignment problem. This suggests a symmetry-aware polynomial neural layer or an identifiability regularizer that fingerprints a learned polynomial map using correlated replicas rather than independent random inputs. The approach is most credible for tensorized polynomial networks, polynomial feature maps, and neural operators with low-degree local expansions, rather than arbitrary ReLU networks.