The Independence-Preserving Property and Planar Web Geometry

What the math gives to ML

The paper turns independence preservation under a two-dimensional analytic diffeomorphism into an explicit logarithmic-density functional equation, and identifies admissible density families through Abelian relations of a planar four-web. The transferable asset is the structural constraint that an invertible nonlinear map can preserve factorization only when its coordinate functions and log-Jacobian satisfy a low-dimensional additive relation. This suggests a constrained invertible latent module whose transformation is trained to preserve independence for several product-measure probes simultaneously, using the web residual as a differentiable regularizer. The stated rank bound gives a principled low-dimensional capacity constraint instead of relying only on a generic discriminator or mutual-information estimator.