Quotient-and-Radical Feature Split

Use this in a polynomial feature layer or tensorized MLP whose hidden representation contains homogeneous components of degrees \(k\). Choose \(u\) and focus on degrees \(u<k\le2u\). Represent a degree-\(k\) feature tensor \(f_k\) explicitly, or approximate it with a low-rank symmetric tensor network. Construct the radical branch by taking a lower-degree feature \(f_{2u-k}\) and multiplying it by the quadratic invariant \(x^2\) exactly \(k-u\) times: \(R(f_{2u-k})=(x^2)^{k-u}f_{2u-k}\). Construct the quotient branch by applying a learned linear map after removing radical coordinates; for an explicit monomial basis, use a fixed mask, while for an implicit tensor representation approximate the quotient projection with QR or SVD on a minibatch feature matrix. Train with \(\mathcal{L}_{\mathrm{quot}}\) on the quotient output and a separate \(\mathcal{L}_{\mathrm{rad}}\) on the transported branch. The optional orthogonality penalty uses the ordinary pairing \(\langle a,b\rangle=a^\top b\) and discourages leakage between branches. Initialize the radical gate at zero and sweep \(\lambda_{\mathrm{rad}}\) and \(\lambda_{\mathrm{orth}}\). The paper determines the radical dimension and transport map; the engineer estimates only the quotient basis or low-rank projection from data. First test a degree-4 or degree-6 polynomial MLP on synthetic parity and hierarchical-composition tasks, comparing one singular branch, pseudoinverse whitening, and the split model at equal width. Measure condition number, gradient norm variance, training failures, and test accuracy. Success is stable training at larger learning rates and improved accuracy when labels depend on both quotient and radical components; if labels ignore the radical, the split model should match the quotient-only baseline without hurting calibration.