A Non-Convex Method for Polynomial Manifold Learning

We study fitting low-dimensional polynomial manifolds to data by alternating two simple steps: a closed-form least-squares update of an analytic polynomial decoder and Gauss--Newton updates of the latent codes using the decoder’s analytic Jacobian. For intrinsic dimension $k{=}1$, the encoder reduces to enumerating stationary points of a univariate polynomial objective, and for $k{>}1$, each example solves only a small damped $k\times k$ system. The method is scalable—each B-step solves a least squares on the monomial design, and the A-step entails a handful of tiny solves per point. We test our method on four 2D curves and a 3D surface.