Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE
For machine learning on high-dimensional data lying on low-dimensional manifolds, IGL provides a method that avoids the curse of dimensionality and automatically discovers intrinsic dimension.
Intrinsic Green's Learning (IGL) models a target function on a manifold as the solution to a linear PDE with a learned source, achieving near-optimal classification on MNIST and automatic recovery of intrinsic dimension with cost linear in the intrinsic dimension.
We introduce Intrinsic Green's Learning (IGL), a framework that models a target function on a manifold as the solution to a linear PDE whose source term is learned from data. Rather than approximating the target directly, IGL learns a source and integrates it against a Green's kernel. An encoder discovers a low-dimensional coordinate chart on the manifold where both the source and the kernel decompose as low-rank tensors, collapsing a high-dimensional integral into independent one-dimensional integrals with cost linear in the intrinsic dimension. A two-stage algorithm separates coordinate discovery from source fitting, a near-convex linear solve, preventing the dimensional collapse of joint training. Learnable gates on each coordinate automatically discover the intrinsic dimension of the manifold. We validate IGL on synthetic manifolds and on MNIST, where it simultaneously achieves near-optimal classification and automatic recovery of the intrinsic dimension.