Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE

Alexandre Quemy

Keywords: CP decomposition, linear PDE, manifold learning

TL;DR: Learn a coordinate chart where a PDE source and its Green's kernel tensor-decompose, collapsing high-dimensional integration into independent 1D integrals

Abstract:

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.

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Submission Number: 39