INTRINSIC GREEN’S LEARNING: SUPERVISED LEARNING ON MANIFOLDS VIA INVERSE PDE

Supervised learning often assumes that high-dimensional data lies on a lower-dimensional intrinsic manifold. However, discovering this structure while simultaneously learning a target function is difficult, often leading to dimensional collapse or poor sample efficiency. This paper asks: can we leverage the mathematical structure of partial differential equations (PDEs) to learn on manifolds more effectively?

The author introduces Intrinsic Green's Learning (IGL), which frames supervised learning as an inverse PDE problem. Instead of approximating a target function directly, IGL models it as the solution to a linear PDE ($Lu = f$), where $f$ is a learned source term. The framework uses an encoder to map data into a low-dimensional coordinate chart where both the source and the Green's kernel decompose into low-rank tensors. This decomposition allows the high-dimensional integral solution to be computed as a series of independent one-dimensional integrals, reducing computational complexity to be linear in the intrinsic dimension ($d$). A critical component is the two-stage training algorithm: Stage 1 learns the coordinate mapping, while Stage 2 solves for the source weights via a near-convex linear regression, preventing the dimensional collapse common in joint training.

IGL demonstrates significant advantages in both synthetic and real-world settings. On synthetic manifolds, the two-stage training prevents dimensional collapse and enables the automatic discovery of the intrinsic dimension through learnable gates. In regression tasks, IGL exhibits a sharp sample-efficiency phase transition, outperforming standard baselines as the number of samples increases. When applied to MNIST, IGL acts as an effective latent-space regularizer, achieving near-optimal classification accuracy while simultaneously identifying the intrinsic dimension of the dataset (approximately 12 dimensions) and maintaining superior label smoothness compared to standard autoencoders.