通过逆PDE学习流形上的目标函数，实现高效监督学习。
Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE
- 将目标函数建模为线性PDE解，从数据中学习源项并积分求解。
- 在MNIST上实现近最优分类，同时自动恢复流形的内在维度。
- 低秩张量分解使高维积分降为一维独立积分，计算成本线性于内维。
我们提出内在格林学习（IGL），一种在流形上建模目标函数的框架，将其视为线性偏微分方程（PDE）的解，其中源项由数据学习得到。IGL不直接逼近目标函数，而是学习源项，并将其与格林核进行积分。编码器在流形上发现低维坐标图，使得源项和核均可分解为低秩张量，将高维积分转化为独立的一维积分，计算成本与内在维度呈线性关系。采用两阶段算法，先发现坐标再拟合源项，通过近凸线性求解避免联合训练中的维度坍缩。每个坐标上的可学习门控机制可自动识别流形的内在维度。我们在合成流形和MNIST数据集上验证了IGL的有效性，在MNIST上实现了近最优分类性能，同时自动恢复了内在维度。
原文摘要 · Abstract (English)
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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