The Geometric Structure of Models Learning Sparse Data
The manifold hypothesis (MH) is often used to explain how machine learning can overcome the curse of dimensionality. However, the MH is only applicable in regimes where the training data provides a sufficiently dense sample of the underlying low-dimensional data manifold, or where such a low-dimensional manifold is conceivably present. We describe the regimes where the MH is not applicable as sparse. In this paper, we demonstrate that models succeed in the sparse regime by exploiting a highly structured local geometry, a property we formalize as normal alignment. We prove that normal-aligned classifiers -- whose input-output Jacobians are rank-one and align perfectly with the training dat...
| Authors | Thomas Walker, T. Mitchell Roddenberry, Ahmed Imtiaz Humayun et al. |
|---|---|
| Primary category | cs.LG |
| Published | 8 May 2026 |
| Last updated at source | 15 May 2026 |
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