The Geometric Structure of Models Learning Sparse Data
For machine learning practitioners, the paper provides a theoretical framework and practical regularization strategies to improve training dynamics and robustness in sparse data regimes.
The paper identifies regimes where the manifold hypothesis fails (sparse data) and shows that models succeed by exploiting a structured local geometry called normal alignment. The authors prove that normal-aligned classifiers minimize training objectives under norm constraints and achieve maximal local robustness, and they introduce GrokAlign and RFAMs, which accelerate training and improve adversarial robustness, respectively.
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 data -- minimize the training objective under norm constraints and achieve maximal local robustness under a non-zero Jacobian constraint. For continuous piecewise-affine deep networks, normal alignment manifests geometrically as centroid alignment within the network's induced power diagram partition and results from the feature-learning regime. Motivated by these theoretical insights, we introduce GrokAlign, a regularization strategy that actively induces normal alignment. We demonstrate that GrokAlign significantly accelerates the training dynamics of deep networks relevant to the grokking phenomenon. Furthermore, we apply the principle of normal alignment to Recursive Feature Machines (RFMs) to introduce Recursive Feature Alignment Machines (RFAMs). We show that RFAMs exhibit greater adversarial robustness compared to RFMs when trained on tabular data.