Data field theory: a geometric framework for learning on Riemannian manifolds with synthetic validation and limitation analysis.

Abstract

Introduction: Conventional machine learning treats learning as parameter optimization, lacking a first-principles framework for phenomena like criticality, generalization, and causal structure. We introduce Data Field Theory (DFT), a mathematical framework modelling learning as the evolution of a data field governed by stochastic partial differential equations on Riemannian manifolds. This work aims to validate DFT's core predictions in settings where its geometric assumptions hold, while honestly assessing its empirical limitations.

Methods: We formulate learning as a field $\phi :M\times {ℝ}_{\ge 0}\to {ℝ}^k$ evolving on a spherical manifold. To test DFT, we implement a hierarchical classification task using synthetic data drawn from von Mises-Fisher distributions, ensuring match with the manifold geometry. We derive four key predictions: (1) critical exponents near concept formation, (2) a spectral robustness law linking Eigen gaps to out-of-distribution (OOD) error, (3) finite-speed causal propagation from hyperbolic regularization, and (4) approximate rotational equivariance via a Ward identity. We also conduct a preliminary real-data experiment projecting MNIST digits onto the sphere.

Results: Synthetic experiments validate all four predictions: (1) Correlation length diverges as $\xi \left(t\right)~|t-t_c{|}^{-\nu }$ with ν = 0.63 ± 0.04, accompanied by 1/f fluctuations; (2) OOD generalization error scales as ${ϵ}_{\text{OOD}}\propto m_{\text{gap}}^{-2}$ (ρ = -0.78, p < 10^-6); (3) Causal propagation speed c _eff = 0.98 ± 0.03 (theory maximum c _max = 1.0) under hyperbolic regularization; and (4) Ward identity residual R = 0.0032 ± 0.0008 converging as R∝h ^1.02. However, on real-world MNIST-sphere data, DFT achieves only 15.7% accuracy versus 51.7% for k-NN, revealing critical limitations.

Discussion: DFT successfully predicts emergent phenomena criticality, spectral robustness, bounded causality, and approximate equivariance under ideal geometric conditions, supporting its theoretical validity. The poor real-data performance highlights key gaps: the current framework lacks adaptive metric learning, noise robustness, and hierarchical feature extraction present in real images. These results establish DFT as a principled mathematical foundation for learning as field dynamics while clearly delineating necessary extensions for practical applicability.