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

Area of Science:

• Theoretical Machine Learning
• Mathematical Physics
• Geometric Deep Learning

Background:

• Conventional machine learning lacks a first-principles framework for phenomena like criticality and causal structure.
• Existing models often treat learning solely as parameter optimization.
• A need exists for a more comprehensive mathematical model of the learning process.

Purpose of the Study:

• To introduce and validate Data Field Theory (DFT), a novel mathematical framework for machine learning.
• To model learning as the evolution of a data field governed by stochastic partial differential equations.
• To assess DFT's predictions in geometrically consistent settings and identify empirical limitations.

Main Methods:

• Formulated learning as a field evolving on a spherical manifold.
• Implemented a hierarchical classification task using synthetic data from von Mises-Fisher distributions.
• Derived and tested four key predictions: critical exponents, spectral robustness law, causal propagation speed, and rotational equivariance.

Main Results:

• Synthetic experiments validated all four DFT predictions with high statistical significance.
• Observed phenomena include diverging correlation length, scaling of out-of-distribution error with spectral gaps, and bounded causal propagation.
• Preliminary real-data experiments on MNIST-sphere data showed significantly lower accuracy compared to k-NN, highlighting limitations.

Conclusions:

• DFT provides a principled mathematical foundation for learning as field dynamics, successfully predicting emergent phenomena under ideal geometric conditions.
• The framework's current limitations on real-world data necessitate extensions in adaptive metric learning, noise robustness, and hierarchical feature extraction.
• DFT establishes a theoretical basis for future advancements in machine learning grounded in physical principles.