p-Adic Statistical Field Theory and Deep Belief Networks
arXiv:2207.13877 · doi:10.1016/j.physa.2023.128492
Abstract
In this work we initiate the study of the correspondence between p-adic statistical field theories (SFTs) and neural networks (NNs). In general quantum field theories over a p-adic spacetime can be formulated in a rigorous way. Nowadays these theories are considered just mathematical toy models for understanding the problems of the true theories. In this work we show these theories are deeply connected with the deep belief networks (DBNs). Hinton et al. constructed DBNs by stacking several restricted Boltzmann machines (RBMs). The purpose of this construction is to obtain a network with a hierarchical structure (a deep learning architecture). An RBM corresponds to a certain spin glass, we argue that a DBN should correspond to an ultrametric spin glass. A model of such a system can be easily constructed by using p-adic numbers. In our approach, a p-adic SFT corresponds to a p-adic continuous DBN, and a discretization of this theory corresponds to a p-adic discrete DBN. We show that these last machines are universal approximators. In the p-adic framework, the correspondence between SFTs and NNs is not fully developed. We point out several open problems.
The introduction was fully revisited. The condition p>2 was removed
References in corpus (9)
- p-Adic Mathematical Physics
- Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals
- Nonperturbative renormalization for the neural network-QFT correspondence
- Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions
- p-adic Cellular Neural Networks
- Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks
- Eigen's Paradox and the Quasispecies Model in a Non-Archimedean Framework
- Self-organized criticality in neural networks
- Euclidean Quantum Field Formulation of p-Adic open string amplitudes