p-Adic Statistical Field Theory and Deep Belief Networks

Authors

Wilson A. Zuniga-Galindo, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

2-13-2023

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.

Comments

Original published version available at https://doi.org/10.1016/j.physa.2023.128492

Recommended Citation

Zuniga-Galindo, Wilson A. "p-adic statistical field theory and deep belief networks." Physica A: Statistical Mechanics and its Applications (2023): 128492. https://doi.org/10.1016/j.physa.2023.128492

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Physica A: Statistical Mechanics and its Applications

DOI

https://doi.org/10.1016/j.physa.2023.128492