An Introduction to the Theory of Local Zeta Functions from Scratch

León-Cardenal, Edwin; Zúñiga–Galindo, W. A.

doi:10.18273/revint.v37n1-2019004

Cited by 4 publications

(2 citation statements)

References 54 publications

(81 reference statements)

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“…For a detailed exposition on p-adic analysis the reader may consult [3], [86], [87]. For a quick review of p-adic analysis the reader may consult [12], [61].…”

Section: Basic Facts On P-adic Analysismentioning

confidence: 99%

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p-Adic Statistical Field Theory and Deep Belief Networks

Zúñiga–Galindo¹

2022

Preprint

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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 a certain spin glass, thus a DBN should correspond to an ultrametric (hierarchical) spin glass. A model of such 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..

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“…For a detailed exposition on p-adic analysis the reader may consult [3], [86], [87]. For a quick review of p-adic analysis the reader may consult [12], [61].…”

Section: Basic Facts On P-adic Analysismentioning

confidence: 99%

“…In a few occasions we use the two-dimensional Haar measure dxdy of the additive group (Q p × Q p , +) normalized by the condition Zp Zp dxdy = 1. For a quick review of the integration in the p-adic framework the reader may consult [12], [61] and the references therein.…”

Section: Basic Facts On P-adic Analysismentioning

confidence: 99%