p-Adic statistical field theory and convolutional deep Boltzmann machines
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Publication:6115474
DOI10.1093/PTEP/PTAD061zbMath1528.81187arXiv2302.03817MaRDI QIDQ6115474
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Publication date: 12 July 2023
Published in: Unnamed Author (Search for Journal in Brave)
Abstract: Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories (QFTs) has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks (NNs). In this case, the fields are real-valued functions defined on an infinite regular rooted tree with valence p, a fixed prime number. This infinite tree provides the topology for a continuous deep Boltzmann machine (DBM), which is identified with a statistical field theory (SFT) on this infinite tree. In the p-adic framework, there is a natural method to discretize SFTs. Each discrete SFT corresponds to a Boltzmann machine (BM) with a tree-like topology. This method allows us to recover the standard DBMs and gives new convolutional DBMs. The new networks use O(N) parameters while the classical ones use O(N^{2}) parameters.
Full work available at URL: https://arxiv.org/abs/2302.03817
Artificial neural networks and deep learning (68T07) Trees (05C05) Model quantum field theories (81T10) Neural nets and related approaches to inference from stochastic processes (62M45) Turing machines and related notions (03D10) Boltzmann equations (35Q20)
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