Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks. 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 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 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(N2) parameters.
W A Zúñiga-Galindo, C He, B A Zambrano-Luna, p-Adic statistical field theory and convolutional deep Boltzmann machines, Progress of Theoretical and Experimental Physics, Volume 2023, Issue 6, June 2023, 063A01, https://doi.org/10.1093/ptep/ptad061
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Progress of Theoretical and Experimental Physics