<i>p</i>-Adic statistical field theory and convolutional deep Boltzmann machines

Understanding how deep learning architectures work is a central scientific problem. Recently, correspondence between neural networks (NNs) and Euclidean quantum field theories (QFTs) has been proposed. This investigates this in the framework of p-adic statistical (SFTs) (NNs). In case, fields are real-valued functions defined on an infinite regular rooted tree with valence p, fixed prime number. provides topology for continuous Boltzmann machine (DBM), which identified theory (SFT) tree. framework, there natural method to discretize SFTs. Each discrete SFT corresponds (BM) tree-like topology. allows us recover standard DBMs gives new convolutional DBMs. The use O(N) parameters while classical ones O(N^{2}) parameters.