p-adic statistical field theory and deep belief networks

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

“…There is consensus about the need of developing a theoretical framework to understand how deep learning architectures work. Recently, physicists have proposed the existence of a correspondence between neural networks (NNs) and quantum field theories (QFTs), more precisely statistical field theory (SFT), see [3]- [12], and the references therein. This correspondence takes different forms depending on the architecture of the networks involved.…”

“…In [12], the study of the above-mentioned correspondence was initiated in the framework of the non-Archimedean statistical field theory (SFT). In this case, the background space (the set of real numbers) is replaced by the set of p-adic numbers, where p is a fixed prime number.…”

“…The p-adics are organized in a tree-like structure; this feature facilitates the description of hierarchical architectures. In [12], a p-adic version of the convolutional deep Boltzmann machines is introduced where only binary data is considered and with no implementation. By adapting the mathematical techniques introduced by Le Roux and Benigio in [13], the author shows that these machines are universal approximators for binary data tasks.…”

“…In this article, we continue discussing the correspondence between STFs and NNs, in the p-adic framework. Compare with [12] here we consider more general architectures and data types. We note that dealing with general data is challenging both in theory and implementation practice.…”