IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v612y2023ics037843712300047x.html
   My bibliography  Save this article

p-adic statistical field theory and deep belief networks

Author

Listed:
  • Zúñiga-Galindo, W.A.

Abstract

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 to a certain spin glass, we argue that a DBN should correspond to an ultrametric spin glass. A model of such a 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.

Suggested Citation

  • Zúñiga-Galindo, W.A., 2023. "p-adic statistical field theory and deep belief networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 612(C).
    • Handle: RePEc:eee:phsmap:v:612:y:2023:i:c:s037843712300047x
    DOI: 10.1016/j.physa.2023.128492
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037843712300047X
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2023.128492?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Khrennikov, A.Yu. & Kozyrev, S.V., 2006. "Replica symmetry breaking related to a general ultrametric space—II: RSB solutions and the n→0 limit," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 241-266.
    2. Khrennikov, A.Yu. & Kozyrev, S.V., 2006. "Replica symmetry breaking related to a general ultrametric space I: Replica matrices and functionals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 222-240.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Khrennikov, Andrei Yu., 2009. "Gene expression from polynomial dynamics in the 2-adic information space," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 341-347.
    2. Khrennikov, A.Yu. & Kozyrev, S.V., 2007. "Replica symmetry breaking related to a general ultrametric space III: The case of general measure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 283-298.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:612:y:2023:i:c:s037843712300047x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.