<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e250" altimg="si8.svg"><mml:mi>p</mml:mi></mml:math>-adic cellular neural networks: Applications to image processing

Zambrano-Luna, B. A.; Zúñiga–Galindo, W. A.

doi:10.1016/j.physd.2023.133668

Cited by 11 publications

(7 citation statements)

References 19 publications

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“…Figure 4 shows the graph of a test function corresponding to an image from the MNIST data set. For further details, the reader may consult the Appendix in [18]. We note that in a p-adic discrete deep RBM, the visible and hidden states are functions on a finite tree.…”

Section: Resultsmentioning

confidence: 99%

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p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

Zúñiga–Galindo¹,

He²,

Zambrano-Luna³

2023

Preprint

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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.

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Section: Resultsmentioning

confidence: 99%

“…These ideas have been used to develop non-Archimedean models of brain activity and mental processes [16]. In [17,18], p-adic versions of the cellular neural networks were studied. These models involved abstract evolution equations.…”

Section: Introductionmentioning

confidence: 99%

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

Zúñiga–Galindo¹,

He²,

Zambrano-Luna³

2023

Preprint

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“…The p -adic analysis has shown to be the right tool in the construction of a wide variety of models of complex hierarchic systems; see, e.g., [ 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ] and the references therein. Many of these models involve abstract evolution equations of the type

.…”

Section: Introductionmentioning

confidence: 99%

“…Many of these models involve abstract evolution equations of the type

. In these models, the discretization of operator

is an ultrametric matrix

, where

is a finite rooted tree with l levels and

branches; here, p is a fixed prime number (see the numerical simulations in [ 27 , 28 ]). Locally, connection matrices look very similar to matrices

.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Wilson–Cowan Models and Connection Matrices

Zúñiga–Galindo

Zambrano-Luna

2023

Entropy

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This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.

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“…These ideas have been used to develop non-Archimedean models of brain activity and mental processes [50]. In [51]- [52], p-adic versions of the cellular neural networks were studied. These models involved abstract evolution equations.…”

Section: Introductionmentioning

confidence: 99%

P-Adic Statistical Field Theory and Deep Belief Networks

Zúñiga–Galindo¹

2022

SSRN Journal

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p-adic cellular neural networks: Applications to image processing

Cited by 11 publications

References 19 publications

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

Hierarchical Wilson–Cowan Models and Connection Matrices

P-Adic Statistical Field Theory and Deep Belief Networks

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