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Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing

W. A. Zúñiga-Galindo, B. A. Zambrano-Luna, Baboucarr Dibba

TL;DR

A new type of p-adic reaction–diffusion cellular neural network with delay is introduced and the stability of these networks is studied and numerical simulations of their responses are provided.

Abstract

The first goal of this article is to introduce a new type of p-adic reaction-diffusion cellular neural network with delay. We study the stability of these networks and provide numerical simulations of their responses. The second goal is to provide a quick review of the state of the art of p-adic cellular neural networks and their applications to image processing.

Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing

TL;DR

A new type of p-adic reaction–diffusion cellular neural network with delay is introduced and the stability of these networks is studied and numerical simulations of their responses are provided.

Abstract

The first goal of this article is to introduce a new type of p-adic reaction-diffusion cellular neural network with delay. We study the stability of these networks and provide numerical simulations of their responses. The second goal is to provide a quick review of the state of the art of p-adic cellular neural networks and their applications to image processing.
Paper Structure (38 sections, 15 theorems, 119 equations, 13 figures)

This paper contains 38 sections, 15 theorems, 119 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Motivations and Preliminaries
  3. PDE based models for neural fields
  4. Neurons geometry and small-world networks
  5. Cellular neural Networks and complex multi-level systems
  6. $p$-Adic numbers and tree-like structures
  7. $p$-Adic integers
  8. Tree-like structures
  9. The Bruhat-Schwartz space in the unit ball
  10. The Haar measure
  11. Other function spaces
  12. $p$-Adic continuous CNNs
  13. Discretization of $p$-adic continuous CNNs
  14. Stability of $p$-adic continuous CNN
  15. A Numerical simulation
  16. ...and 23 more sections

Key Result

Proposition 1

Zambrano-Zuniga-1 Let $\tau$ be a fixed positive real number. Then, for each $X_{0}\in\mathcal{X}_{\infty}$ there exists a unique $X\in C([0,\tau],\mathcal{X}_{\infty})$ which satisfies where The function $X(x,t)$ is differentiable in $t$ for all $x$, and it is a solution of equation (Continuous_CNN) with initial datum $X_{0}$.

Figures (13)

  • Figure 1: Based upon Chistyakov, we construct an embedding $\mathfrak{f}:\mathbb{Z}_{p}\rightarrow\mathbb{R}^{2}$. The figure shows the images of $\mathfrak{f}(\mathbb{Z}_{2})$ and $\mathfrak{f}(\mathbb{Z}_{3})$. This computation requires a truncation of the $p$-adic integers. We use $\mathbb{Z}_{2}/2^{14}\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}/3^{10}\mathbb{Z}_{3}$, respectively.
  • Figure 2: The rooted tree associated with the group $\mathbb{Z}_{2}/2^{3}\mathbb{Z}_{2}$. The elements of $\mathbb{Z}_{2}/2^{3}\mathbb{Z}_{2}$ have the form $i=i_{0}+i_{1}2+i_{2}2^{2}$,$\;i_{0}$, $i_{1}$, $i_{2}\in\{0,1\}$. The distance satisfies $-\log_{2}\left\vert i-j\right\vert _{2}=$level of the first common ancestor of $i$, $j$. Figure taken from Zuniga et al
  • Figure 3: A discrete $2$-adic CNN with $8$ cells: $C_{3}=\{0,1,2,3,4,5,7\}\subset\mathbb{Z}_{2}/2^{3}\mathbb{Z}_{2}\subset 2^{-3}\mathbb{Z}_{2}/2^{3}\mathbb{Z}_{2}$. We set $\mathbb{B}=0$ and $\mathbb{A}(i,j)=\left[ a_{i,j}\right]$, with $a_{i,j}\neq0$ if $|i-j|_{2}=1/2$ and $i$, $j\in C_{3}$; $a_{i,j}=0$ otherwise. Taken from Zambrano-Zuniga-1
  • Figure 4: Heat map of $U(x)$. The position of each neuron corresponds with a leave of $G_{5}$. Time $25$ and step $\delta_{t}$$=0.05$.
  • Figure 5: Heat map of $X(x,t)$. Time $25$ and step $\delta_{t}$$=0.05$.
  • ...and 8 more figures

Theorems & Definitions (19)

  • Proposition 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • Corollary 10
  • ...and 9 more