On the structure of multi-layer cellular neural networks: Complexity between two layers

TL;DR

This paper investigates the relationships and factor maps between hidden spaces of multi-layer cellular neural networks, focusing on their topological entropies and Hausdorff dimensions, revealing new insights into their structural complexity.

Contribution

It demonstrates the existence of factor maps between hidden spaces with different topological entropies and provides formulas for calculating their Hausdorff dimensions.

Findings

Existence of factor maps between hidden spaces with distinct entropies.

Explicit formulas for Hausdorff dimensions of hidden spaces.

Relationships between dimensions of hidden spaces via factor maps.

Abstract

Let $\mathbf{Y}$ be the solution space of an $n$-layer cellular neural network, and let $\mathbf{Y}^{(i)}$ and $\mathbf{Y}^{(j)}$ be the hidden spaces, where $1 \leq i, j \leq n$. ($\mathbf{Y}^{(n)}$ is called the output space.) The classification and the existence of factor maps between two hidden spaces, that reaches the same topological entropies, are investigated in [Ban et al., J.~Differential Equations \textbf{252}, 4563-4597, 2012]. This paper elucidates the existence of factor maps between those hidden spaces carrying distinct topological entropies. For either case, the Hausdorff dimension $\dim \mathbf{Y}^{(i)}$ and $\dim \mathbf{Y}^{(j)}$ can be calculated. Furthermore, the dimension of $\mathbf{Y}^{(i)}$ and $\mathbf{Y}^{(j)}$ are related upon the factor map between them.