Cellular Neural Networks: Mosaic Pattern and Spatial Chaos

Abstract

We consider a cellular neural network (CNN) with a bias term z in the integer lattice Z 2 on the plane R 2. We impose a symmetric coupling between nearest neighbors, and also between next-nearest neighbors. Two parameters, a and ε, are used to describe the weights between such interacting cells. We study patterns that can exist as stable equilibria. In particular, the relationship between mosaic patterns and the parameter space (z, a; ε) can be completely characterized. This, in turn, addresses the so-called learning problem in CNNs. The complexities of mosaic patterns are also addressed.

Spatial disorder of Cellular Neural Networks

“…h(r) Notably, the result of (i) in the main theorem is consistent with [21] at r = 0, i.e., h(r) is continuous at r = 0.…”

“…Their corresponding pattern y can thus be called a mosaic and a defect pattern, respectively. Recently, Juang and Lin [21,22] and Hsu and Lin [19] and Hsu et al [20] considered mathematical results about the complexity of stable stationary solutions and the multiplicity of traveling wave solutions. For convenience, f will be denoted by fr , with f = r and m = 1 in the following of this…”

Pattern generation problems arising in multiplicative integer systems

“…Among them, the authors studied pattern generation problems on multidimensional shifts of finite type and developed some efficient means of studying the generation of admissible patterns, and then computing the topological entropy; see [1,2,3,4,5,6,7,21,22,24,29]. This study shows that these methods can be used to study the multi-dimensional decoupled systems and one-dimensional coupled systems of multiplicative integers, including X 0 2,3 .…”

Hausdorff Dimension of Multi-Layer Neural Networks

“…As being demonstrated in [7][8][9]13], the solution space is a so-called shift of finite type (SFT, also known as a topological Markov shift) and the output space is a sofic shift. More specifically, a SFT can be represented as a directed graph and a sofic shift can be represented as a labeled graph for some labeling and finite alphabet  .…”