Complexity of neural networks on Fibonacci-Cayley tree

Journal of Algebra Combinatorics Discrete Structures and Applications

article

Jung-Chao Ban¹  Chih-Hung Chang³ 

[1] Department of Mathematical Sciences, National Chengchi University;Math. Division, National Center for Theoretical Science, National Taiwan University;Department of Applied Mathematics, National University of Kaohsiung

关键词: Neural networks;    Learning problem;    Cayley tree;    Separation property;    Entropy;   

DOI  :  10.13069/jacodesmath.560410

学科分类：社会科学、人文和艺术（综合）

来源: Yildiz Technical University

【 摘 要 】

This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree via an invariant called entropy. We demonstrate that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. What is more, the entropy of a Fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. We apply the result to neural networks defined on Fibonacci-Cayley tree, which reflect those neural systems with neuronal dysfunction. Aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on Fibonacci-Cayley tree, we address the formula of the boundary in the parameter space.

【 授权许可】

CC BY