細胞神經網絡的空間複雜度

Abstract

本文旨在研究一維的多層細胞神經網絡所產生的花樣其空間複雜度以及二維非均勻空間細胞神經網路其拓樸熵的稠密性。在多層細胞神經網絡部分， 如果我們在輸出空間給予適當的符號， 輸出空間會等價於一個 sofic 移動空間。我們給出兩個物理上的不變量， 拓樸熵和動態 $\zeta$-函數的公式。 這同時給了抽象的 sofic 移動空間一個實際上可以應用的例子。除此之外， 我們發現了當考慮多層細胞神經網絡時， 拓樸熵的對稱性會被破壞掉。 這也再一次證明多層細胞神經網絡和單層細胞神經網絡是兩個性質極端不同的系統。更進一步地， 當我們考慮非均勻空間的細胞神經網絡系統， 其拓樸熵會稠密的分佈在 $[0, \log 2]$ 這個封閉區間之中。 換句話說， 非均勻空間的細胞神經網絡系統有著極為豐富的物理現象蘊含在裡面。

This dissertation consists two parts. The first part investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input; the second part focus on the dense entropy of two-dimensional inhomogeneous cellular neural networks with/without input. For the first part, applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input. The second part is strongly related to the learning problem (or inverse problem); the necessary and sufficient conditions for the admissibility of local patterns must be characterized. The entropy function is dense in $[0, \log 2]$ with respect to the parameter space and the radius of the interacting cells, indicating that, in some sense, such system exhibits a wide range of phenomena.