細胞類神經網路的花樣與空間熵

Abstract

在本論文中，我們探討一維與二維的細胞類神經網路的花樣與其複雜性。 首先，我們根據合適局部花樣來切割參數空間。參數空間可被切割成有限個不相交的區域。在同一區域參數的細胞類神經網路， 其可能花樣的集合是一樣的。在此同時， 我們可完全解決細胞類神經網路的直接學習問題。關於花樣的複雜性， 我們分別對一維與二維的細胞類神經網路來考慮。在一維的細胞類神經網路，我們提出一個演算法來計算花樣的空間熵的確切值。這個演算法是利用合適局部花樣所決定的轉移矩陣，其最大特徵值取自然對數後， 即是空間熵。 但在二維的細胞類神經網路， 因為垂直方向與水平方向的轉移矩陣有相容性的問題，我們無法求出確切的花樣空間熵。我們可取而代之地求花樣空間熵的上界與下界。 我們提除一個演算法，利用貼的技巧， 來求在有限格子點上的花樣總數， 上界因此而得到。 另外， 我們求出最小尺寸的幅貼花樣磚的總數， 來計算下界。

In this thesis, We investigate the mosaic patterns and patial entropy of cellular neural networks(CNNs). First, we partition the parameter space into finitely many disjoint regions such that in each region the CNN has the same mosaic patterns. The direct problem is then completely solved. After that, the spatial entropy is investigated for one-dimensional case and two-dimensional case respectively. In a one-dimensional case, we propose an algorithm to compute the exact spatial entropy of mosaic patterns for general templates. This algorithm is based on the feasible local pattern and the corresponding transition matrix. By computing the maximum eigenvalue of the transition matrix, the exact spatial entropy is then obtained. As for the two-dimensional case, for which we cannot obtain the exact spatial entropy in general, is markedly different with the one-dimensional case because of the compatibility problem between horizontal and vertical transition matrices. Instead of the exact spatial entropy, we give the upper and lower bound. We present an algorithm to attain the mosaic patterns in finite lattice and the upper bound is then obtained. On the other hand, the lower bound is obtained by constructing the maximal set of patching blocks of minimal size(depend on the characteristic of the global patterns).