遗传算法在矩阵置换对称性求解中的应用

Application of Genetic Algorithms in Searching for Matrix's Permutation Symmetries

对称性是减少问题的自由度的一个强有力的工具 .但在实际的应用中 ,系统变换操作的总数目将随系统维数的增加而急剧上升 ,这给高维系统对称性的计算带来了极大的不便 ,从而使得对称性方法的应用受到了很大的限制 .本文以全互连结构的神经网络为例 ,提出一种基于遗传算法的搜索方法 ,在对称群Sn 中寻找网络的对称置换操作 ,给出了计算机上的模拟结果 ,并与传统的遍历搜索方法作比较 ,分析了各自的优缺点 .结果表明 ,这种基于遗传算法的搜索方法能够在极短的时间内找到网络的大部分对称置换操作 .这使得对称性方法在高维神经网络研究及设计中的应用成为可能 .

Symmetry is a powerful tool to reduce the number of degrees of freedom of a problem. But calculating the symmetry of a high-dimensional system would be very difficult since the total number of transformations increases dramatically with the dimension of the system, which places many restrictions on the application of the symmetry method. A novel approach based on genetic algorithms is proposed to search for the permutation symmetries of the weight matrix of full-connected neural networks within the symmetry group Sn. Searching results for several different dimensional matrixes are given and compared to that of the ergodic searching method. It turns out that the searching method based on genetic algorithms can find the majority of symmetric permutations of the matrix within a short time, which makes it possible to study and design of the high-dimensional neural networks by the symmetry tool.