Stiefel流形上沿测地线搜索的自适应主(子)分量分析对偶学习算法

Adaptive dual learning algorithm for principal(minor) component analysis along geodesic on Stiefel manifold

神经网络在线提取子分量并不成功。基于Oja-Brockett-Xu并行神经网络拓扑结构,通过紧致Stiefel流形上加权Rayleigh商目标函数的优化框架,提出一个通过改变搜索方向并行提取主分量和子分量的自适应对偶学习算法。在正交矩阵群上采用基于右平移不变的Killing度量,通过在单位元处基于指数映射的测地线搜索,得到Stiefel流形上主(子)分量分析的对偶学习算法,提出的算法通过简单的变换步长参数符号,从主分量分析切换至子分量分析,权值矩阵在任意迭代时刻保持正交归一性。数值仿真验证了该算法的有效性。

Using the same topology as that of Oja-Brockett-Xu parallel neural network,a novel dual purpose adaptive algorithm for principal and minor component extraction was proposed by the optimization framework of a weighted Ray-leigh quotient on the compact Stiefel manifold. By taking the right translation invariant Killing metric on orthogonal ma-trix group and search along the geodesic emanating from identity by means of exponential map,a novel dual learning al-gorithm for principal and minor component analysis was proposed. The proposed algorithm could switch from PCA （Principal Component Analysis）to MCA（Minor Component Analysis）with a simple sign change of its stepsize pa-rameter. Moreover,orthonormality of the weight matrix was guaranteed at any iteration step. The effectiveness of the proposed algorithm was further verified in the section of numerical simulation.