主成分分析的一个黎曼几何随机算法

一个典型的求解主成分问题的方法是Oja-Sanger算法，但其不能保证迭代矩阵列的单位列正交性，实际计算时矩阵列甚至是无界的．将主成分问题等价变换为Stiefel流形上的一个二次优化问题，采用黎曼几何算法思想，获得求解主成分分析(PCA)的一个黎曼几何随机算法(自适应算法)．该方法可确保迭代矩阵列的单位列正交性．数值模拟结果表明，本文算法优于Oja-Sanger算法．

A Riemannian Geometry Underlying Stochastic Algorithm for Adaptive Principal Component Analysis

A typical algorithm to solve principal component analysis (PCA) problem is the so-called Oja-Sanger method. But its resulting matrix sequence cannot ensure the unitary column orthogonality. In practical applications, even the boundedness of the matrix sequence cannot be ensured. This paper converted the PCA problem into an (equivalent) optimization problem with the objective function being second order polynomials, restricted on a Stiefel manifold. Then a Riemannian geometry underlying stochastic algorithm was constructed to solve the problem, applying the basic ideas of Riemannian geometry algorithms. The new algorithm has good mathematical properties. It can ensure unitary column orthogonality for the resulting matrix sequence. The numerical simulation shows that the new method is superior to Oja-Sanger method.