Stiefel流形上的梯度算法及其在特征提取中的应用

摘要

为了提高系统特征提取算法的计算效率、减少占用的存储空间和简化程序设计，该文基于Riemann流形上优化算法的几何框架，提出了改进的Stiefel流形上的梯度下降算法。根据不同要求采用不同的测地线计算公式，并使用多项式逼近测地线方程，同时采用了秦九韶-Horner多项式算法及线搜索、变步长的方法。以主分量分析问题为例，详细讨论了 Stiefel 流形上的梯度算法在其中的应用。理论分析和实验结果均表明，此方法可以在确保迭代矩阵列向量单位正交性的同时获得更好的计算效率和收敛速度，并且更容易实现。%To improve the computational efficiency of system feature extraction, reduce the occupied memory space, and simplify the program design, a modified gradient descent method on Stiefel manifold is proposed based on the optimization algorithm of geometry frame on the Riemann manifold. Different geodesic calculation formulas are used for different scenarios. A polynomial is also used to lie close to the geodesic equations. JiuZhaoQin-Horner polynomial algorithm and the strategies of line-searching technique and change of the step size of iteration are also adopted. The gradient descent algorithm on Stiefel manifold applied in Principal Component Analysis (PCA) is discussed in detail as an example of system feature extraction. Theoretical analysis and simulation experiments show that the new method can achieve superior performance in both the convergence rate and calculation efficiency while ensuring the unitary column orthogonality. In addition, it is easier to implement by software or hardware.