This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

翻译：本文针对配备非标准度量的广义Stiefel流形，提出了两种基于Cayley变换的创新向量传输算子。具体而言，我们引入了微分收缩算子以及Cayley变换对微分矩阵指数的近似形式。研究证明，这些向量传输算子在非标准度量下满足Ring-Wirth非扩张条件，且其中一种算子具有等距特性。基于这些新型向量传输算子，我们将改进的Polak-Ribi$\grave{e}$re-Polyak（PRP）共轭梯度法推广至广义Stiefel流形。在非单调线搜索条件下，我们证明了该算法能全局收敛至稳定点。通过广义特征值问题和典型相关分析的数值实验，实证验证了所提向量传输算子的计算效率。