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This paper proposes two innovative vector transport operators, leveraging theCayley transform, for the generalized Stiefel manifold embedded with anon-standard inner product. Specifically, it introduces the differentiatedretraction and an approximation of the Cayley transform to the differentiatedmatrix exponential. These vector transports are demonstrated to satisfy theRing-Wirth non-expansive condition under non-standard metrics while preservingisometry. Building upon the novel vector transport operators, we extend themodified Polak-Ribi$\acute{e}$re-Polyak (PRP) conjugate gradient method to thegeneralized Stiefel manifold. Under a non-monotone line search condition, weprove our algorithm globally converges to a stationary point. The efficiency ofthe proposed vector transport operators is empirically validated throughnumerical experiments involving generalized eigenvalue problems and canonicalcorrelation analysis.

Two novel vector transports for generalized Stiefel manifold with non-standard metrics and its application to Riemannian conjugate gradient method