Riemannian Submersion Metrics on Grassmann Manifolds with Applications to Preconditioned Optimization

A family of Riemannian submersion metrics on the real Grassmann manifold, parameterized by smooth maps from the Stiefel manifold to the manifold of symmetric positive definite matrices that satisfy an invariance property, is investigated. These maps are strongly related to a known family of metrics on the Stiefel manifold. Endowed with metrics from these families, the Stiefel manifold becomes a Riemannian submersion over the Grassmann manifold. An explicit formula for the projection onto the horizontal bundle is derived and horizontal lifts are investigated. This leads to explicit expressions for Riemannian gradients and Hessians of smooth functions on the Grassmann manifold. The formulas are applied to the optimization of a well-known cost on the Grassmann manifold, the generalized Rayleigh quotient. Based on an estimate of the condition number of the Riemannian Hessian at a critical point of the cost, a construction of certain Riemannian metrics adapted to the cost is proposed. This gives rise to Riemannian preconditioning schemes. In addition, all those differential geometric quantities are explicitly derived to implement a geometric conjugate gradient algorithm on the Grassmann manifold endowed with a submersion metric.