A Global Cayley Parametrization Of Stiefel Manifold\\For Direct Utilization Of Optimization Mechanisms Over Vector Spaces

Optimization problem with orthogonality constraints, whose feasible region is called the Stiefel manifold, has rich applications in data sciences. The severe non-linearity of the Stiefel manifold has hindered the utilization of optimization mechanisms developed specially over a vector space for the problem. In this paper, we present a global parametrization of the Stiefel manifold entirely by a single fixed vector space with the Cayley transform, say Global Cayley Parametrization (G-CP), to solve the problem through optimization over a vector space. The G-CP has key properties for solving the problem with G-CP and for applications to orthogonality constraint stochastic/distributed optimization problems. A numerical experiment shows that G-CP strategy outperforms the standard strategy with a retraction [Absil-Mahony-Sepulchre, 08].