A generalized canonical metric for optimization on the indefinite Stiefel manifold

Abstract: Various tasks in scientific computing can be modeled as an optimization problem on the indefinite Stiefel manifold. We address this using the Riemannian approach, which basically consists of equipping the feasible set with a Riemannian metric, preparing geometric tools such as orthogonal projections, formulae for Riemannian gradient, retraction and then extending an unconstrained optimization algorithm on the Euclidean space to the established manifold. The choice for the metric undoubtedly has a great influence on the method. In the previous work [D.V. Tiep and N.T. Son, A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold, arXiv:2410.22068v2[math.OC]], a tractable metric, which is indeed a family of Riemannian metrics defined by a symmetric positive-definite matrix depending on the contact point, has been used. In general, it requires solving a Lyapunov matrix equation every time when the gradient of the cost function is needed, which might significantly contribute to the computational cost. To address this issue, we propose a new Riemannian metric for the indefinite Stiefel manifold. Furthermore, we construct the associated geometric structure, including a so-called quasi-geodesic and propose a retraction based on this curve. We then numerically verify the performance of the Riemannian gradient descent method associated with the new geometry and compare it with the previous work.