Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton's methods

TL;DR

This paper develops the first explicit second-order Riemannian geometry and Newton's methods on the symplectic Stiefel manifold, enabling efficient optimization for quantum physics and scientific computing problems.

Contribution

It introduces tractable Riemannian metrics, derives second-order geometric tools, and proposes Newton's methods with preconditioning and saddle point solutions on the symplectic Stiefel manifold.

Findings

New second-order geometry for symplectic Stiefel manifold

Development of Riemannian Newton schemes with preconditioning

Numerical validation demonstrating efficiency and convergence

Abstract

Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit second-order geometry and Newton's methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operator-valued formulas for computing the Riemannian Hessian of a~cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton's methods, directly via transforming it into a~saddle point problem followed by vectorization, or iteratively via applying any matrix-free iterative method either to the operator Newton equation or its saddle point formulation. Finally, we propose a hybrid Riemannian Newton optimization algorithm that enjoys both global convergence and quadratic/superlinear local convergence at the final stage. Various numerical experiments are presented to validate the proposed methods.