Second-order geometry and Riemannian Newton's method for optimization on the indefinite Stiefel manifold

TL;DR

This paper develops a detailed second-order geometric framework and an efficient Riemannian Newton's method for optimization on the indefinite Stiefel manifold, demonstrating fast convergence and practical efficiency.

Contribution

It provides a comprehensive implementation of Newton's method on the indefinite Stiefel manifold, including second-order geometry analysis and computational strategies.

Findings

Fast local convergence of the proposed Newton's method

Efficient computation of the Hessian and Levi-Civita connection

Numerical experiments validate practical efficiency

Abstract

This paper first presents a detailed implementation of Newton's method on the indefinite Stiefel manifold. To this end, an intensive analysis of the second-order geometry of the manifold is performed. Specifically, given the two types of Riemannian metrics on the manifold proposed in existing studies, the Levi-Civita connection is derived using Koszul's formula. To reduce the computational cost, the Hessian of a function on the manifold is computed analytically in detail. Since the resulting Newton's equation is generally difficult to solve in closed form, it is solved in the tangent space using the linear conjugate gradient method. Numerical experiments demonstrate the expected fast local convergence of Newton's method and validate the practical efficiency of the proposed implementation.