A second-order method on the Stiefel manifold via Newtonnicode{x2013}Schulz

arXiv:2605.02838v1 Announce Type: cross Abstract: Retraction-free approaches offer attractive low-cost alternatives to Riemannian methods on the Stiefel manifold, but they are often first-order, which may limit the efficiency under high-accuracy requirements. To this end, we propose a second-order method landing on the Stiefel manifold without invoking retractions, which is proved to enjoy local quadratic (or superlinear for its inexact variant) convergence. The update consists of the sum of (i) a component tangent to the level set of the constraint-defining function that aims to reduce the objective and (ii) a component normal to the same level set that reduces the infeasibility. Specifically, we construct the normal component via Newtonnicode{x2013}Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newtonnicode{x2013}Schulz iteration and Stiefel manifolds, in which Newtonnicode{x2013}Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newtonnicode{x2013}Schulz. Numerical experiments on the orthogonal Procrustes problem, principal component analysis, and real-data independent component analysis illustrate that the proposed method performs better than the existing methods.

Source: arXiv cs.LG | 2026-05-05