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 Newton$\unicode{x2013}$Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newton$\unicode{x2013}$Schulz iteration and Stiefel manifolds, in which Newton$\unicode{x2013}$Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newton$\unicode{x2013}$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.

翻译：无收缩方法为Stiefel流形上的黎曼方法提供了低成本的替代方案，但这些方法通常仅为一阶，因而在高精度需求下效率受限。为此，我们提出一种无需调用收缩操作、直接着陆于Stiefel流形的二阶方法，并证明其具有局部二次收敛性（其非精确变体则具有超线性收敛性）。该更新由两部分组成：(i) 切于约束定义函数水平集的分量，旨在降低目标函数；以及(ii) 正交于该水平集的分量，用于减少不可行性。具体而言，我们通过牛顿-舒尔茨（一种用于正交化的不动点迭代）构造法向分量。此外，我们建立了牛顿-舒尔茨迭代与Stiefel流形之间的几何联系，其中牛顿-舒尔茨沿法向空间移动。对于切向分量，我们推导出融合牛顿-舒尔茨的修正牛顿方程。针对正交Procrustes问题、主成分分析及真实数据独立成分分析的数值实验表明，所提方法优于现有方法。