A Form-Invariant Analysis of Newton’s Method for Optimization on Riemannian Manifolds
A Form-Invariant Analysis of Newton’s Method for Optimization on Riemannian Manifolds
WuJun Che1
作者信息
- 1. Institute of Automation, CAS
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摘要
Newton’s method, a cornerstone algorithm for root-finding in Euclidean spaces, has been extended to Riemannian manifolds to tackle an increasingly broad class of optimization problems. While its computational characteristics and convergence properties have been extensively studied in the literature, the underlying algorithmic behavior of Riemannian Newton’s method remains inadequately characterized by existing theoretical frameworks—despite its proven practical effectiveness. This paper presents a novel theoretical perspective grounded in the principle of form invariance. We establish that the second-order Taylor expansion of a real-valued function possesses form invariance in local coordinates on a Riemannian manifold, which naturally gives rise to a fully quadratic model for Newton’s method. The resultant Newton step formula is identical to its Euclidean counterpart, preserving computational simplicity while accommodating the intrinsic curvature of the manifold. This work not only provides a rigorous explanation for the algorithm’s behavior but also paves the way for more efficient computational implementations.
Abstract
Riemannian Newtons method is a cornerstone of second-order numerical optimization on manifolds. A common empirical observation is that this method, especially when equipped with approximate Riemannian Hessians, exhibits nearly identical computational forms and numerical behaviors to its Euclidean counterpart, even on curved manifolds with nonvanishing curvature. Existing theoretical frameworks fail to furnish rigorous geometric reasoning underlying this striking local equivalence. This paper bridges this theoretical gap by establishing the form invariance of the second-order Taylor expansion for real-valued functions defined on Riemannian manifolds. A pivotal geometric discovery of this work is the exact mutual cancellation of Christoffel-symbol correction terms within the derived Taylor expansion, a phenomenon arising from the intrinsic geometric compatibility between tangent-space linearization and manifold-valued Taylor approximation that has never been explicitly identified or systematically elaborated in prior literature. This intrinsic cancellation eliminates coordinate-dependent curvature corrections and underpins the aforementioned form invariance, which precisely characterizes the formal consistency between Euclidean and Riemannian Newton steps and delivers a rigorous theoretical justification for their local equivalence. The Newton update rule deduced from this invariant Taylor expansion is fully metric-independent, bypasses the costly explicit evaluation of Riemannian Hessians, retains the concise algebraic structure of Euclidean Newton iterations, and inherently adapts to arbitrary manifold curvature. The theoretical results established herein consolidate the foundational theory of Riemannian Newtons method and offer a unified geometric viewpoint for second-order optimization over general curved manifolds. Notably, all optimization algorithms built upon second-order Taylor expansion models can be locally and seamlessly generalized from Euclidean spaces to arbitrary smooth Riemannian manifolds via this form-invariant geometric framework.关键词
Newton’s method/ Riemannian manifold/ Form Invariance/ Optimization/ Second-Order Taylor ExpansionKey words
Newton’s method/ Riemannian manifold/ Form Invariance/ Optimization/ Second-Order Taylor Expansion引用本文复制引用
WuJun Che.A Form-Invariant Analysis of Newton’s Method for Optimization on Riemannian Manifolds[EB/OL].(2026-07-27)[2026-07-30].https://chinaxiv.org/abs/202602.00216.学科分类
数学
