arXiv:2601.08039v1 Announce Type: new 
Abstract: In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric $g$ is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To address this challenge, we construct structure-preserving metrics that are geodesically complete while ensuring that every stationary point under the new metric remains stationary under the original one. Building on this foundation, we revisit the classical symmetric two-point zeroth-order estimator and analyze its mean-squared error from a purely intrinsic perspective, depending only on the manifold's geometry rather than any ambient embedding. Leveraging this intrinsic analysis, we establish convergence guarantees for stochastic gradient descent with this intrinsic estimator. Under additional suitable conditions, an $\epsilon$-stationary point under the constructed metric $g'$ also corresponds to an $\epsilon$-stationary point under the original metric $g$, thereby matching the best-known complexity in the geodesically complete setting. Empirical studies on synthetic problems confirm our theoretical findings, and experiments on a practical mesh optimization task demonstrate that our framework maintains stable convergence even in the absence of geodesic completeness.

دراسة حديثة تقدم تقدمًا في تحسين الزرث-أوردر ريمان، مع التركيز على تقريب النقاط الثابتة في الفضاءات غير المكتملة جغرافيًا. يقترح المؤلفون مقاييس تحافظ على الهيكل تضمن بقاء النقاط الثابتة تحت المقياس الجديد ثابتة تحت المقياس الأصلي، مما يعزز تحليل متوسط مربع الخطأ لمقدّر النقاط الثابتة المتماثل.

Un estudio reciente presenta avances en la optimización de zeroth-order riemanniana, centrándose en la aproximación de puntos estacionarios en variedades geodésicamente incompletas. Los autores proponen métricas que preservan la estructura, asegurando que los puntos estacionarios bajo la nueva métrica permanezcan estacionarios bajo la métrica original, mejorando así el análisis del error cuadrático medio del estimador simétrico de dos puntos.

Une étude récente présente des avancées dans l'optimisation de zeroth-order riemannienne, en se concentrant sur l'approximation des points stationnaires dans des variétés géodésiquement incomplètes. Les auteurs proposent des métriques préservant la structure qui garantissent que les points stationnaires sous la nouvelle métrique restent stationnaires sous la métrique originale, améliorant ainsi l'analyse de l'erreur quadratique moyenne de l'estimateur symétrique à deux points.

A recent study presents advancements in Riemannian zeroth-order optimization, focusing on approximating stationary points in geodesically incomplete manifolds. The authors propose structure-preserving metrics that ensure stationary points under the new metric remain stationary under the original metric, enhancing the classical symmetric two-point zeroth-order estimator's mean-squared error analysis.

Riemannian Zeroth-Order Gradient Estimation with Structure-Preserving Metrics for Geodesically Incomplete Manifolds

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