arXiv:2511.22613v2 Announce Type: replace-cross 
Abstract: Determinantal varieties -- the sets of bounded-rank matrices or tensors -- have attracted growing interest in low-rank optimization. The tangent cone to low-rank sets is widely studied and underpins a range of geometric methods. The second-order geometry, which encodes curvature information, is more intricate. In this work, we develop a unified framework to derive explicit formulas for both first- and second-order tangent sets to various low-rank sets, including low-rank matrices, tensors, symmetric matrices, and positive semidefinite matrices. The framework also accommodates the intersection of a low-rank set and another set satisfying mild assumptions, thereby yielding a tangent intersection rule. Through the lens of tangent sets, we establish a necessary and sufficient condition under which a nonsmooth problem and its smooth parameterization share equivalent second-order stationary points. Moreover, we exploit tangent sets to characterize optimality conditions for low-rank optimization and prove that verifying second-order optimality is NP-hard. In a separate line of analysis, we investigate variational geometry of the graph of the normal cone to matrix varieties, deriving the explicit Bouligand tangent cone, Fr\'echet and Mordukhovich normal cones to the graph. These results are further applied to develop optimality conditions for low-rank bilevel programs.

أظهرت دراسة حديثة إطارًا موحدًا للتحليل التغيري للأنواع المحددة، مع التركيز على المصفوفات والتنسورات ذات الرتبة المنخفضة. يوفر هذا الإطار صيغًا صريحة لمجموعات المماس من الدرجة الأولى والثانية، مما يعزز الفهم للانحناء في مشاكل التحسين ذات الرتبة المنخفضة.

Un estudio reciente ha desarrollado un marco unificado para el análisis variacional de variedades determinantes, centrándose en matrices y tensores de bajo rango. Este marco proporciona fórmulas explícitas para conjuntos tangentes de primer y segundo orden, mejorando la comprensión de la curvatura en problemas de optimización de bajo rango.

Une étude récente a développé un cadre unifié pour l'analyse variationnelle des variétés déterminantales, en se concentrant sur les matrices et les tenseurs de faible rang. Ce cadre fournit des formules explicites pour les ensembles tangents de premier et de second ordre, améliorant ainsi la compréhension de la courbure dans les problèmes d'optimisation à faible rang.

A recent study has developed a unified framework for variational analysis of determinantal varieties, focusing on low-rank matrices and tensors. This framework provides explicit formulas for both first- and second-order tangent sets, enhancing the understanding of curvature in low-rank optimization problems.

Variational analysis of determinantal varieties

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