Variational analysis of determinantal varieties

2025-11-27

AI Summary

Characterizing first- and second-order tangent cones to low-rank sets—encompassing matrices, tensors, symmetric matrices, and positive semidefinite matrices—is fundamental yet challenging due to their nonconvex, nonsmooth geometry. Method: Leveraging Bouligand tangent cones and Mordukhovich normal cones, the authors derive explicit analytical representations and establish a general intersection rule for tangent cones of low-rank sets intersected with arbitrary constraint sets. They further analyze the structure of normal cone graphs for matrix families and develop second-order optimality conditions for low-rank bilevel programming. Contribution/Results: This work provides the first complete analytic characterization of second-order gradient tangent cones for low-rank sets; establishes equivalence between second-order stationary points of nonsmooth low-rank optimization problems and those of their smooth parameterizations; and proves that verifying second-order optimality is NP-hard. Collectively, these results constitute a unified variational geometric framework, delivering foundational tools and theoretical guarantees for low-rank structured optimization.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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, Fréchet and Mordukhovich normal cones to the graph. These results are further applied to develop optimality conditions for low-rank bilevel programs.

Problem

• Derive explicit formulas for first- and second-order tangent sets to low-rank sets
• Establish conditions for equivalent stationary points in nonsmooth and smooth parameterizations
• Characterize optimality conditions and prove NP-hardness of verifying second-order optimality

Innovation

• Unified framework for first- and second-order tangent sets
• Tangent intersection rule for low-rank set intersections
• Characterizing optimality conditions via tangent sets and normal cones