Variational analysis of determinantal varieties

TL;DR

This paper develops a comprehensive geometric framework to analyze the tangent and normal cones of low-rank matrix and tensor varieties, providing new insights into second-order optimality and complexity in low-rank optimization.

Contribution

It introduces explicit formulas for tangent sets and a tangent intersection rule for low-rank varieties, advancing the understanding of second-order geometry in low-rank optimization.

Findings

Derived explicit formulas for tangent and normal cones of low-rank sets.

Established a necessary and sufficient condition for second-order stationarity equivalence.

Proved that verifying second-order optimality is NP-hard.

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.