On the continuity of the tangent cone to the determinantal variety

TL;DR

This paper investigates the continuity properties of tangent and normal cones to the determinantal variety, which are crucial for optimization algorithms involving low-rank matrix constraints, by analyzing limits and stratifications.

Contribution

It provides new insights into the continuity of tangent and normal cone mappings for the determinantal variety, including results on Whitney stratification and $a$-regularity.

Findings

Established the continuity of tangent cone correspondence at points in the determinantal variety.

Derived the continuity of the normal cone correspondence from tangent cone results.

Connected the geometric properties to the algebraic structure of the determinantal variety.

Abstract

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set $\mathbb{R}_{\leq r}^{m \times n}$ of all $m \times n$ real matrices of rank at most $r$. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each $X \in \mathbb{R}_{\leq r}^{m \times n}$ to the tangent cone to $\mathbb{R}_{\leq r}^{m \times n}$ at $X$. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the $a$-regularity of the Whitney stratification of $\mathbb{R}_{\leq r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety.