Identifiability and Error Bound: Metric and Geometric Perspectives

Identifiability means that iterates generated by optimization algorithms are eventually confined to an identifiable set. This property is computationally useful because minimizing a nonsmooth function near a critical point reduces to minimizing its smooth restriction on the corresponding identifiable manifold. Motivated by this reduction, we study the Error Bound (EB) property from both ambient and manifold viewpoints. Under mild assumptions in Euclidean space, we prove that local EB on $(\mathbb{R}^n,d)$ is equivalent to local EB on an identifiable manifold $(\mathcal{M},d)$. We establish this result from two complementary perspectives: a metric analysis based on slope and linear growth away from $\mathcal{M}$, and a geometric analysis based on subdifferentials, partial smoothness, and $\mathcal{VU}$-theory. As an application, we recover the EB equivalence for $\ell_1$-regularized optimization in the literature.