Identifiability and Error Bounds: Metric and Geometric Perspectives

Identifiability and partial smoothness are important notions in optimization, linking differential geometry with variational analysis and providing a foundation for classical active-set methods, sensitivity analysis, and optimality conditions. These notions capture the local structure of nonsmooth optimization problems and often reduce their local analysis to that of a smooth restriction on an identifiable manifold. Motivated by this reduction, we study the error-bound property (EB) in the ambient space ${\mathbb{R}}^n$ and on an identifiable manifold $\mathcal{M}$. Using a slope-based formulation, we prove that local EB on $({\mathbb{R}}^n,d)$ is equivalent to local EB on $(\mathcal{M},d)$ under identifiability. We further establish this equivalence under $C^1$ partial smoothness and the nondegeneracy condition. A key ingredient is a novel linear-growth result, which shows that $C^2$ regularity of $\mathcal{M}$ is not required. In addition, we provide a complementary geometric analysis based on $\mathcal{V}\mathcal{U}$-theory. As an application, we recover the EB equivalence for ${\mathrm{\ell }}_1$-regularized optimization previously established in the literature.