Nuclear-norm curvature bounds for improved dimension dependence

Establish whether imposing a gradient-domination assumption on the nuclear norm of the Hessian can improve the dimension dependence of random-walk Metropolis acceptance bounds in cases where operator-norm curvature estimates obscure favourable averaging effects.

Background

The paper’s acceptance analysis controls local curvature through the operator norm of the Hessian. The authors explain that this choice can be non-sharp because it fails to capture averaging effects across coordinates, particularly in product-target examples where the general curvature-force argument gives a more conservative proposal scale than classical optimal-scaling theory.

The authors report that cursory calculations suggest replacing or supplementing the operator-norm condition with a comparable gradient-domination assumption formulated using the nuclear norm of the Hessian. They leave this concrete refinement for future work, with the intended goal of obtaining improved dimension dependence in settings where coordinate-wise averaging is beneficial.