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Laplace-type results characterize the limit of sequence of measures$(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure$(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto\exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$.If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers ofthe potential $U$. Classical results require the invertibility of the Hessianof $U$ in order to establish such asymptotics. In this work, we study theparticular case of norm-like potentials $U$ and establish quantitative boundsbetween $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order$1$ under an invertibility condition of a generalized Jacobian. One key elementof our proof is the use of geometric measure theory tools such as the coareaformula. We apply our results to the study of maximum entropy models(microcanonical/macrocanonical distributions) and to the convergence of theiterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at lowtemperatures for non-convex minimization.

On quantitative Laplace-type convergence results for some exponential probability measures, with two applications