Stochastic Analysis Seminar – Federico Cornalba (University of Bath)

Priors with non-smooth log densities have been widely used in Bayesian inverse problems, particularly in imaging, due to their sparsity-inducing properties. To date, the majority of algorithms for handling such densities are based on incoportating the proximal operator into the Langevin dynamics or replacing the non-smooth part by a smooth approximation known as the Moreau envelope. On the other hand, there has been substantial interest in the use of the Hadamard parameterization or quadratic variational form for handling the $\ell_1$-prior in the optimization and machine learning literature. In this work, we develop a novel approach for sampling densities with $\ell_1$-priors based on a Hadamard product parameterization. This builds upon the idea that the Laplace prior has a Gaussian mixture representation and our method can be seen as a form of over-parametrization: by increasing the number of variables, we construct a density from which one can directly recover the original density. This is fundamentally different from proximal-type approaches and our approach does not rely on convexity. For our new density, we present its Langevin dynamics in continuous time and establish well-posedness and geometric ergodicity. We also present a discretization scheme for the continuous dynamics and prove convergence as the time-step diminishes.