This paper proposes two new Markov chain Monte Carlo algorithms to simulate samples from high-dimensional non-differentiable log-concave distributions. The methods combine stochastic simulation with non-smooth convex optimization and generalize the Metropolis adjusted Lanvegin algorithm to a wide range of non-differentiable distributions. This is achieved by considering discretizations of Langevin diffusions associated with smooth Moreau approximations of the target density. The resulting algorithms exploit the proximity mappings of the target density to efficiently explore the parameter space, in a similar manner to how the conventional MALA uses gradients. In addition, for differentiable target densities the proposed algorithms coincide with a variant of MALA that uses a split-step backward Euler approximation of the Langevin diffusion, instead of the classic forward Euler approximation. In this case the proximity mappings provide a computationally efficient means to compute the proposal without having to solve implicit equations. Finally, the proposed methodology is demonstrated on relevant high-dimensional examples such as high-dimensional LASSO models and low-rank matrix denoising.
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