Split-and-augmented Gibbs sampler - Application to large-scale inference problems
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Recently, a new class of Markov chain Monte Carlo (MCMC) algorithms took advantage of convex optimization to build efficient and fast sampling schemes from high-dimensional distributions. Variable splitting methods have become classical in optimization to divide difficult problems in simpler ones and have proven their efficiency in solving high-dimensional inference problems encountered in machine learning and signal processing. This paper derives two new optimization-driven sampling schemes inspired from variable splitting and data augmentation. In particular, the formulation of one of the proposed approaches is closely related to the alternating direction method of multipliers (ADMM) main steps. The proposed framework enables to derive faster and more efficient sampling schemes than the current state-of-the-art methods and can embed the latter. By sampling efficiently the parameter to infer as well as the hyperparameters of the problem, the generated samples can be used to approximate maximum a posteriori (MAP) and minimum mean square error (MMSE) estimators of the parameters to infer. Additionally, the proposed approach brings confidence intervals at a low cost contrary to optimization methods. Simulations on two often-studied signal processing problems illustrate the performance of the two proposed samplers. All results are compared to those obtained by recent state-of-the-art optimization and MCMC algorithms used to solve these problems.
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