Symmetric Metropolis-within-Gibbs algorithm for lattice Gaussian sampling

As a key sampling scheme in Markov chain Monte Carlo (MCMC) methods, Gibbs sampling is widely used in various research fields due to its elegant univariate conditional sampling, especially in tacking with multidimensional sampling systems. In this paper, a Gibbs-based sampler named as symmetric Metropolis-within-Gibbs (SMWG) algorithm is proposed for lattice Gaussian sampling. By adopting a symmetric Metropolis-Hastings (MH) step into the Gibbs update, we show the Markov chain arising from it is geometrically ergodic, which converges exponentially fast to the stationary distribution. Moreover, by optimizing its symmetric proposal distribution, the convergence efficiency can be further enhanced.