Enhanced sampling schemes for MCMC based blind Bernoulli-Gaussian deconvolution

TL;DR

This paper introduces two novel sampling schemes for sparse deconvolution in a blind Bernoulli-Gaussian model, demonstrating improved convergence efficiency over traditional Gibbs sampling through theoretical validation and simulation results.

Contribution

The paper proposes and analyzes two new MCMC sampling schemes, the $K$-tuple Gibbs and partially marginalized samplers, enhancing convergence efficiency in blind Bernoulli-Gaussian deconvolution.

Findings

Both schemes outperform plain Gibbs in convergence time.

The partially marginalized sampler converges in fewer iterations.

The $K$-tuple Gibbs has lower per-iteration computational cost.

Abstract

This paper proposes and compares two new sampling schemes for sparse deconvolution using a Bernoulli-Gaussian model. To tackle such a deconvolution problem in a blind and unsupervised context, the Markov Chain Monte Carlo (MCMC) framework is usually adopted, and the chosen sampling scheme is most often the Gibbs sampler. However, such a sampling scheme fails to explore the state space efficiently. Our first alternative, the $K$-tuple Gibbs sampler, is simply a grouped Gibbs sampler. The second one, called partially marginalized sampler, is obtained by integrating the Gaussian amplitudes out of the target distribution. While the mathematical validity of the first scheme is obvious as a particular instance of the Gibbs sampler, a more detailed analysis is provided to prove the validity of the second scheme. For both methods, optimized implementations are proposed in terms of computation and storage cost. Finally, simulation results validate both schemes as more efficient in terms of convergence time compared with the plain Gibbs sampler. Benchmark sequence simulations show that the partially marginalized sampler takes fewer iterations to converge than the $K$-tuple Gibbs sampler. However, its computation load per iteration grows almost quadratically with respect to the data length, while it only grows linearly for the $K$-tuple Gibbs sampler.