Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $\nu$ satisfies log-Sobolev inequality (LSI) and $f$ has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in R\'enyi divergence of order $q > 1$ when the biased limit satisfies either LSI or Poincar\'e inequality

Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach. Part I: Methodology and Experiments

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori, and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularisation parameters directly from the observed data by maximum marginal likelihood estimation, and can simultaneously estimate multiple regularisation parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimisation algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimisation techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, non-blind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the L1, total-variation, total-variation and L1, and total-generalised-variation pseudo-norms. A detailed theoretical analysis of the proposed method is presented in the companion paper arXiv:2008.05793.Comment: 37 pages - SIIMS 202