Efficient sampling of high-dimensional Gaussian fields: the non-stationary / non-sparse case

TL;DR

This paper introduces a novel sampling method for high-dimensional Gaussian fields that works beyond sparse or circulant structures, using a perturbation-optimization approach suitable for complex inverse problems.

Contribution

It presents a general sampling technique based on perturbation-optimization, applicable to non-stationary and non-sparse Gaussian fields in inverse problems.

Findings

Effective in hyperparameter estimation for super-resolution.

Applicable to non-convolutive linear observation models.

Outperforms existing samplers in complex inverse problems.

Abstract

This paper is devoted to the problem of sampling Gaussian fields in high dimension. Solutions exist for two specific structures of inverse covariance : sparse and circulant. The proposed approach is valid in a more general case and especially as it emerges in inverse problems. It relies on a perturbation-optimization principle: adequate stochastic perturbation of a criterion and optimization of the perturbed criterion. It is shown that the criterion minimizer is a sample of the target density. The motivation in inverse problems is related to general (non-convolutive) linear observation models and their resolution in a Bayesian framework implemented through sampling algorithms when existing samplers are not feasible. It finds a direct application in myopic and/or unsupervised inversion as well as in some non-Gaussian inversion. An illustration focused on hyperparameter estimation for super-resolution problems assesses the effectiveness of the proposed approach.