Data-informed posterior approximation for Bayesian linear inverse problems
For practitioners solving large-scale Bayesian inverse problems, this work provides a computationally efficient, matrix-free approach that leverages intrinsic low-dimensional structure.
The authors developed a data-informed framework for Bayesian linear inverse problems that reduces posterior computation to a low-dimensional data-informed subspace, enabling efficient hyperparameter estimation and posterior approximation. Numerical experiments validated the method's effectiveness.
Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub--Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method.