Joint Signal Recovery and Uncertainty Quantification via the Residual Prior Transform

Conventional priors used for signal recovery are often limited by the assumption that the type of a signal's variability, such as piecewise constant or linear behavior, is known and fixed. This assumption is problematic for complex signals that exhibit different behaviors across the domain. The recently developed {\em residual transform operator} effectively reduces such variability-dependent error within the LASSO regression framework. Importantly, it does not require prior information regarding structure of the underlying signal. This paper reformulates the residual transform operator into a new prior within a hierarchical Bayesian framework. In so doing, it unlocks two powerful new capabilities. First, it enables principled uncertainty quantification, providing robust credible intervals for the recovered signal, and second, it provides a natural framework for the joint recovery of signals from multimodal measurements by coherently fusing information from disparate data sources. Numerical experiments demonstrate that the residual prior yields high-fidelity signal and image recovery from multimodal data while providing robust uncertainty quantification.