Joint Signal Recovery and Uncertainty Quantification via the Residual Prior Transform

TL;DR

This paper introduces a residual prior transform within a hierarchical Bayesian framework that enhances signal recovery and uncertainty quantification, especially for complex signals with variable behaviors, using multimodal data.

Contribution

It reformulates the residual transform operator as a new prior, enabling principled uncertainty quantification and joint multimodal signal recovery.

Findings

High-fidelity signal and image recovery from multimodal data

Robust credible intervals for recovered signals

Effective fusion of disparate data sources

Abstract

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.