Simultaneous Frequentist Calibration of Confidence Regions for Multiple Functionals in Constrained Inverse Problems

Many scientific analyses require simultaneous comparison of multiple functionals of an unknown signal at once, calling for multidimensional confidence regions with guaranteed simultaneous frequentist under structural constraints (e.g., non-negativity, shape, or physics-based). This paper unifies and extends many previous optimization-based approaches to constrained confidence region construction in linear inverse problems through the lens of statistical test inversion. We begin by reviewing the historical development of optimization-based confidence intervals for the single-functional setting, from "strict bounds" to the Burrus conjecture and its recent refutation via the aforementioned test inversion framework. We then extend this framework to the multiple-functional setting. This framework can be used to: (i) improve the calibration constants of previous methods, yielding smaller confidence regions that still preserve frequentist coverage, (ii) obtain tractable multidimensional confidence regions that need not be hyper-rectangles to better capture functional dependence structure, and (iii) generalize beyond Gaussian error distributions to generic log-concave error distributions. We provide theory establishing nominal simultaneous coverage of our methods and show quantitative volume improvements relative to prior approaches using numerical experiments.