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.

翻译：许多科学分析需要同时比较未知信号的多个泛函，这就要求在结构约束（如非负性、形态或基于物理的约束）下构建具有保证同时频率性质的多维置信域。本文通过统计检验反演的视角，统一并扩展了线性逆问题中基于优化的约束置信域构建方法。我们首先回顾了单泛函情形下基于优化的置信区间历史发展——从“严格界限”到Burrus猜想及其近期通过前述检验反演框架的反驳。随后将该框架扩展至多泛函情形。该框架可用于：（i）改进现有方法的校准常数，在保持频率覆盖性质的同时获得更小的置信域；（ii）构建无需局限于超矩形结构的可处理多维置信域，以更好地捕捉泛函依赖结构；（iii）将高斯误差分布推广至一般对数凹误差分布。我们建立了所提方法的名义同时覆盖理论，并通过数值实验展示了相较于现有方法的定量体积改进。