标题： 约束反问题中多个泛函的置信区域的联合频数校准 显示英文标题

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

摘要： 许多科学分析需要同时比较未知信号的多个泛函，这需要在结构约束（例如非负性、形状或基于物理的约束）下具有保证的同时频率置信区域。本文通过统计检验反演的视角，统一并扩展了许多先前基于优化的方法，在线性逆问题中构建约束置信区域。我们首先回顾了单泛函设置下基于优化的置信区间的的历史发展，从“严格界限”到Burrus猜想及其通过上述检验反演框架最近的否定。然后我们将该框架扩展到多泛函设置。该框架可用于：(i) 改进之前方法的校准常数，产生更小的置信区域，同时保持频率覆盖，(ii) 获得易于处理的多维置信区域，不需要是超矩形以更好地捕捉泛函依赖结构，以及(iii) 超出高斯误差分布，推广到一般的对数凹误差分布。我们提供了理论证明我们的方法具有名义上的同时覆盖，并通过数值实验展示了相对于先前方法的定量体积改进。 显示英文摘要

摘要： 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.