CIF: Small: The Interplay Between Convex Feasibility Problems and Minimization Problems in Signal Recovery

信号恢复包含了大量的反问题，在这些反问题中，信号要从与之相关的物理或数学测量组成的数据观测中恢复或重建。这一领域的重要性源于其在科学和工程的众多领域的普遍存在，包括医学成像、地球物理学、天文学、电子显微镜、无损检测、地震学、电信、社交媒体分析和国土安全。该项目研究了信号恢复问题作为凸优化问题的基本原则，并开发了数据处理的新策略和方法，显著提高了现有技术的效率，拓宽了它们的范围。本文主要研究信号恢复中相对独立共存的两个突出框架，即凸可行性问题和凸最小化问题之间的相互作用。这两种方法采用不同的原理来利用先验知识和数据，它们的数学形式化导致不同的不动点范式，并且用于解决它们的算法不依赖于相同的技术。研究者表明，尽管存在这些深刻的分歧，但两种形式之间可以建立富有成效的联系，这是互利的，并提出了新的模型和算法。本研究的一个重要成果是一个松弛模型，它弥合了可行性和最小化公式之间的差距。另一个亮点是一种新的近端几何框架，用于解决结构化最小化问题，该框架使用了一种适应于凸可行性算法的深切割平面技术。本研究的理论发现和新算法的影响通过具体的信号恢复问题的应用来说明。

Signal recovery encompasses the large body of inverse problems in which a signal is to be restored or reconstructed from the observation of data consisting of measurements physically or mathematically related to it. The importance of this field stems from its pervasiveness in numerous areas of science and engineering, including medical imaging, geophysics, astronomy, electron microscopy, nondestructive testing, seismology, telecommunications, social media analysis, and homeland security. This project investigates foundational principles guiding the formulation of signal recovery problems as convex optimization problems and develops new strategies and methodologies for data processing that significantly improve the efficiency of existing techniques and broadens their scope.This research focuses on the interplay between two prominent frameworks that coexist in relative independence in signal recovery, namely convex feasibility problems and convex minimization problems. These two approaches employ different principles to exploit the prior knowledge and the data, their mathematical formalizations lead to distinct fixed point paradigms, and the algorithms used to solve them do not rely on the same techniques. The investigator shows that, despite these profound divergences, fruitful connections can be established between the two formalisms, that are mutually beneficial and suggest new models and algorithms. An important outcome of this research is a relaxation model that bridges the gap between feasibility and minimization formulations. Another highlight is a novel proximal geometric framework for solving structured minimization problems using a deep cutting plane technology adapted from convex feasibility algorithms. The impact of the theoretical findings and of the new algorithms resulting from this research is illustrated through applications to concrete signal recovery problems.