CIF: Small: Signal Recovery Beyond Minimization: A Monotone Inclusion Framework

从数据中提取信息的问题是信号处理和机器学习中许多任务的核心。这个问题的重要性源于它在许多科学和工程领域的普遍性，包括医学成像，物理学，天文学，预测，无损检测，地震学，电信，社交媒体分析，语音分析，医疗保健和国土安全。该项目研究控制信号恢复和机器学习问题的数学公式的基本原则，并开发新的数据处理策略和方法，以显着提高现有技术的效率并扩大其范围。用于制定信息提取任务的最流行的方法是将损失函数与每一条先验知识和每一次观察相关联，并最小化这些函数的集合。近年来，出现了越来越多的问题配方，这不能自然地减少到易于处理的最小化问题，这是最好的捕捉更一般的概念的平衡。该项目的主要目标是奠定基于单调算子理论的框架的理论和计算基础，以建模和聚合数据处理问题中的先验知识和观察。建议的框架包括标准的最小化设置以及各种形式的均衡。它利用单调算子的广泛建模能力，其丰富的理论，以及单调算子分裂算法的强大机制，提供强大而有效的数值求解方法。通过具体的信号恢复和机器学习问题的应用说明了理论研究结果和新方法的影响。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

The problem of extracting information from data is at the core of many tasks in signal processing and machine learning. The importance of this problem stems from its pervasiveness in numerous areas of science and engineering, including medical imaging, geophysics, astronomy, forecasting, nondestructive testing, seismology, telecommunications, social media analysis, speech analysis, healthcare, and homeland security. This project investigates foundational principles governing the mathematical formulation of signal recovery and machine learning problems and develops new strategies and methodologies for data processing that significantly improve the efficiency of existing techniques and broadens their scope. The most prevalent methodology that has been used to formulate information-extraction tasks has been to associate a loss function with each piece of prior knowledge and each observation, and to minimize an aggregate of these functions. In recent years, an increasing number of problem formulations have emerged, which cannot be naturally reduced to tractable minimization problems and which are best captured by more general notions of equilibria. The broad goal of this project is to lay out the theoretical and computational foundations of a framework based on monotone-operator theory to model and aggregate prior knowledge and observations in data processing problems. The proposed framework encompasses the standard minimization setting as well as various forms of equilibria. It exploits the broad modeling capabilities of monotone operators, their rich theory, and the powerful machinery of monotone operator splitting algorithms to provide robust and efficient numerical solution methods. The impact of the theoretical findings and of the new methodologies resulting from this research is illustrated through applications to concrete signal recovery and machine-learning problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.