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存在，包括医学成像、地球物理学、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。该项目研究了控制信号恢复和机器学习问题的数学公式的基本原理，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并拓宽了其范围。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个先验知识和每个观察相关联，并最小化这些函数的总和。近年来，出现了越来越多的问题形式，这些问题不能自然地简化为可处理的最小化问题，它们最好被更一般的平衡概念所捕获。这个项目的主要目标是建立一个基于单调算子理论的框架的理论和计算基础，以模拟和汇总数据处理问题中的先验知识和观察结果。所提出的框架包括标准最小化设置以及各种形式的平衡。它利用单调算子广泛的建模能力、丰富的理论和单调算子分裂算法的强大机制来提供鲁棒和高效的数值求解方法。本研究产生的理论发现和新方法的影响通过应用于具体的信号恢复和机器学习问题来说明。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.