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

North Carolina State University, Raleigh NC

Investigators

Abstract

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.