Denoising, Decomposition, and Deconvolution of Moment Sequences by Convex Optimization

从不完整的、有噪声的测量中恢复信号是信号与系统理论中的基本问题之一。在各种应用中，从雷达到光谱学，信号是几个基本正弦波的组合。但是，为了从系统中提取信息，必须首先从嘈杂的、不完整的采集样本中识别这些正弦波的频率。不幸的是，大多数流行的信号分析技术在存在噪声的情况下提供很少的保证，并且需要关于待估计的信号的结构的大量先验知识。本研究通过应用当代应用数学的关键见解来解决这些问题，将信号分析作为统一方法的一部分，将系统分解为简单的构建块。受研究者最近从高度不完整的信息中恢复信号的工作的启发，本项目以基于凸优化的现代视角重新审视信号处理中的这些基本代数问题。 本研究将原子范数最小化理论应用于信号、系统和控制中矩混合去噪的实际问题。 本研究发展了原子范数去噪的抽象理论和计算均方估计率的通用程序。 这项工作的重点是频谱估计问题，并将在以前的子空间为基础的方法的缺点进行评估。 最后，本研究计划将原子范数框架应用于控制理论中的汉克尔算子问题，研究系统识别和模型降阶中开放问题的新方法。

Recovering signals from incomplete, noisy measurements is one of the foundational problems in signals and systems theory. In a variety of applications, from RADAR to spectroscopy, signals are combinations of a few basic sine waves. But, to extract information from the system, one must first identify the frequencies of these sine waves from a noisy, incomplete collection of acquired samples. Unfortunately, most popular techniques for signal analysis provide few guarantees in the presence of noise and require a great deal of prior knowledge about the structure of the signal to be estimated. This research addresses these problems by applying key insights from contemporary applied mathematics, analyzing signals as part of a unified approach to decompose systems into simple building blocks.Motivated by the investigator's recent work on recovering signals from highly incomplete information, this project revisits these fundamental, algebraic problems in signal processing with a modern perspective based on convex optimization. This research applies the theory of atomic norm minimization to the practical problems of denoising mixtures of moments in signals, systems and controls. This study develops an abstract theory of atomic norm denoising and a general program for computing mean-square-estimation rates. This work focuses on spectrum estimation problems and will be evaluated in terms of the shortcomings of previous subspace-based approaches. Finally, this research program applies the atomic norm framework to Hankel operator problems in control theory, investigating new approaches to open problems in system identification and model-order reduction.