Theory and Algorithms of Transformed L1 Minimization with Applications in Data Science

这项研究项目研究从部分观测中恢复信号(图像)的计算方法，这是一种称为压缩传感的技术。该项目解决了压缩传感中的一个悬而未决的问题，即冗余或限制阻碍了已有技术的成功使用。这项研究研究了一类在稳健条件下促进稀疏性的函数，这些函数可以用高效或快速的算法最小化。该项目研究的计算工具将增强在天文和雷达成像、医学成像、威胁检测和推荐系统等应用中的传感能力。该项目为研究生提供系统的培训，以获得计算数学的高级学位。该项目开发的计算方法将成为信息技术和数据科学的宝贵工具，在数字时代造福国家和普通公众。稀疏信号恢复中广泛使用的凸函数是L1，它会引入偏差。本课题研究一族无偏非凸稀疏提升函数，称为变换L1(TL1)。线性约束下的TL1极小化问题可以用具有闭合阈值函数的迭代阈值方法来求解。目标是当约束是病态的或不满足成功应用L1方法的理论保证时，在鲁棒感知条件下改进L1。

This research project studies computational methods to recover signals (images) from partial observations, a technique known as compressed sensing. The project addresses an outstanding issue in compressed sensing, in which redundancies or restrictions preclude the successful use of established techniques. The research investigates a class of functions that promote sparsity under robust conditions and can be minimized with efficient or fast algorithms. The computational tools studied in the project will enhance sensing capabilities in applications such as imaging in astronomy and radar, medical imaging, threat detection, and recommender systems. The project provides systematic training of graduate students towards advanced degrees in computational mathematics. The computational methods developed in the project will serve as a valuable tool for information technology and data sciences, benefitting the country and the general public in the digital age. The widely used convex function for sparse signal recovery is L1, which is known to introduce bias. This project studies a family of unbiased non-convex sparsity promoting functions called the transformed L1 (TL1). The TL1 minimization under a linear constraint can be solved by iterative thresholding methods with closed-form thresholding functions. The goal is to improve on L1 under robust sensing conditions when the constraint is ill-conditioned or the theoretical guarantees for successful application of the L1 method are not satisfied.