CAREER: Nonlinear Models and Regularization for Infinite-Dimensional Inverse Problems

职业：无限维反问题的非线性模型和正则化

基本信息

批准号：
1943201
负责人：
Kiryung Lee
金额：
$ 53.09万
依托单位：
Ohio State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1943201&HistoricalAwards=false
关键词：
CAREER Nonlinear Models Regularization Infinite

项目摘要

In recent years, numerous data-driven applications have produced significant improvements in the quality of life across society. The resulting deluge of big data has given rise to computational and algorithmic challenges that are not addressed by traditional statistical paradigms. Data science is now providing new perspectives on how to tackle these challenges, particularly with respect to model-based inference and data acquisition. Yet, in many applications, there exists a nontrivial gap between the mathematical modeling of a physical phenomenon and the model approximation used to facilitate computations. This research project seeks to narrow this gap through a disciplined approach that combines new signal models and new optimization problem formulations that would lead to improved numerical algorithms. The project is expected to have an impact on many applications in signal processing, imaging science, and statistics. The principal investigator will mentor students at all levels through various outreach activities, and will proactively encourage participation from underrepresented groups.This research addresses fundamental questions in important data science applications which are described by infinite-dimensional models, such as in super-resolution imaging and non-parametric density estimation. In the first phase, a sampling theory will be developed together with provably robust and efficient algorithms for a class of piecewise polynomials, such a framework being sufficiently flexible to cover a variety of practical applications. Learning this model from limited observations will be formulated as a regularized optimization problem; its non-asymptotic theory will be established by leveraging insights from geometric functional analysis, high-dimensional probability, and convex optimization. In the second phase, by leveraging these piecewise polynomial models, an optimization theory will be established to solve a set of selected infinite-dimensional inverse problems without incurring the distortion traditionaly due to discretization. The effectiveness of the developed methods will be demonstrated over a set of imaging data measurements.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.

近年来，大量数据驱动的应用程序显着改善了整个社会的生活质量。由此产生的大数据洪流带来了传统统计范式无法解决的计算和算法挑战。数据科学现在为如何应对这些挑战提供了新的视角，特别是在基于模型的推理和数据采集方面。然而，在许多应用中，物理现象的数学建模与用于促进计算的模型近似之间存在着巨大的差距。该研究项目旨在通过一种严格的方法来缩小这一差距，该方法结合了新的信号模型和新的优化问题公式，从而改进了数值算法。该项目预计将对信号处理、成像科学和统计学领域的许多应用产生影响。首席研究员将通过各种外展活动指导各级学生，并积极鼓励代表性不足的群体参与。这项研究解决了由无限维模型描述的重要数据科学应用中的基本问题，例如超分辨率成像和非参数密度估计。在第一阶段，将开发采样理论以及可证明稳健且高效的一类分段多项式算法，这样的框架足够灵活，可以覆盖各种实际应用。从有限的观察中学习这个模型将被表述为正则化的优化问题；其非渐近理论将利用几何泛函分析、高维概率和凸优化的见解来建立。在第二阶段，通过利用这些分段多项式模型，将建立优化理论来解决一组选定的无限维反问题，而不会产生传统上由于离散化而造成的失真。所开发方法的有效性将通过一组成像数据测量得到证明。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Low-Rank Matrix Estimation from Rank-One Projections by Unlifted Convex Optimization

通过未提升凸优化从一阶投影进行低阶矩阵估计

DOI：
10.1137/20m1330099
发表时间：
2021
期刊：
SIAM Journal on Matrix Analysis and Applications
影响因子：
1.5
作者：
Bahmani, Sohail;Lee, Kiryung
通讯作者：
Lee, Kiryung

Sub-NYQUIST Multichannel Blind Deconvolution

DOI：
10.1109/icassp39728.2021.9413856
发表时间：
2021-06
期刊：
ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
影响因子：
0
作者：
S. Mulleti;Kiryung Lee;Yonina C. Eldar
通讯作者：
S. Mulleti;Kiryung Lee;Yonina C. Eldar

Sketching Low-Rank Matrices With a Shared Column Space by Convex Programming