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

近年来，大量数据驱动的应用程序显著提高了整个社会的生活质量。随之而来的大数据洪流引发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获取方面。然而，在许多应用中，物理现象的数学建模和用于简化计算的模型近似之间存在着巨大的差距。这一研究项目试图通过一种有纪律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，将导致改进的数值算法。该项目预计将对信号处理、成像科学和统计学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，并将积极鼓励代表不足的群体参与。本研究旨在解决由无限维模型描述的重要数据科学应用中的基本问题，如超分辨率成像和非参数密度估计。在第一阶段，将发展采样理论和一类分段多项式的证明稳健和高效的算法，这样的框架足够灵活，以涵盖各种实际应用。从有限观测中学习这个模型将被描述为一个正则化的优化问题；它的非渐近理论将通过利用几何泛函分析、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，将建立一个优化理论来求解一组选定的无限维反问题，而不会引起传统上由于离散化而导致的失真。开发的方法的有效性将通过一组成像数据测量得到证明。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.