Learning Algorithms for Inverse Problems from Data: Statistical and Computational Foundations

在图像分析、地球科学、计算基因组学和许多其他领域中出现的逆问题的解决方法是基于人类分析员对问题背后的结构的详细理解而设计的。该项目旨在开发新的数据驱动的方法，以学习反问题的解决方法，并发展相关的统计基础。具体地说，该项目将提供一种以数据为导向的学习正则化设计的新方法，可以在特定的计算预算内进行计算或优化，并提供统计保证。这项研究将涉及研究生和本科生，并将通过新课程的开发传播给更广泛的受众。正则化技术被广泛应用于模型选择和统计反问题的解决中，因为它们在解决因不适定、仅获得少量观测数据或要推断的信号或模型的高维而造成的困难方面有效。在它们最常见的表现形式中，这些方法采取在基于优化的公式中添加到目标上的惩罚函数的形式。惩罚函数的设计是基于关于手头的特定模型选择或逆问题的先前特定领域的专业知识，以期在解中促进所需的结构。这个项目将开发一个框架来构建推理问题的算法，以解决以下问题-如果我们由于缺乏详细的领域知识而事先不知道我们在解决方案中寻求的结构怎么办？我们能否直接从数据而不是人工提供的专业知识中找到合适的正规化方法？就样本复杂性和这种框架所需的计算资源数量而言，基本的限制是什么？从统计学上讲，我们如何为位于正规化集合中的点数估计提供置信限？该奖项反映了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。

Methods for the solution of inverse problems arising in domains such as image analysis, the geosciences, computational genomics, and many others are designed based on a detailed understanding by a human analyst of the structure underlying the problem. This project aims to develop new data-driven approaches to learning solution methods for inverse problems and to develop the associated statistical foundations. Specifically, the project will provide a new approach to data-driven design of learning regularizers, which can be computed or optimized within a specified computational budget, and come with statistical guarantees. The research will engage both graduate and undergraduate students and will be disseminated to a broader audience through the development of new courses.Regularization techniques are widely employed in the solution of model selection and statistical inverse problems because of their effectiveness in addressing difficulties due to ill-posedness, access to only a small number of observations, or the high dimensionality of the signal or model to be inferred. In their most common manifestation, these methods take the form of penalty functions added to the objective in optimization-based formulations. The design of the penalty function is based on prior domain-specific expertise about the particular model selection or inverse problem at hand, with a view to promoting a desired structure in the solution. This project will develop a framework for the construction of algorithms for inferential problems so as to address the following questions – What if we do not know in advance the structure we seek in our solution due to a lack of detailed domain knowledge? Can we identify a suitable regularizer directly from data rather than human-provided expertise? What are the fundamental limitations in terms of sample complexity and the amount of computational resources required in such a framework? Statistically, how do we provide confidence bounds for point estimates that lie in a collection of regularizers?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.