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反问题的解决方法是基于人类分析人员对问题底层结构的详细理解而设计的。该项目旨在开发新的数据驱动方法来学习反问题的解决方法，并发展相关的统计基础。具体来说，该项目将为学习正则器的数据驱动设计提供一种新方法，它可以在指定的计算预算内进行计算或优化，并具有统计保证。这项研究将涉及研究生和本科生，并将通过开发新课程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逆问题的解决，因为它们可以有效地解决由于不适定性、只访问少量观测值或要推断的信号或模型的高维数而导致的困难。这些方法最常见的表现形式是在基于优化的公式中将惩罚函数添加到目标中。惩罚函数的设计基于对特定模型选择或手头逆问题的先验领域特定专业知识，以促进解决方案中的理想结构。该项目将为推理问题的算法构建开发一个框架，以解决以下问题：如果由于缺乏详细的领域知识，我们无法事先知道我们在解决方案中寻求的结构，该怎么办？我们能否直接从数据而不是人类提供的专业知识中确定合适的正则化器？在这样一个框架中，样本复杂性和所需计算资源的数量方面的基本限制是什么？在统计上，我们如何为正则化集合中的点估计提供置信限？该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.