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

用于解决在诸如图像分析、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方法是基于人类分析师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来学习逆问题的解决方法，并开发相关的统计基础。具体来说，该项目将提供一种新的方法来设计学习正则化器的数据驱动设计，可以在指定的计算预算内进行计算或优化，并提供统计保证。该研究将吸引研究生和本科生，并将传播到更广泛的受众通过开发新的courses.Regularization技术被广泛采用的模型选择和统计逆问题的解决方案，因为它们在解决困难的有效性，由于不适定性，访问只有少量的观察，或高维的信号或模型进行推断。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的惩罚函数的形式。惩罚函数的设计是基于关于特定模型选择或手头的逆问题的先前特定领域的专业知识，以促进解决方案中的期望结构。这个项目将开发一个框架的推理问题的算法建设，以解决以下问题-如果我们不知道在我们的解决方案中，由于缺乏详细的领域知识，我们寻求的结构提前？我们能否直接从数据而不是人类提供的专业知识中识别合适的正则化器？在这样一个框架中，样本复杂性和所需计算资源量方面的基本限制是什么？从统计学上讲，我们如何为正则化器集合中的点估计提供置信界？该奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.