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

从数据中学习反问题的算法：统计和计算基础

• 批准号: 2113724
• 负责人: Venkat Chandrasekaran
• 金额: $ 20万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2113724&HistoricalAwards=false
• 关键词: Learning; Algorithms; Inverse; Problems; Data

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.

解决图像分析、地球科学、计算基因组学等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结构的详细理解而设计的。该项目旨在开发新的数据。具体来说，该项目将提供一种新的学习正则化器的数据驱动设计方法，该方法可以在指定的计算预算内进行计算或优化，并附带。研究将提供统计保证。吸引研究生和本科生，并将通过新课程的开发向更广泛的受众传播。正则化技术广泛应用于模型选择和统计逆问题的解决，因为它们可以有效解决由于不适定性、访问性等原因造成的困难在最常见的表现形式中，这些方法采用在基于优化的公式中添加到目标中的罚函数的形式。函数是基于先验的关于特定模型选择或当前逆问题的特定领域专业知识，以促进解决方案中所需的结构。该项目将开发一个用于构建推理问题算法的框架，以解决以下问题：什么。如果由于缺乏详细的领域知识而无法提前知道我们在解决方案中寻求的结构，我们可以直接从数据而不是人类提供的专业知识中识别出合适的正则化器吗？这样的框架需要多少计算资源？从统计上看，我们如何为正则化集合中的点估计提供置信区间？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。