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.

在图像分析，地球科学，计算基因组学等领域中产生的反问题解决方案的方法是基于人类对问题基础结构的详细理解。该项目旨在开发新的数据驱动方法来学习反问题的学习解决方案方法并开发相关的统计基础。具体而言，该项目将提供一种新的方法，用于学习正规化器的数据驱动设计，可以在指定的计算预算中计算或优化，并提供统计保证。这项研究将吸引研究生和本科生，并通过新课程的开发将更广泛的受众传播到更广泛的受众群体中。验证技术被广泛用于模型选择和统计逆问题的解决方案，因为它们在解决难度不足的情况下，由于无法访问量的难度，仅访问少量的观察值，或者是少量的观察值，或者是较高的信号或模型。在最常见的表现中，这些方法采用了基于优化公式的目标添加到目标中的惩罚函数的形式。惩罚函数的设计基于有关特定模型选择或手头反问题的先前领域特定专业知识，以促进解决方案中的理想结构。该项目将开发一个用于构建推论问题算法的框架，以便解决以下问题 - 如果由于缺乏详细的领域知识，我们在解决方案中不知道我们在解决方案中寻求的结构怎么办？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规化程序吗？样本复杂性和这种框架所需的计算资源量的基本限制是什么？从统计上讲，我们如何为估计的积分估计提供信心界限？该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，通过评估来诚实地获得支持。