Collaborative Research: Machine Learning and Inverse Problems in Discrete and Continuous Settings

该项目的目标是推动数据在科学和应用中的原则使用。通过严格的数学分析，pi打算揭示看似不相关的学习问题和方法的隐藏统一，促进理论和计算发展的转移，并统一日益增长的应用文献。提出的工作旨在部分满足社会和科学需要，建立结合数据和复杂数学模型的范式，以在考虑不确定性量化的同时获得更准确的预测。pi打算解决一些新的挑战，这些挑战是随着模型复杂性的增加和数据集规模的扩大，为机器学习和反问题的优化和贝叶斯方法的基础带来的。这个项目将强调统计一致性和算法可扩展性之间的联系：一致性问题通常在计算上是可处理的，可扩展算法设计的一个关键原则是利用任何存在的统计一致性。将进行的具体研究项目有五个总体主题：i)分析在随机数据上定义的离散对象的连续极限。ii)新的正则化技术的研究。iii)可扩展采样算法的设计与分析。iv)使用复杂模型的离散近似。v)解决方案不确定度的量化。要对这一范围广泛的主题作出实质性贡献，就需要各私人机构之间的密切合作。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。

The goal of this project is to push forward the principled use of data in science and applications. By means of rigorous mathematical analysis, the PIs intend to uncover the hidden unity of seemingly unrelated learning problems and methodologies, facilitating the transfer of theoretical and computational developments and unifying the growing applied literature. The proposed work intends to partially satisfy the societal and scientific need to build paradigms that combine data and complex mathematical models to obtain more accurate predictions while accounting for uncertainty quantification.The PIs intend to address some of the new challenges that the increasing complexity of models and the growing size of data sets have brought to the foundations of optimization and Bayesian approaches to machine learning and inverse problems. This project will emphasize the connection between statistical consistency and algorithmic scalability: consistent problems are often computationally tractable, and a key principle for the design of scalable algorithms is to exploit statistical consistency wherever present. The specific research projects that will be pursued have five overarching themes: i) The analysis of continuum limits of discrete objects defined on random data.ii) The study of new regularization techniques. iii) The design and analysis of scalable sampling algorithms. iv) The use of discrete approximations of complex models. v) The quantification of uncertainty in the solutions. Contributing in a substantial manner to this wide range of themes will require close collaboration between the PIs.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.