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

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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.