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

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.