CAREER: Statistical Inference in the Presence of Group Actions: Theory, Method, and Application

In the rapidly expanding field of data science, the ability to understand group actions in data analysis is pivotal for a broad spectrum of scientific tasks. In mathematical terms, a “group” is a collection of elements combined with an operation that links any two elements to form a third, adhering to closure, associativity, identity, and invertibility principles.

A “group action” involves applying elements of a group to another set’s elements, transforming them in structured ways, such as through rotations or reflections. These transformations are crucial in many data processing applications, including cryo-electron microscopy (cryo-EM), image registration, and multi-reference alignment. Each observation in these problems involves a common, unknown signal and an unknown group element, with the primary goal being to infer both the signal and the group elements accurately.

This project aims to significantly advance statistical understanding and develop effective methodologies for handling data influenced by group actions. The wide existence of such data ensures that the progress we make towards our objectives will have a great impact not only on the statistics and machine learning community but also on a much broader scientific community, including fields such as structural biology, computer vision, and signal processing.

This project will have educational outcomes that result in curriculum development, teaching, and outreach activities, including activities to K-12 students through the University of Pennsylvania Data Science Academy. The project will advance applications in image recognition and time series alignment, which have broad application in areas like medical imaging.

This project is structured around three main aims, each designed to tackle distinct aspects of group actions. First, the PI will improve the accuracy of orbit recovery in scenarios where the prior distributions of group elements are non-uniform, developing computationally efficient procedures that are effective under realistic conditions. Second, the PI will develop theories and methods for group synchronization problems, particularly under high noise levels and in situations with incomplete data, aiming to reduce the error of group recovery and provide entrywise inference.

Third, the PI will address theoretical and computational challenges in the multi-reference alignment problem, developing procedures specifically designed for the cyclic structural nature of data, thereby enabling more precise uncertainty quantification. Together, these aims will not only enhance the theoretical understanding of and the ability to analyze group actions but also lead to the development of accurate and computationally efficient algorithms designed to tackle real-world challenges in data analysis where group actions are integral.

This research project will have impacts more broadly, in that it will result in software development and in the education of technical experts. These experts will use this software to advance applications in image recognition and time series alignment, which have broad application in areas like medical imaging. These activities will then advance applications in image recognition and time series alignment, which have broad application in areas like medical imaging.

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.