RUI: Commutativity in Numerical Computation

Randolph-Macon College, Ashland VA

Investigators

Abstract

This project concerns consistency in scientific inferences. Multiple observations of the same phenomenon rarely produce identical results. Statistical noise may be the first complication that comes to mind, but there are plenty of other possible discrepancies as well: different antenna arrangements; different camera angles; different modalities, e.g., images and text; and more. Inferring the true underlying phenomenon requires sophisticated mathematics. This project focuses on matrix methods. Hiding within the simple numerical entries of a matrix are intricate patterns called eigenvectors that are connected to the harmonics of musical instruments. Even though different kinds of measurements may produce different numerical values, they may share the same eigenvectors when observing the same objects. One of the goals of the project is to pull consistency out of inconsistent data. In theory, measurements on the same object from different instruments should agree in certain ways, but because of statistical noise and other sources of error and uncertainty, each instrument provides an inexact view of the world, and the two instruments will not agree perfectly with each other. The goal is to bring the measurements into alignment using mathematics and therefore likely into better agreement with the true state of the universe. The project will be completed at a Primarily Undergraduate Institution, and introducing undergraduate students to research in STEM fields is a primary objective. The mathematical focus of the project is simultaneous diagonalization of commuting Hermitian matrices. In theory, commuting Hermitian matrices share a common eigenvector matrix. However, numerically-computed eigenvectors may differ wildly in the presence of ill conditioning. A numerical method for simultaneous diagonalization seeks, either explicitly or implicitly, small perturbations of the input matrices that commute exactly. The project objectives include contributions to theory, method, and application. First, a reconsideration of "near commutativity" is planned, in which eigenvalues are fixed in place and attention shifts from the space of Hermitian matrices to the unitary group of eigenvector matrices. Second, the project will pursue new numerical methods for simultaneous diagonalization, motivated in part by relatively recent spectral divide-and-conquer methods. Third, the new approaches will be applied to applications such as independent component analysis and unsupervised machine learning with multimodal data. 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.