Distinguished Seminar in Computational Science and Engineering
September 26, 2024, 12-1PM
Structured Matrix Approximations via Tensor Decompositions
Misha Kilmer
William Walker Professor of Mathematics
Professor, Computer Science
Department of Mathematics
Tufts University
Abstract:
We provide a new computational framework for approximating a class of structured matrices; here, the term structure is very general. The key is our matrix-to-tensor invertible mapping that allows us to pose the matrix-approximation problem as a tensor-approximation problem. Mapping the tensor approximant back to matrix space, we obtain a structured matrix approximation that can be expressed as a sum of structured Kronecker products or written in block low-rank form, thereby leading to compressed representations that can be exploited in applications. We present theory that shows that the error in the matrix approximation determined by our approach is equal to the error of the tensor approximation. We illustrate the ability of our method to uncover block-low-rank format in applications such as system identification and space-time covariance estimation. To further demonstrate the versatility of our method, we use it to uncover sum of structured Kronecker products structure on matrices from the SuiteSparse collection and from the RestoreTools image reconstruction toolbox. This is joint work with Arvind Saibaba (NCSU).
Bio:
Misha Kilmer is the William Walker Professor of Mathematics at Tufts University and is concurrently serving as Deputy Director of the Institute for Computational and Experimental Research in Mathematics Brown University. In 2019, Prof. Kilmer was named a Fellow of the Society for Industrial and Applied Mathematics (SIAM) “for her fundamental contributions to numerical linear algebra and scientific computing, including ill-posed problems, tensor decompositions, and iterative methods.” She served as a Section Editor for SIAM Review, SIAM’s flagship journal, for five years. Prof. Kilmer had the honor of being a Kirk Distinguished Visiting Fellow at the Isaac Newton Institute for Mathematical Sciences at Cambridge University in Spring 2023. She is an inaugural member of the Tufts Chapter of the National Academy of Inventors, the 2023 recipient of the Tufts Faculty Distinguished Scholar Award, and earned the unique distinction of being promoted directly from Assistant to Full Professor at Tufts in 2005.
Structured Matrix Approximations via Tensor Decompositions
Misha Kilmer
Tufts University