[PAST EVENT] [CANCELED] Steven Goldenberg, Computer Science - Ph.D. Dissertation Proposal
The Singular Value Decomposition (SVD) is one of the most fundamental matrix factorizations in linear algebra. As a generalization of the eigenvalue decomposition, the SVD is essential for a wide variety of fields including statistics, signal and image processing, chemistry, quantum physics and even weather prediction. The methods for numerically computing the SVD mostly fall under three main categories: direct, iterative, and streaming. Direct methods focus on solving the SVD in its entirety, making them suitable for smaller dense matrices where the computation cost is tractable. On the other end of the spectrum, streaming methods were created to provide an "on-line" algorithm that computes an approximate SVD as data is created or read-in over time. Consequently, they can also work on extremely large datasets that can not fit within memory. To do this, they attempt to obtain only a few singular values and rely on probabilistic guarantees which limit their overall accuracy. Iterative SVD solvers fill in the large gap between these two extremes by providing accurate solutions for a subset of singular values on large (often sparse) matrices.
In this dissertation, we focus on the creation of flexible and robust iterative SVD solvers that provide fast convergence to high precision. We first introduce a novel iterative solver based on the Golub-Kahan and Davidson methods named GKD. GKD provides fast, high-precision SVD solutions for large sparse matrices as demonstrated through comparisons with the PRIMME software package. Then, we investigate the use of flexible stopping criteria for GKD and other SVD solvers that are tailored to specific applications. Finally, we present our future research plans which include optimizations to matrix completion algorithms utilizing many SVD calls and randomization techniques for increased robustness in the presence of multiplicities.
Steven Goldenberg has been working on his Ph.D. degree in the Department of Computer Science at William & Mary since Fall 2015. He is working with Dr. Andreas Stathopoulos in the field of numerical linear algebra. Steven Goldenberg received his M.S. degree from William & Mary in 2017, and his B.A. degree in Mathematics from Johns Hopkins University in 2010.