Instance Optimal Iterative Methods for Matrix Function Approximation

Explore a 32-minute lecture on instance-optimal iterative methods for matrix function approximation, delivered by Cameron Musco from the University of Massachusetts Amherst at the Simons Institute. Delve into the conjugate gradient (CG) method's application in solving positive definite linear systems, examining its convergence rates and instance-optimality bounds. Investigate the Lanczos method's extension to approximating arbitrary matrix functions, including matrix square roots and exponentials. Analyze recent progress in proving instance-optimality for rational matrix functions and other key function classes. Consider open questions in the field and potential faster methods through alternative approaches like stochastic gradient methods. Gain insights from joint work with Noah Amsel, Tyler Chen, Anne Greenbaum, and Christopher Musco in this optimization and algorithm design-focused talk.