Momentum accelerated algorithms for nonsymmetric matrices and complex approximation theory

Data Science Seminar

Nicholas Marshall
Oregon State University

Abstract

In this talk, we present a new approach to accelerating algorithms for nonsymmetric matrices, inspired by momentum methods in optimization. In particular, we focus on accelerating the power method for nonsymmetric matrices using higher-order momentum terms. Analyzing this algorithm motivates the development of new complex approximation theory. We define a family of polynomials related to Faber polynomials (a generalization of Chebyshev polynomials) and use this family to give a constructive proof that z^n is approximately a polynomial of degree ~sqrt(n) within certain regions of the complex plane. We illustrate the developed theory and algorithms through numerical examples and discuss generalizations of the presented framework.