Random Walks, Faber Polynomials and Accelerated Power Methods

TL;DR

This paper introduces polynomial families derived from random walks that efficiently approximate power functions and enhance iterative linear algebra methods, including dynamic momentum power iterations.

Contribution

It develops new polynomial families linked to random walks and Faber polynomials, enabling accelerated power methods for non-symmetric matrices.

Findings

Polynomials approximate z^n with degree ~√n in complex domains.

Constructed polynomials exhibit rapid growth properties.

Applications include arbitrary-order dynamic momentum power iterations.

Abstract

In this paper, we construct families of polynomials defined by recurrence relations related to mean-zero random walks. We show these families of polynomials can be used to approximate $z^n$ by a polynomial of degree $\sim \sqrt{n}$ in associated radially convex domains in the complex plane. Moreover, we show that the constructed families of polynomials have a useful rapid growth property and a connection to Faber polynomials. Applications to iterative linear algebra are presented, including the development of arbitrary-order dynamic momentum power iteration methods suitable for classes of non-symmetric matrices.