Scaled Relative Graph: Nonexpansive operators via 2D Euclidean Geometry

TL;DR

This paper introduces the scaled relative graph (SRG), a geometric tool that maps nonlinear operators to 2D subsets, enabling intuitive and rigorous convergence analysis of fixed-point iterations.

Contribution

The paper presents the SRG framework, a novel geometric approach for analyzing contractive and nonexpansive operators in fixed-point algorithms.

Findings

SRG provides a visual and analytical method for convergence proofs.

The approach simplifies the analysis of nonlinear operators.

It bridges operator theory and Euclidean geometry.

Abstract

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence.