# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1902.09788