Scaled Relative Graphs in Normed Spaces

TL;DR

This paper generalizes the SRG framework from Hilbert to normed spaces, introducing directional SRGs that help certify operator properties like contraction and monotonicity through geometric tests.

Contribution

It extends the SRG framework to normed spaces using a regular pairing, enabling new geometric containment tests and calculus rules for operator analysis.

Findings

Directional SRGs provide geometric tests for contraction and monotonicity.

Calculus rules for SRGs under various operations are derived.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to normed spaces. Our extension replaces the inner product with a regular pairing, whose asymmetry gives rise to directional angles and, in turn, directional SRGs. Directional SRGs are shown to provide geometric containment tests certifying key operator properties, including contraction and monotonicity. Calculus rules for SRGs under scaling, inversion, addition, and composition are also derived. The theory is illustrated by numerical examples, including a graphical contraction certificate for Bellman operators.