Scaled Relative Graphs in Normed Spaces
This work provides a theoretical extension of SRGs to normed spaces, enabling geometric analysis of operators in more general settings, which is incremental as it generalizes existing results.
The paper extends the Scaled Relative Graph (SRG) framework from Hilbert spaces to normed spaces, introducing directional SRGs that provide geometric containment tests for operator properties like contraction and monotonicity. The theory is illustrated with numerical examples, including a graphical contraction certificate for Bellman operators.
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.