Scaled Relative Graphs in Normed Spaces

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.

Figures (4)

• Figure C1: Directional angles in $\ell^1$ (left), $\ell^2$ (center), and $\ell^\infty$ (right) between unit-norm vectors $x$ (solid) and $y$ (dashdotted) in $\mathbb{R}^2$. The outward normals at $x$ are $\operatorname{sign}(x)$ in $\ell^1$, $x/\left\lVert x \right\rVert_2$ in $\ell^2$, and $\operatorname{sign}(x_{m_x})e_{m_x}$ in $\ell^\infty$ with the min-index pairing. In $\ell^1$, an angle encodes transitions between sign patterns on the unit cross-polytope; in $\ell^2$, it encodes continuous rotations on the unit sphere; in $\ell^\infty$, it encodes transitions between active facets of the unit hypercube.
• Figure E1: Left of $A_1$ (top) and $A_\infty$ (bottom) in $\ell^1$ (left), $\ell^2$ (center), and $\ell^\infty$ (right; max pairing). The left half-plane marks monotonicity violations: $A_1$ is $\ell^1$-monotone and $A_\infty$ is $\ell^\infty$-monotone. Neither is $\ell^2$-monotone.
• Figure E2: Left of $F_{1}$ (top) and $F_{\infty}$ (bottom) in $\ell^1$ (left), $\ell^2$ (center), and $\ell^\infty$ (right; max pairing). The left half-plane marks monotonicity violations: $F_{1}$ is $\ell^1$-monotone and $F_{\infty}$ is $\ell^\infty$-monotone. Neither is $\ell^2$-monotone.
• Figure E3: Left (max pairing) in $\ell^\infty$ of policy evaluation operator $T_\pi$ and regularized policy evaluation operator ${T}_{\pi,\alpha}$, with ${\alpha=0.25}$, for a randomly generated $8$-state with ${\gamma=0.7}$. Left:$\mathrm{SRG}_L({T}_\pi)$ lies in a disk of radius ${\gamma}$, confirming ${T}_\pi$ is $\gamma$-contractive. Right:$\mathrm{SRG}_L({T}_{\pi,\alpha})$ lies in a disk of radius ${\ell \approx 0.89}$, thus improving on the Lipschitz bound ${\alpha + \gamma = 0.95}$.

Theorems & Definitions (15)

• example 1: Sign, max, and min-index pairings in $\mathbb{R}^2$
• definition 1: Left cosines and angles
• lemma 1: Directional cosine identities
• example 2: Directional cosines in $\mathbb{R}^2$
• remark 1: Duality
• proposition 1: Gain--phase decomposition
• lemma 2: Phase characterization of monotonicity
• definition 2: Left SRG
• theorem 1: Operator properties from directional
• theorem 2: calculus