pith. machine review for the scientific record. sign in

arxiv: 2604.02407 · v1 · submitted 2026-04-02 · 🧮 math.OC · cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Scaled Relative Graphs in Normed Spaces

Alberto Padoan
Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:05 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords scaled relative graphsnormed spacesdirectional anglesoperator contractionmonotonicityBellman operatorsfixed point theoryoptimization
0
0 comments X

The pith

Directional scaled relative graphs certify contraction and monotonicity of operators in normed spaces via geometric containment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the scaled relative graph framework from Hilbert spaces to general normed spaces by replacing the inner product with a regular pairing. This replacement produces directional angles, which in turn define directional SRGs that test whether an operator satisfies contraction or monotonicity by checking if its graph stays inside specific regions. If the tests hold, they certify the properties geometrically without needing to compute distances or inner products directly. The work also supplies rules for how these graphs transform under scaling, inversion, addition, and composition, and it demonstrates the approach on Bellman operators through numerical examples. The extension matters because many problems in optimization and control are naturally posed in spaces that lack an inner product structure.

Core claim

Replacing the inner product with a regular pairing on a normed space yields directional angles and directional scaled relative graphs. These graphs certify that an operator is a contraction or monotone by verifying geometric containment conditions. Calculus rules are derived for the graphs under scaling, inversion, addition, and composition, and the results are illustrated by a graphical certificate for Bellman operators.

What carries the argument

Directional scaled relative graphs, formed from directional angles induced by a regular pairing, that serve as geometric containment tests for operator properties such as contraction and monotonicity.

If this is right

  • An operator whose directional SRG lies in the appropriate region is guaranteed to be a contraction.
  • Monotonicity of an operator is certified by a similar containment condition on its directional SRG.
  • SRGs of composite operators can be obtained from the individual SRGs via the derived addition and composition rules.
  • Graphical verification becomes possible for Bellman operators in spaces without inner products.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could support new fixed-point algorithms that rely on visual or computational checks in Banach spaces rather than Hilbert spaces.
  • Similar containment tests might extend to other operator classes such as averaged or cocoercive mappings.
  • Software implementations could render the graphs to automate property verification during algorithm design.

Load-bearing premise

A regular pairing exists on the normed space that is sufficiently well-behaved to define directional angles yielding valid containment tests.

What would settle it

An operator in a normed space that satisfies contraction yet whose directional SRG fails to lie inside the predicted containment region for contractions.

Figures

Figures reproduced from arXiv: 2604.02407 by Alberto Padoan.

Figure 1
Figure 1. Figure 1: Directional angles in ℓ 1 (left), ℓ 2 (center), and ℓ∞ (right) between unit-norm vectors x (solid) and y (dashdotted) in R2 . The outward normals at x are sign(x) in ℓ 1 , x/ ∥x∥2 in ℓ 2 , and sign(xmx )emx in ℓ∞ with the min-index pairing. In ℓ 1 , an angle encodes transitions between sign patterns on the unit cross-polytope; in ℓ 2 , it encodes continuous rotations on the unit sphere; in ℓ∞, it encodes t… view at source ↗
Figure 2
Figure 2. Figure 2: shows the SRGs of A1 (top) and A∞ (bottom) in ℓ 1 (left), ℓ 2 (center), and ℓ∞ (right), with the max pairing in ℓ∞. For A1, the ℓ 1 SRG (top-left) lies in the right half-plane, confirming ℓ 1 -monotonicity. By contrast, the ℓ 2 and ℓ∞ SRGs extend into the left half-plane, so monotonicity does not hold in these norms. For A∞, a dual picture emerges: the ℓ∞ SRG (bottom-right) lies in the right half-plane, wh… view at source ↗
Figure 4
Figure 4. Figure 4: (left) confirms this for a randomly generated 8-state MDP with γ = 0.7. The left SRG in ℓ∞ (with max pairing) is contained in the disk of radius γ centered at the origin. b) Regularized policy evaluation: In approximate dy￾namic programming, a common strategy to incorporate prior knowledge is to regularize the Bellman operator [22], [36]. Consider the regularized policy evaluation operator Tπ,αv = Tπv + αφ… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to general normed spaces by replacing the inner product with a regular pairing. This yields directional SRGs whose geometric containment tests are claimed to certify operator properties including contraction and monotonicity. Calculus rules are derived for SRGs under scaling, inversion, addition, and composition, with the theory illustrated by numerical examples including a graphical contraction certificate for Bellman operators.

Significance. If the directional SRG containment tests hold without additional restrictions on the space or pairing, the work would provide a useful geometric certificate tool beyond Hilbert spaces, extending prior SRG results to Banach-space settings common in optimization and operator theory. The explicit calculus rules and numerical illustrations add practical value for verifying properties such as monotonicity.

major comments (2)
  1. Abstract and introduction: the claim that directional SRGs certify contraction and monotonicity via containment tests rests on the regular pairing inducing directional angles that preserve the required geometric implications under scaling, inversion, addition, and composition. In normed spaces the pairing is asymmetric, and no explicit conditions are given to guarantee that the containment tests remain valid without extra assumptions on the space or operator; this is load-bearing for the central extension.
  2. The weakest assumption (regular pairing sufficiently well-behaved for directional angles) is not accompanied by a counter-example check or a minimal set of axioms on the pairing that would ensure the containment tests survive the asymmetry; without this the certification claims for general normed spaces remain conditional.
minor comments (2)
  1. The numerical examples (Bellman operator) would benefit from explicit reporting of the computed directional angles or containment radii so readers can reproduce the graphical certificate.
  2. Notation for the regular pairing and directional angle should be introduced with a short comparison to the Hilbert-space inner-product case to aid readers familiar with the original SRG work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable feedback on our work extending Scaled Relative Graphs to normed spaces. We address the major comments below, clarifying the role of the regular pairing and committing to revisions that strengthen the presentation of the assumptions and their implications.

read point-by-point responses

  1. Referee: Abstract and introduction: the claim that directional SRGs certify contraction and monotonicity via containment tests rests on the regular pairing inducing directional angles that preserve the required geometric implications under scaling, inversion, addition, and composition. In normed spaces the pairing is asymmetric, and no explicit conditions are given to guarantee that the containment tests remain valid without extra assumptions on the space or operator; this is load-bearing for the central extension.

    Authors: We appreciate this observation. The regular pairing is defined in Section 2 to satisfy specific properties (positivity, absolute homogeneity, and boundedness) that ensure the directional angles preserve the geometric relations needed for the containment tests in Theorems 3.1–3.4. These properties are chosen precisely to handle the asymmetry in normed spaces while maintaining the implications for contraction and monotonicity under the calculus operations. To make this more explicit, we will revise the abstract and introduction to reference these properties directly and add a dedicated remark in Section 2 listing the minimal axioms required for the tests to hold without additional assumptions on the space or operator. revision: yes

  2. Referee: The weakest assumption (regular pairing sufficiently well-behaved for directional angles) is not accompanied by a counter-example check or a minimal set of axioms on the pairing that would ensure the containment tests survive the asymmetry; without this the certification claims for general normed spaces remain conditional.

    Authors: We agree that explicitly stating the minimal axioms would improve clarity. The definition of a regular pairing already provides these axioms, which are sufficient for the directional SRG to certify the properties as proven in the main theorems. We will expand Definition 2.1 to list the axioms separately and include a brief discussion explaining why they ensure the containment tests remain valid despite asymmetry. A counter-example check is not included because the proofs are direct from the axioms, but we will add a sentence noting that violation of regularity would invalidate the angle interpretation, thereby addressing the conditional nature of the claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension derives independently from regular pairing

full rationale

The paper extends the prior Hilbert-space SRG framework by defining a regular pairing on normed spaces to induce directional angles and directional SRGs. Containment tests for contraction and monotonicity, along with calculus rules under scaling/inversion/addition/composition, are derived directly from the pairing's properties and the resulting geometric constructions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on explicit derivations from the new pairing axioms rather than renaming or smuggling prior results. The work is self-contained against the stated assumptions on the pairing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a regular pairing in an arbitrary normed space and on standard properties of angles and containment in the plane; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Every normed space admits a regular pairing that can replace the inner product for defining angles.
    Invoked to extend the SRG definition beyond Hilbert spaces.

pith-pipeline@v0.9.0 · 5377 in / 1057 out tokens · 33134 ms · 2026-05-13T21:05:54.681325+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

  1. [1]

    Regeneration theory,

    H. Nyquist, “Regeneration theory,”Bell System Technical Journal, vol. 11, no. 1, pp. 126–147, 1932

    work page 1932

  2. [2]

    H. W. Bode,Network Analysis and Feedback Amplifier Design. New York, NY , USA: Van Nostrand, 1945

    work page 1945

  3. [3]

    Absolute stability of nonlinear systems of automatic control,

    V . M. Popov, “Absolute stability of nonlinear systems of automatic control,”Automation and Remote Control, vol. 22, no. 8, pp. 857– 875, 1961

    work page 1961

  4. [4]

    On the input-output stability of time-varying nonlinear feedback systems—Part I: Conditions derived using concepts of loop gain, conicity, and positivity,

    G. Zames, “On the input-output stability of time-varying nonlinear feedback systems—Part I: Conditions derived using concepts of loop gain, conicity, and positivity,”IEEE Transactions on Automatic Con- trol, vol. 11, no. 2, pp. 228–238, 1966

    work page 1966

  5. [5]

    System analysis via integral quadratic constraints,

    A. Megretski and A. Rantzer, “System analysis via integral quadratic constraints,”IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819–830, 1997

    work page 1997

  6. [6]

    Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,

    E. K. Ryu, R. Hannah, and W. Yin, “Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,”Mathematical Pro- gramming, vol. 194, pp. 569–619, 2022

    work page 2022

  7. [7]

    Facchinei and J

    F. Facchinei and J. S. Pang,Finite-Dimensional Variational Inequali- ties and Complementarity Problems. New York, NY , USA: Springer Verlag, 2003

    work page 2003

  8. [8]

    Beck,First-Order Methods in Optimization

    A. Beck,First-Order Methods in Optimization. Philadelphia, PA, USA: SIAM, 2017

    work page 2017

  9. [9]

    E. K. Ryu and W. Yin,Large-Scale Convex Optimization: Algorithms and Analyses via Monotone Operators. Cambridge University Press, 2022

    work page 2022

  10. [10]

    H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed. Cham, Switzerland: Springer, 2017

    work page 2017

  11. [11]

    Splitting algorithms for the sum of two nonlinear operators,

    P. L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”SIAM J. Numerical Analysis, vol. 16, no. 6, pp. 964–979, 1979

    work page 1979

  12. [12]

    Graphical nonlinear systems analysis,

    T. Chaffey, F. Forni, and R. Sepulchre, “Graphical nonlinear systems analysis,”IEEE Trans. Autom. Control, vol. 68, no. 10, pp. 6073–6088, 2023

    work page 2023

  13. [13]

    R. Pates. (2021) The scaled relative graph of a linear operator. ArXiv:2106.05650

    work page arXiv 2021

  14. [14]

    A rolled-off passivity theorem,

    T. Chaffey, “A rolled-off passivity theorem,”Systems & Control Letters, vol. 162, p. 105198, 2022

    work page 2022

  15. [15]

    Circuit model reduction with scaled relative graphs,

    T. Chaffey and A. Padoan, “Circuit model reduction with scaled relative graphs,” inProc. 61st IEEE Conf. Decision and Control, Cancún, Mexico, 2022, pp. 6530–6535

    work page 2022

  16. [16]

    Scaled relative graph analysis of Lur’e systems and the generalized circle criterion,

    J. P. J. Krebbekx, R. Tóth, and A. Das, “Scaled relative graph analysis of Lur’e systems and the generalized circle criterion,” inProc. European Control Conf., Thessaloniki, Greece, 2025, pp. 1213–1218

    work page 2025

  17. [17]

    On phase in scaled graphs,

    S. van den Eijnden, C. Chen, K. Scheres, T. Chaffey, and A. Lanzon, “On phase in scaled graphs,” inProc. 64th IEEE Conf. Decision and Control, Rio de Janeiro, Brazil, 2025, pp. 3595–3600

    work page 2025

  18. [18]

    Mixed small gain and phase theorem: A new view using scaled relative graphs,

    E. Baron-Prada, A. Anta, A. Padoan, and F. Dörfler, “Mixed small gain and phase theorem: A new view using scaled relative graphs,” inProc. European Control Conf., Thessaloniki, Greece, 2025, pp. 114–119

    work page 2025

  19. [19]

    Baron-Prada, A

    E. Baron-Prada, A. Padoan, A. Anta, and F. Dörfler. (2025) Stability results for MIMO LTI systems via scaled relative graphs

    work page 2025

  20. [20]

    Computing scaled relative graphs of discrete- time LTI systems from data,

    T. Nauta and R. Pates, “Computing scaled relative graphs of discrete- time LTI systems from data,” 2026, arXiv:2603.05239

    work page arXiv 2026

  21. [21]

    M. A. Dahleh and I. J. Diaz-Bobillo,Control of Uncertain Systems: A Linear Programming Approach. Englewood Cliffs, NJ, USA: Prentice-Hall, 1994

    work page 1994

  22. [22]

    D. P. Bertsekas,Dynamic Programming and Optimal Control, 4th ed. Belmont, MA, USA: Athena Scientific, 2012

    work page 2012

  23. [23]

    R. S. Sutton and A. G. Barto,Reinforcement Learning: An Introduc- tion, 2nd ed. Cambridge, MA, USA: MIT Press, 2018

    work page 2018

  24. [24]

    Zeidler,Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems

    E. Zeidler,Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. New York, NY , USA: Springer, 1986

    work page 1986

  25. [25]

    C. E. Chidume,Iterative Methods for Nonlinear Functional Equations. London, U.K.: Springer, 2009

    work page 2009

  26. [26]

    Non-Euclidean contraction theory for robust nonlinear stability,

    A. Davydov, S. Jafarpour, and F. Bullo, “Non-Euclidean contraction theory for robust nonlinear stability,”IEEE Transactions on Automatic Control, vol. 67, no. 12, pp. 6667–6681, 2022

    work page 2022

  27. [27]

    A. V . Proskurnikov and F. Bullo. (2025) Regular pairings for non-quadratic Lyapunov functions and contraction analysis. ArXiv:2504.14407

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [28]

    Bullo,Contraction Theory for Dynamical Systems, 1.3 ed

    F. Bullo,Contraction Theory for Dynamical Systems, 1.3 ed. Kindle Direct Publishing, 2026. [Online]. Available: https://fbullo.github.io/ ctds

    work page 2026

  29. [29]

    Non- Euclidean Monotone Operator Theory and Applications,

    A. Davydov, S. Jafarpour, A. V . Proskurnikov, and F. Bullo, “Non- Euclidean Monotone Operator Theory and Applications,”J. Machine Learning Research, vol. 25, pp. 1–33, 2024

    work page 2024

  30. [30]

    On inner products in linear metric spaces,

    P. Jordan and J. von Neumann, “On inner products in linear metric spaces,”Ann. Math., vol. 36, no. 2, pp. 719–723, 1935

    work page 1935

  31. [31]

    Semi-inner-product spaces,

    G. Lumer, “Semi-inner-product spaces,”Trans. Amer. Math. Soc., vol. 100, no. 1, pp. 29–43, 1961

    work page 1961

  32. [32]

    Classes of semi-inner-product spaces,

    J. R. Giles, “Classes of semi-inner-product spaces,”Trans. Amer. Math. Soc., vol. 129, no. 3, pp. 436–446, 1967

    work page 1967

  33. [33]

    Orthogonality and linear functionals in normed linear spaces,

    R. C. James, “Orthogonality and linear functionals in normed linear spaces,”Trans. Amer. Mathematical Society, vol. 61, no. 2, pp. 265– 292, 1947

    work page 1947

  34. [34]

    Sur le semi-produit scalaire dans quelques espaces vectoriels normés,

    P. M. Mili ˇci´c, “Sur le semi-produit scalaire dans quelques espaces vectoriels normés,”Matematiˇ cki Vesnik, vol. 23, no. 4, pp. 181–185, 1971

    work page 1971

  35. [35]

    C. A. Desoer and M. Vidyasagar,Feedback Systems: Input-Output Properties. New York, NY , USA: Academic Press, 1975

    work page 1975

  36. [36]

    A theory of regularized markov decision processes,

    M. Geist, B. Scherrer, and O. Pietquin, “A theory of regularized markov decision processes,” inProc. 36th Int. Conf. Machine Learn- ing, vol. 97, Long Beach, CA, USA, 2019, pp. 2160–2169. APPENDIX Proof of Lemma 1.(a) Fixx∈X\{0}. By compatibility of the regular pairing with the norm,Jx, xK=∥x∥ 2, so cosL(x, x) = ∥x∥2 ∥x∥∥x∥ = 1. The straight angle propert...

    work page 2019