Nonlinear Joint Spectral Radius

Francesco Tudisco; Nicola Guglielmi; Piero Deidda

REVIEW 2 major objections 5 minor 74 references

Nonlinear Joint Spectral Radius

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper defines a nonlinear joint spectral radius on cones and proves it is the exact threshold for worst-case stability of switched subhomogeneous monotone systems: $\rho_K(F)<1$ if and only if the system is asymptotically stable.

desk verdict Solid core stability theory with honest disclosure of prior work; the computability claim rests on a subcone hypothesis that fails in the paper's own examples, so the exactness framing needs tightening. read the letter →

arxiv 2507.11314 v1 pith:GZAQG3EI submitted 2025-07-15 math.DS

Piero Deidda , Nicola Guglielmi , Francesco Tudisco This is my paper · ORCID

classification math.DS MSC 37B2547H0715A18

keywords jointspectralradiusnonlinearPerron-Frobeniustheoryswitcheddynamicalsystemsorder-preservingmapssubhomogeneousThompsonmetricasymptoticstabilitydeepneuralnetworks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a nonlinear joint spectral radius for families of subhomogeneous, order-preserving maps acting on a cone, and argues that it is the correct worst-case growth indicator for switched discrete-time systems. The central result is a threshold theorem: the family is asymptotically stable under every switching sequence if and only if this nonlinear joint spectral radius is strictly less than one. A companion estimate shows that the same number controls how fast two nearby trajectories separate, giving a computable Lipschitz-type bound along every composition. The authors also develop dual characterizations in terms of monotone prenorms, compare the joint spectral radius with a generalized version built from cone spectral radii of compositions, and give a polytopal-type algorithm that can compute the value exactly under spectral-maximizing conditions. A reader should care because this extends a classical linear tool to nonlinear Perron-Frobenius type systems and gives concrete stability and robustness certificates for switched networks, including nonnegative deep neural networks.

What carries the argument

The load-bearing object is the cone joint spectral radius $\rho_K(F)$, defined through the induced norm of subhomogeneous maps on a cone; its threshold behavior is carried by the order-preserving property, which makes each map non-expansive in Thompson's metric and connects Euclidean trajectory growth to the metric via standard cone-norm bounds. Two associated objects do most of the work: the asymptotic homogeneous maps $f_0(x)=\lim_{c\to 0} f(cx)/c$ and $f_\infty(x)=\lim_{c\to\infty} f(cx)/c$, whose joint spectral radii bracket $\rho_K(F)$ for a subhomogeneous family (Theorem 4.7); and the generalized cone joint spectral radius $\hat\rho_K(F)$ based on cone spectral radii of compositions, whose equality with $\rho_K(F)$ is established under the interior-subcone eigenvector condition and used in the polytopal-type algorithm. The algorithm itself builds an extremal finitely generated monotone prenorm whose vertices are images of a conjectured spectrum-maximizing product, mirroring the linear polytope method (Theorems 5.10, 6.4, Lemma 6.2).

What would settle it

Run the paper's Example 5.24: a bounded subhomogeneous family on $\mathbb{R}_+^2$ with $\rho_K(F)=1$ and $\hat\rho_K(F)=1/2$. The generalized-radius test would incorrectly certify asymptotic stability, while Theorem 3.4 says $\rho_K=1$ means the family is not asymptotically stable; any proposed relaxation of the interior-subcone condition must explain this example. A more decisive test would be to check whether equality can fail for a bounded family that is additionally equicontinuous — the open question the authors flag.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the classical joint spectral radius threshold survives nonlinearity: for a bounded family $F$ of continuous subhomogeneous maps on a solid closed cone $K$, $\rho_K(F)<1$ if and only if the switched system $x_{k+1}=f_{\sigma(k)}(x_k)$ is asymptotically stable for every switching sequence (Theorem 3.4). When the maps are also order-preserving and thus non-expansive in Thompson's metric, the same radius bounds trajectory separation: $\|f(x)-f(y)\| \leq C(x,y,\epsilon)(\rho_K(F)+\epsilon)^k\|x-y\|$ for every composition of length $k$ (Theorem 3.6). The paper further proves that in the homogeneous case the nonlinear JSR has a dual formula as the infimum over monotone prenorms of their induced suprema, and that it coincides with the generalized cone JSR — the limsup of $k$-th roots of cone spectral radii of length-$k$ compositions — whenever a fixed subcone inside the interior contains a dominant eigenvector of every composition (Corollary 5.22). Two explicit examples show that boundedness of the family alone is not enough for that equality, in contrast to the linear Berger-Wang theorem.

Load-bearing premise

The main extension beyond the linear theory assumes that all long compositions of the maps share a dominant eigenvector lying in a fixed region strictly inside the cone; without this assumption, boundedness alone does not make the easy spectral-radius-based test correct.

Editorial extensions

If this is right

If $\rho_K(F)<1$, every trajectory of the switched system converges to zero under any admissible switching rule; if $\rho_K(F)\geq 1$, the family is not asymptotically stable at the uniform rate.
The trajectory-separation bound (Theorem 3.6) turns the nonlinear JSR into a worst-case Lipschitz certificate: a small change in an interior input cannot be amplified by more than $(\rho_K(F)+\epsilon)^k$ up to a constant, across all layer choices.
For subhomogeneous families, the JSR is sandwiched between the JSRs of the two homogeneous asymptotic families, so stability bounds can be computed from homogeneous maps without simulating all switching sequences.
When the interior-subcone condition holds, $\rho_K(F)=\hat\rho_K(F)$, so the radius can be certified by checking spectral radii of length-$k$ compositions, and the proposed algorithm terminates finitely for finite families with a dominant spectrum-maximizing product.
Applied to nonnegative neural networks, the framework yields conditions for fixed-point attraction in deep equilibrium models and bounds on gradient growth, stated as a necessary condition $\rho_K<1$ for global convergence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical consequence not stated in the paper is that the interior-subcone eigenvector condition is the real gatekeeper for spectral-radius certificates: without it, the easy test can be off by an arbitrary factor, as in the paper's Example 5.24 where $\rho_K(F)=1$ while $\hat\rho_K(F)=1/2$.
The framework suggests a testable design rule for neural network architectures: keep activations and maps such that all compositions share an invariant interior subcone, which would make the generalized JSR computable and the Lipschitz bound tight, extending what the paper proves for perturbed families in Example 5.25.
The open equicontinuity question could be probed numerically: if a bounded, equicontinuous, homogeneous order-preserving family with $\hat\rho_K<\rho_K$ exists, the analogy with the linear Berger-Wang theorem fails more strongly than the examples shown; the paper leaves this unresolved.
Because $f_0$ can degenerate to infinity when constant terms dominate, applying the framework to networks with biases requires shifting the cone or rescaling around a working point; the paper does not develop that variant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core stability theorem (Thm 3.4) is sound and the paper is honest about its debt to prior work: Remark 5.5 discloses that the nonlinear JSR is a particular case of the competitive spectral radius of Akian, Gaubert and Marchesini. That is handled properly, not buried. The genuinely new material is Theorem 4.7 (the two-sided bound via the homogeneous limits f0 and f∞), the monotone-prenorm duality in Theorem 5.10, the generalized-JSR equality conditions around Corollary 5.22, the continuity results in 5.31, and the polytopal algorithm. I read the main proofs with the nonlinear Perron-Frobenius toolkit and they hold together; in particular there is no circularity in Theorem 5.10, which constructs the extremal prenorm from the semigroup rather than assuming it.

The soft spot is the computability claim. The advertised "computable threshold" is carried by Cor 5.22, asserting ρK(F) = bρK(F) under the existence of a subcone K′ contained in Int(K) containing a dominant eigenvector of every map in Σ(F). That hypothesis is load-bearing and is not checkable from the generator family alone. Example 5.24 is the smoking gun: a bounded family with bρK = 1/2 and ρK = 1. Use bρK as a stability certificate without extra conditions and you can be off by a factor of two in the wrong direction. Example 5.27 shows the hypothesis can fail even when every generator has an interior eigenvector, because a product only has boundary eigenvectors. The authors are candid about both examples, but they never give a verifiable characterization of when K′ exists; Theorem 5.26 offers a Hilbert-contractive sufficient condition, but its K′′ is asserted to be a subcone without proof, and the non-contractive case is left open. So the exactness statements in Cor 5.22 and the finite-time claims in Theorem 6.4 are conditional in a way the abstract does not convey.

The neural-network section overreaches: "tight," "sharper than L2 bounds," and uniform Jacobian bounds are asserted, not derived or tested, and the algorithm ships no code. Theorem 6.4 is proved via three sketched claims, which I could not fully verify line by line.

Who this is for: people working on switched monotone systems and nonlinear Perron-Frobenius theory. The central stability result and the negative examples are valuable; the exactness framing needs revision. This deserves a serious referee: accept the theory, require the authors to flag the subcone hypothesis as an open condition or prove it from the data, and strip the unverified engineering claims.

Referee Report

2 major / 5 minor

Summary. The paper introduces a nonlinear joint spectral radius ρ_K(F) for switched systems x_{k+1}=f_{σ(k)}(x_k) whose maps f_i are subhomogeneous and order-preserving on a cone K. The main mathematical results are: Theorem 3.4, which characterizes asymptotic stability of the switched semigroup by ρ_K(F)<1; Theorem 3.6, which bounds trajectory separation under Thompson nonexpansiveness in terms of ρ_K(F); Theorem 4.7, which sandwiches ρ_K(F) between the JSRs of two homogeneous limit families; Theorem 5.10, a dual representation of the homogeneous JSR via monotone prenorms; and Section 5.2, which introduces a generalized JSR bρ_K(F) and proves equality with ρ_K(F) under a strong interior-subcone hypothesis. A polytopal-type algorithm is proposed in Section 6 with finite-time convergence conditions. The paper is notably transparent about limitations, including counterexamples where bρ_K≠ρ_K (Examples 5.24 and 5.27) and where continuous extremal prenorms fail (Examples 5.16 and 5.19).

Significance. If the main theorems are correct, the paper gives a coherent and natural threshold object for stability of switched monotone cone systems, extending the linear joint spectral radius to nonlinear Perron-Frobenius settings. The core stability theorem and the dual prenorm construction are nontrivial and appear internally consistent. A particular strength is the collection of counterexamples that delineate where the linear Berger-Wang equality fails in the nonlinear world; these examples are an honest and useful contribution. The practical significance is tempered, however, by the fact that the equality ρ_K(F)=bρ_K(F) and the finite-time convergence of the algorithm rest on hypotheses that are not checkable from the generator family alone and can fail in simple cases; the paper would benefit from stating this limitation at the level of the abstract and introduction.

major comments (2)

[§5.2, Cor. 5.22; §6] The equality ρ_K(F)=bρ_K(F) and, through it, the exactness and finite-time convergence claims of the polytopal algorithm are proved only under the assumption that there is a closed subcone K'⊂Int(K) containing a dominant eigenvector of every f∈Σ(F). This hypothesis is not checkable from the generator family alone, and the manuscript's own examples show that it cannot be replaced by natural weaker assumptions: Example 5.24 gives a bounded homogeneous family with bρ_K(F)=1/2 and ρ_K(F)=1, and Example 5.27 gives two generators each with an interior dominant eigenvector whose product has only boundary eigenvectors. Since Section 6's algorithm is advertised as computing or certifying ρ_K(F), the paper should state prominently that this computability is conditional on K', explain how one could certify such a subcone in concrete applications, and describe what can be guaranteed when K' is not known to exist. Without such discussion, the algorithmic section overstates the scope of the results.
[§3, 'Fixed points and deep equilibrium models'] The claim that Theorem 3.4 yields 'a necessary condition for global convergence to a fixed point' is not correct as stated. Theorem 3.4 concerns asymptotic stability in the sense of convergence to zero, whereas the fixed points of deep equilibrium models and of affine maps with bias are generally nonzero. A concrete counterexample within the paper's own framework is f(x)=x/2+1 on K=R_+. This map is continuous, subhomogeneous, and order-preserving, has the unique globally attracting fixed point x*=2, but ρ_K({f})=1 because ∥f^k∥→2 on the unit ball. Thus ρ_K(F)<1 is not necessary for global convergence to a nonzero fixed point. The application paragraph should be revised to refer instead to the error dynamics around a fixed point or to a separate stability notion for nonzero equilibria.

minor comments (5)

[§3, proof of Theorem 3.4] In the converse direction, the normality constant δ of the cone is omitted when passing from ∥f(r_1x)∥≤α^k r_2 to a bound on ∥f(x)∥; the argument still works if one writes Cα^k and then notes that (Cα^k)^{1/k}→α, but the displayed chain of inequalities is not correct as written.
[§5.2, Eq. (16)] The statement that the lim sup in the definition of bρ_K(F) can be replaced by a supremum is only plausible because any word can be repeated; the text says this follows from Proposition A.7, but Proposition A.7 alone does not give this replacement. A short explicit argument using ρ_K(f^m)=ρ_K(f)^m would improve the presentation.
[§5.2, proof of Cor. 5.22] The proof applies Theorem 3.4 to the scaled family F_ε, but Theorem 3.4 assumes boundedness; in the present setting boundedness follows from Remark 5.23 under the same K' hypothesis. This chain of reasoning should be stated explicitly so the corollary is self-contained.
[§6, Example 5.19] In Example 5.19 the computation of the supremum over Σ_k(F) is slightly confusing because it is written at the point x_0=(1,1) with ∥x_0∥_1=2, whereas the operator norm is an extremum over the unit ball. The final value ρ_K(F)=1/2 is correct, but the intermediate display would be clearer if the unit-normalized point were used.
[Throughout] Several typographical errors should be corrected: 'controlloed' in Section 4, 'ans assume' in Proposition 4.2, 'Schouder' in Theorem 5.26, 'Algortihm' in Section 6, and 'sepctrum' in the proof of Theorem 6.4. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nonlinear JSR is defined independently by a limsup formula, the stability and duality theorems are proved by direct constructions from the semigroup, and the only non-trivial equality (rho_K = b_rho_K) is explicitly conditional on a subcone hypothesis that the paper states, tests, and does not disguise as unconditional.

full rationale

The central object rho_K(F) is introduced in Eq. (2) as a limsup of sup-norms over compositions, with no fitted parameters, and Theorem 3.4 proves asymptotic stability iff rho_K(F) < 1 by exhibiting U,V and the scaling factor from the definition, and conversely bounding rho_K by alpha from asymptotic stability; this is a direct derivation rather than a restatement. Theorem 3.6 uses the definition of rho_K together with Thompson nonexpansiveness and the normality lemma, yielding an explicit constant C(x,y,epsilon); the Lipschitz-type estimate is not folded into the definition of rho_K. Section 4 bounds rho_K(F) between rho_K(F_infty) and rho_K(F_0) using the pointwise limit maps f_0, f_infty and monotonicity of norms, so the comparison is derived, not assumed. The homogeneous duality Theorem 5.10 is constructive: it defines Theta_epsilon(x) = sup_{f in Sigma(F_epsilon)} psi(f(x)) and proves the key inequality Theta_epsilon(f_i) <= rho_K(F)+epsilon from semigroup closure, with non-degeneracy obtained from the JSR definition itself. This is the standard Rota-Strang argument, not a circular one. The equality rho_K(F) = b_rho_K(F) in Corollary 5.22 is conditional on the explicit hypothesis that a closed subcone K' subset Int(K) contains dominant eigenvectors of every composition; Theorem 5.21 proves the direction used, and the paper honestly gives Example 5.24 showing that boundedness alone is insufficient and Example 5.27 showing that generator-level interior eigenvectors need not survive composition. These are limitations of a sufficient condition, not circularity. The numerical algorithm in Section 6 terminates under stated hypotheses (contractivity and a dominant spectrum-maximizing product); Lemma 6.2 gives an if-and-only-if certificate whose proof uses the definitions of rho_K and M(x/z), and the numerical examples verify rather than fit the theory. Self-citations appear in the machine-learning motivation and in references to polytopal algorithms, but no load-bearing theorem is imported solely from the authors' prior work, and no uniqueness claim is used to forbid alternatives. The paper also explicitly flags an open question about equicontinuity, further indicating that unproved equivalences are not being smuggled in. Therefore no circular step meets the quoted-evidence threshold.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The central claim rests on standard results from nonlinear Perron-Frobenius theory (quoted from [46]) and on the paper's explicit modeling assumptions: solid closed cones, continuous sub-homogeneous order-preserving maps, bounded families, and the strong hypotheses needed for the JSR/gJSR equality (a common interior subcone of dominant eigenvectors) and for finite-time algorithm termination (Hilbert contractivity and a dominant spectral gap). No free parameters are fitted to data; all constants in the examples are stated example specifications, not fitted values.

assumptions (7)

standard math Nonlinear Perron-Frobenius background results quoted as Lemmas A.1-A.9 (normality bounds, Thompson/Hilbert nonexpansivity, perturbation eigenpair approximation, spectral radius continuity, boundary extension for polyhedral cones)
Used in the proofs of Theorems 3.6, 5.6, 5.10, 5.26, 6.4 and Appendix B without reproof; these are established results from Lemmens-Nussbaum [46].
standard math Fekete's lemma and submultiplicativity of the induced operator norm for homogeneous maps
Needed to replace the limsup by an inf/limit in Eq (6), Proposition 5.3, and Theorem 5.21.
standard math Brouwer and Schauder fixed point theorems, Ascoli-Arzela theorem, Dini's theorem
Used in Appendix B for slice eigenvectors and continuous eigenpair curves (Brouwer, Ascoli-Arzela), in Theorem 5.26 (Schauder), and in Proposition 4.2 (Dini).
domain assumption Monotone limits c -> f(cx)/c exist in the Alexandrov compactification K-infinity for sub-homogeneous maps, with uniqueness of limits along monotone chains
Section 4, Eqs (5), needs these limits to define f0 and f_infinity; the argument uses order completeness of the cone, discussed but not proved in the paper.
domain assumption Every map in the semigroup has its dominant eigenvector inside a common subcone K' strictly inside the cone
Hypothesis of Theorem 5.21 and Corollary 5.22 for rho_K = b_rho_K; hard to verify and false in the paper's own Examples 5.24 and 5.27.
domain assumption The cone is polyhedral (condition G) for boundary-continuous extensions of prenorms and limit maps
Propositions 4.4, 5.14, 5.17 and Theorem A.9 assume polyhedrality or condition G.
domain assumption Each map contracts the Hilbert metric with parameter beta < 1 and has an interior dominant eigenvector
Theorem 5.26 and Theorem 6.4 require strict Hilbert contractivity; fails already for cone-preserving linear maps with boundary or Jordan-like behavior (Example 5.27).

invented entities (3)

Nonlinear cone joint spectral radius rho_K(F) independent evidence
purpose: Worst-case asymptotic growth rate over all switching sequences for switched sub-homogeneous cone systems
Defined by an explicit limsup formula (Eq 2); grounded externally as a special case of the competitive spectral radius in [2] (Remark 5.5) and characterized by theorems against the independent notion of asymptotic stability (Theorem 3.4).
Generalized cone joint spectral radius b_rho_K(F) independent evidence
purpose: Spectral-radius-based alternative whose equality with rho_K is shown under an interior-eigenvector subcone condition
Definition (16) mirrors the linear generalized JSR of [10]; its equality with rho_K is proved (Corollary 5.22), and a counterexample (Example 5.24) shows where it strictly drops, so its semantic content is anchored.
Finitely generated monotone prenorms (Definition 6.1)
purpose: Finite-dimensional representation of extremal prenorms used by the polytopal algorithm
An internal construction: the paper proves its properties (continuity, monotonicity, non-degeneracy) and exact extremality under Lemma 6.2's conditions, so it is not a free postulate, but it has no external referent.

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Cite this review

Pith. "Pith review of Nonlinear Joint Spectral Radius." pith.science (2026). https://pith.science/paper/GZAQG3EI

@misc{pith2026250711314,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Joint Spectral Radius},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZAQG3EI}},
  note         = {Machine review of arXiv:2507.11314}
}

read the original abstract

We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems governed by sub-homogeneous and order-preserving maps acting on cones. We show that this nonlinear JSR characterizes both the asymptotic stability of the system and the divergence or convergence rate of trajectories originating from different points within the cone. Our analysis establishes upper and lower bounds on the nonlinear JSR of a sub-homogeneous family via the JSRs of two associated homogeneous families obtained through asymptotic scaling. In the homogeneous case, we develop a dual formulation of the JSR and investigate the equality between the joint spectral radius and the generalized joint spectral radius, extending classical results from linear theory to the nonlinear setting. We also propose a polytopal-type algorithm to approximate the nonlinear JSR and provide conditions ensuring its finite-time convergence. The proposed framework is motivated by applications such as the analysis of deep neural networks, which can be modeled as switched systems with structured nonlinear layers. Our results offer new theoretical tools for studying the stability, robustness, and convergence behavior of such models.

Figures

Figures reproduced from arXiv: 2507.11314 by the authors.

**Figure 2.** Extremal polytope norm (left figure) and prenorm (right figure) generated [PITH_FULL_IMAGE:figures/full_fig_p045_2.png] view at source ↗

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For any c >0, ρc K(f ) = bρc K(f ) = ˇρc K(f )
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There exists a continuous curve c ∈ (0, ∞) → (xc, ρc K(f )), where xc ∈ Ωc is some eigenvector associated to ρc K(f ), such that: xc1 ≤K xc2 , ρ c2 K (fϵ) ≤ ρc1 K (fϵ) ≤ c2 c1 ρc2 K (fϵ), for any 0 < c1 < c2. Proof. Start considering xc ∈ Ωc such that f (xc) = ρc K(f )xc. Then...
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In particular: ρc K(f ) = sup x∈Ωc m(f (x)/x)

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In particular: ρc K(f ) = inf x∈Int(K)∩Ωc M (f (x)/x)

If there exists x ∈ Int(K)∩Ωc such that f (x) ≤K βx, then ρc K(f ) ≤ β. In particular: ρc K(f ) = inf x∈Int(K)∩Ωc M (f (x)/x). Proof. We start by proving the first thesis. Assume x as in the hypothesis and consider y ∈ Int(K), then M (f (x)/y) ≥ M (αx/y) = αM (x/y). In particu...

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