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Continuity and approximability of competitive spectral radii

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that for cone-preserving families satisfying strict positivity, the competitive spectral radius is 1-Lipschitz in the matrix sets and can be approximated to arbitrary accuracy by a discretized eigenproblem.

desk verdict The continuity theorem is real, but the approximation claim needs Assumption IV.1, which is stronger than advertised; fix the framing and verify the example. read the letter →

arxiv 2505.22468 v1 pith:UNUYZTBP submitted 2025-05-28 math.OC cs.NAmath.DSmath.NA

Marianne Akian , Stéphane Gaubert , Loïc Marchesini , Ian Morris This is my paper
classification math.OCcs.NAmath.DSmath.NA MSC 91A5015A6047H0965K15
keywords competitivespectralradiusmatrixmultiplicationgamesjointcone-preservingoperatorsThompsonmetricShapleyoperatorrelativevalueiterationRVI-KMalgorithm
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

The paper studies the competitive spectral radius, a two-player generalization of the joint spectral radius: Min and Max alternately choose matrices from compact sets, and the growth rate of the resulting product is the payoff. The central claim is that for cone-preserving families satisfying a strict-positivity condition, the competitive spectral radius is a 1-Lipschitz function of the matrix sets with respect to the Thompson metric. Under the additional Small Cone assumption, the paper shows the value can be approximated to arbitrary accuracy by solving a discretized nonlinear eigenproblem on a finite grid. The discretized eigenvalue brackets the true value within the mesh size, and the proposed RVI-KM algorithm terminates in $O(1/h^2)$ iterations, for a total complexity of $O(1/h^{2d})$ for a $d$-dimensional cone. This makes non-rectangular matrix multiplication games with positive matrices approximable, a case not covered by the earlier rectangular/entropy-game theory.

What carries the argument

The load-bearing object is the Shapley operator of the escape-rate game, restricted to a cross-section of the cone: $F v(x) = \inf_a \sup_b [\log\langle T_{ab}x,e^*\rangle + v(T_{ab}x/\langle T_{ab}x,e^*\rangle)]$. Its unique additive eigenvalue equals the competitive spectral radius. The approximation scheme discretizes this operator: a finite $h$-net $X_h$ in the Hilbert metric, together with an interpolation operator $I_h^+ v(x)=\min_y [v(y)+\operatorname{Funk}(x,y)]$, produces a finite-dimensional Shapley operator whose eigenvalue is computable and whose error is controlled by $h$. The RVI-KM algorithm is the iterative engine: relative value iteration with averaging, whose convergence and error estimate come from nonexpansive iteration theory.

What would settle it

Take any two compact sets of positive matrices in the same part with Thompson-Hausdorff distance $\varepsilon$ and compute their competitive spectral radii by an independent high-accuracy method; if $|\rho(A,B)-\rho(A',B')|$ exceeds $\varepsilon$, the 1-Lipschitz claim fails. Alternatively, run Algorithm 1 on a family satisfying continuity but not the Small Cone assumption, such as a set of diagonal positive matrices, and check whether the computed eigenvalue converges to the true value with error shrinking like $h$; failure would show Assumption IV.1 is genuinely necessary for Theorem IV.6.

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Extended reading notes

Core claim

The paper establishes the competitive spectral radius $(A,B) \mapsto \rho(A,B)$ is 1-Lipschitz with respect to the Hausdorff distance induced by the Thompson metric on compact subsets of a common part of $\mathrm{End}(\operatorname{Int} C)$. It then proves that under the Small Cone assumption, the additive eigenvalue $\lambda$ of the discretized Shapley operator on a finite grid $X_h$ satisfies $-h+\lambda \le \rho \le \lambda$. The RVI-KM algorithm computes an approximate eigenvalue $\lambda_k$ and stops in $O(1/h^2)$ iterations with the guaranteed enclosure $\rho \in [\lambda_k-3h, \lambda_k+2h]$. The authors present this as answering the approximability question for non-rectangular families of positive matrices.

Load-bearing premise

All the quantitative approximation guarantees depend on the Small Cone assumption: the players' matrices must all map some smaller cone inside the main cone back into itself, so the finite grid stays strictly inside the simplex; if no such invariant subcone exists, the discretized bounds are not proven.

Editorial extensions

If this is right

  • Non-rectangular matrix multiplication games with positive matrices can be approximated to any prescribed accuracy with explicit error bounds, answering in the positive case a question raised for general matrix multiplication games.
  • If the allowed matrix sets are perturbed by at most $\varepsilon$ in Thompson-Hausdorff distance, the competitive spectral radius changes by at most $\varepsilon$.
  • The RVI-KM algorithm gives two-sided bounds on the value, not just a heuristic estimate: at precision $h$ it returns $\lambda_k$ with $\rho \in [\lambda_k-3h, \lambda_k+2h]$.
  • The same continuity and approximation machinery applies to other cone-preserving operator families, such as congruence operators on positive semidefinite matrices, so two-player versions of singular-value growth rates are covered.
  • In the population-dynamics illustration, the computed optimal play enters a periodic cycle and the state converges to a projective fixed point; the paper reports this turnpike behavior as a numerical observation.

Reading between the lines

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

  • Editorial inference: the Small Cone assumption is the real bottleneck. The paper neither gives a constructive criterion for it nor verifies it for the Leslie-matrix example, so the numerical table there is not covered by the proven error bounds unless such a cone is exhibited.
  • Editorial inference: the continuity theorem is stated for general hemi-metric spaces, so the 1-Lipschitz stability should transfer to parameterized families of nonexpansive maps beyond linear cone-preserving operators, for instance switched systems in a Funk-metric setting.
  • Editorial inference: the total cost $O(1/h^{2d})$ makes the method sensitive to dimension; replacing the full interpolation operator $I_h^+$ by a local or adaptive interpolation, which the paper names as a bottleneck, could lower the effective exponent.
  • Editorial inference: the same discretization could be used to approximate worst-case versus best-case growth in population or network models where two players have opposing objectives, providing a computable two-player analogue of the lower spectral radius for positive matrices.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the competitive spectral radius, defined as the value of a two-player matrix multiplication game, for families of linear operators that preserve the interior of a closed convex pointed cone. It proves a continuity theorem: for compact matrix sets contained in a fixed part of End(Int C), the competitive spectral radius is 1-Lipschitz with respect to the Hausdorff distance induced by the Thompson metric (Theorem III.1 and Corollary III.2). It then gives an approximation scheme under an additional small-cone assumption (Assumption IV.1): a cross-section of an invariant subcone is discretized, a finite-dimensional nonlinear eigenproblem for a discretized Shapley operator is solved by relative value iteration with Krasnoselskii-Mann damping, and two-sided error bounds are obtained (Theorems IV.6 and IV.8). The algorithm is illustrated on a three-age population dynamics model with Leslie matrices.

Significance. The continuity result is clean and quantitative: Theorem III.1 gives an explicit Lipschitz bound with no tuned parameters, and the proof is self-contained. The approximation scheme is a genuine step toward computable values of non-rectangular matrix multiplication games, with an explicit O(1/h^2) iteration bound and total complexity O(1/h^{2d}). The use of McShane-Whitney interpolation to obtain two-sided bounds is elegant. However, the advertised scope is larger than what is proved: the paper's front matter says approximation holds under the same assumptions as continuity, whereas the algorithmic theorems require the strictly stronger small-cone Assumption IV.1, which is not verified in the numerical example. This gap does not invalidate the theorems, but it substantially qualifies the central claim.

major comments (4)
  1. [Abstract and §I-B, Assumption IV.1] The Abstract and Section I-B claim that, under the same assumptions that give continuity (the common-part / strict-positivity condition), the competitive spectral radius can be approximated up to arbitrary accuracy. The approximation results in Section IV all depend on Assumption IV.1 (Small Cone), which is strictly stronger. For example, take A = {diag(2,1), diag(1,2)} and B = {I} in R_+^2; this is a compact set contained in a single part of End(Int R_+^2), so Corollary III.2 applies, but no closed cone K ⊂ R_+^2 with K ∩ Δ ⊂ relint Δ is invariant under both diagonal matrices: the orbit of any positive ray under products of these maps accumulates on a boundary ray, so any closed invariant cone must intersect the boundary. Thus Theorems IV.6 and IV.8 are not available for this family. The front matter should either state Assumption IV.1 explicitly or weaken the approximation claim to be conditional on it.
  2. [§V, Assumption IV.1] The numerical application to age-structured population dynamics does not verify Assumption IV.1. The text only checks that the Leslie matrices lie in a common part of End(Int C), which is sufficient for continuity by Corollary III.2 but not for the discretization error bounds. Consequently, the entry in Table I and the reported numerical value (for example 1.3147 at 8646 points) are not certified by Theorems IV.6 and IV.8 unless the small-cone condition is established for this family. The authors should either prove IV.1 for the Leslie example or explicitly label the numerical results as heuristic.
  3. [Theorem IV.7 and following text] Theorem IV.7 as stated is formally incorrect: since v_k belongs to L_xbar and \hat F_hn v takes value 0 at xbar, the equation \hat F_hn v = λ + v forces λ = 0. The intended statement is either \hat F_hn v = v for the normalized operator or \hat F_h^+ v = λ + v for the unnormalized operator; the proof of Theorem IV.8 uses an eigenpair of \hat F_h^+. This is a load-bearing statement in the algorithm section and should be corrected.
  4. [Corollary IV.9] Corollary IV.9 only discretizes the action spaces, using Corollary III.2; it does not by itself provide an approximation of the competitive spectral radius. The state-space discretization, which is the core of the algorithm, still requires Assumption IV.1. The sentence introducing Corollary IV.9 ('Corollary III.2 allows us to reduce to this case, after discretization of the action spaces') could be read as removing the small-cone requirement, which it does not. The presentation should separate the two discretizations and state clearly that the small-cone assumption remains necessary.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'competive spectral radius'.
  2. [Example II.1] The notation 'C = Rn≥0' is typeset incorrectly; it should be R_{≥0}^n.
  3. [Section V] In the sentence 'For β ∈ A and α ∈ B' the roles of α and β appear to be reversed relative to the preceding definition of L(α, β).
  4. [Section V, Table I] The runtime for the 13041-point row (1584.62 s) is two orders of magnitude larger than the 8646-point row; if this is not a typographical error, the authors should explain the superlinear growth.
  5. [Section IV] The same symbol h is used for the state-space mesh size and for the termination threshold in Algorithm 1; this double use is confusing and should be clarified by renaming one of them.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the approximation and continuity claims are proven from the Shapley-operator eigenproblem, not reduced to their own inputs; reliance on the authors' prior work [3] is ordinary theorem use.

full rationale

The paper's central claims are Corollary III.2 (1-Lipschitz continuity of the competitive spectral radius under a common-part/positivity condition) and Theorems IV.6 and IV.8 (approximation of the value by the discretized Shapley eigenvalue and termination of RVI-KM with error bounds). Neither claim is obtained by fitting a parameter to the quantity being predicted, nor is the target quantity defined in terms of the output of the algorithm. The approximation proof derives the bounds −h+λ ≤ ρ ≤ λ directly from the nonexpansiveness and interpolation properties of S and I_h^+ (Theorem IV.6), and the algorithm bound ρ ∈ [λ_k−3h, λ_k+2h] follows from the eigenproblem residual and the Ishikawa/Baillon–Bruck estimates (Theorem IV.8). The paper does rely on the authors' earlier characterization of escape-rate games [3] for the value identities (Theorem II.2 and the dual characterization Theorem II.5), and this is a self-citation that is load-bearing in the sense that those identities are used as a starting point. However, those are prior theorems about existence and characterization of the value, not the continuity or approximability results that are new here, and the present paper does not redefine those results to make the new statements true by construction. A genuine scope issue, but not a circularity, is that the abstract and introduction say approximation is obtained 'under the same assumptions' as continuity, whereas the approximation theorems require the strictly stronger Small Cone Assumption IV.1, which the paper does not verify for the population-dynamics example and which is not implied by the common-part condition of Corollary III.2. This is an overstatement of scope or a possible correctness gap, but it is not a circular step: Theorem IV.6 is conditional on IV.1 and does not assume the conclusion ρ ∈ [−h+λ, λ]. Overall, no step of the derivation reduces by definition or by fitting to its own output.

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

No free parameters are fitted. The paper's theorems depend on standard fixed-point and subadditivity results, on the authors' prior value-characterization theorems from [3], and on two explicitly stated structural assumptions (common part and Small Cone). No new particles, fields, or entities are introduced.

assumptions (7)
  • domain assumption Value and dual characterization of escape rate games (Theorem II.2, Theorem II.5) from [3].
    Invoked in Section II.D and III to identify the competitive spectral radius with additive eigenproblems; not reproved here.
  • standard math Fekete subadditivity for the sequence s_k defining the escape rate.
    Used in Section II.D to define the value as the limit of s_k/k.
  • standard math Schauder-Tychonoff fixed point theorem.
    Used in Theorem IV.3 to produce additive eigenvectors for the Shapley operators.
  • standard math Ishikawa's convergence theorem and Baillon-Bruck error estimate for averaged nonexpansive maps.
    Used in Theorem IV.7 to guarantee convergence and termination rate of the RVI-KM iteration.
  • domain assumption Small Cone Assumption IV.1: existence of a closed cone K with T_ab(K) subset K and K∩Δ compact in relint Δ.
    This is an extra hypothesis, not implied by the part/positivity condition, needed for the discretization error and termination bounds.
  • domain assumption Compact subsets of a common part P of End(Int C), i.e., the strict positivity assumption.
    This is the hypothesis of Corollary III.2 where Lipschitz continuity is proved; it is explicitly stated and not verified for the population example.
  • domain assumption Operators T_ab extend continuously to the closed cone C.
    Needed for the dual characterization (2) in Theorem II.5; for the matrix actions considered, this is automatic when matrices map C into C.

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Pith. "Pith review of Continuity and approximability of competitive spectral radii." pith.science (2026). https://pith.science/paper/UNUYZTBP

@misc{pith2026250522468,
  author       = {Pith},
  title        = {Pith review of: Continuity and approximability of competitive spectral radii},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNUYZTBP}},
  note         = {Machine review of arXiv:2505.22468}
}
read the original abstract

The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximize the growth rate of this product, whereas the other player wishes to minimize it. We show that when the matrices represent linear operators preserving a cone and satisfying a "strict positivity" assumption, the competitive spectral radius depends continuously - and even in a Lipschitz-continuous way - on the matrix sets. Moreover, we show that the competive spectral radius can be approximated up to any accuracy. This relies on the solution of a discretized infinite dimensional non-linear eigenproblem. We illustrate the approach with an example of age-structured population dynamics.

Figures

Figures reproduced from arXiv: 2505.22468 by the authors.

Figure 1
Figure 1. Eigenfunction computed with 8646 points of discretization [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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