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REVIEW 4 major objections 6 minor 26 references

Tight bounds of norms of Wasserstein metric matrix

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims sharply improved norm and condition-number bounds for Wasserstein-1 metric matrices, extending to Kronecker-product two-dimensional cases.

desk verdict Sharper bounds for Wasserstein-1 matrices, but the main theorem is half-proved and one part is false at n=1. read the letter →

arxiv 2506.15326 v1 pith:IFNSWAUI submitted 2025-06-18 math.RA

Aaisha Be , Nachiketa Mishra , Debasisha Mishra This is my paper · ORCID
classification math.RA MSC 15A1815A6015B0565F0565H10
keywords WassersteinmetricmatrixnormsconditionnumbersymmetricToeplitzKroneckerproductnumericalrangeHadamardpositivedefinite
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 establishes tight lower and upper bounds on the 1-, 2-, and infinity-norms of Wasserstein-1 metric matrices, their inverses, and their condition numbers, for both the one-dimensional matrix $Q$ with entries $\lambda^{|i-j|}$ and its two-dimensional Kronecker-product analogue. The key results show that the 1-norm (equivalently the infinity-norm) is bounded above by $1 + 2\lambda(1-\lambda^{n-1})/(1-\lambda)$ and the 1-condition number by $(1+\lambda)(1-\lambda+2\lambda(1-\lambda^{n-1}))/(1-\lambda)^2$, both sharper than previous estimates. It also locates eigenvalues and numerical ranges in explicit intervals and provides decompositions of $Q$ as a Hadamard product and via the Cayley transform. For the Hadamard inverse, exact norm formulas are obtained. If correct, these bounds give near-exact cost and conditioning estimates for the matrix-vector multiplications central to Sinkhorn-type optimal transport algorithms.

What carries the argument

The central object is the symmetric Toeplitz matrix $Q = (\lambda^{|i-j|})$, written as $Q = I + L + L^T$. The argument rests on four mechanisms: (1) the factorization $(I - \lambda N^T) Q (I - \lambda N) = \hat{D}_\lambda$, where $N$ is the subdiagonal nilpotent shift and $\hat{D}_\lambda$ is diagonal with entries $1-\lambda^2$ except the last entry $1$, which reduces inverse-norm and conditioning bounds to diagonal scaling; (2) Gershgorin disks and Perron-Frobenius theory for eigenvalue and spectral-radius bounds; (3) numerical radius inequalities, including the relation $\omega(I+A) = 1+\omega(A)$ applied to the nonnegative symmetric matrix $A = L+L^T$, to bound eigenvalues; and (4) the Hadamard-product representation $Q = A \circ A^T$, combined with a known spectral-norm bound for Hadamard products, to estimate the 2-norm.

What would settle it

Compute the exact 1-norm of $Q$ for a fixed $n$, say $n=4$ and $\lambda=0.5$, by summing row entries and comparing it to the claimed upper bound $1 + 2\lambda(1-\lambda^{n-1})/(1-\lambda)$; if the bound is violated for any $\lambda \in (0,1)$, Theorem 3.1(v) fails. More directly, symbolically expand $(I-\lambda N^T)Q(I-\lambda N)$ for $n=2$ and $n=3$ and check whether it equals $\hat{D}_\lambda$; a single counterexample to this identity for a diagonal entry would overturn the inverse and condition-number bounds.

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Extended reading notes

Core claim

For the one-dimensional Wasserstein-1 metric matrix $Q = (\lambda^{|i-j|})$ with $0<\lambda<1$, the paper claims that its 1-norm and infinity-norm coincide and satisfy $1/(1+\lambda)^2 \leq \|Q\|_1 = \|Q\|_\infty \leq 1 + 2\lambda(1-\lambda^{n-1})/(1-\lambda)$, and that the condition number $\kappa_1(Q)$ satisfies $(1-\lambda)/((1+\lambda)^3(1-\lambda^n)^2) \leq \kappa_1(Q) \leq (1+\lambda)(1-\lambda+2\lambda(1-\lambda^{n-1}))/(1-\lambda)^2$. These upper bounds improve on the earlier estimates $(1+\lambda)/(1-\lambda)$ and $((1+\lambda)/(1-\lambda))^2$. The same pattern is extended to the two-dimensional Kronecker product $Q = Q^{[2]} \otimes Q^{[1]}$, where the bounds multiply. The paper further asserts that eigenvalues of $Q$ lie in $(0,\, 1+2\|L\|_2 \cos(\pi/(n+1))]$ and that the numerical range is contained in $(0,\, (1+\lambda)(1-\lambda^{n-1})/(1-\lambda)]$, with corresponding two-dimensional analogues.

Load-bearing premise

The inverse and condition-number bounds rest on the factorization $(I - \lambda N^T)Q(I - \lambda N) = \hat{D}_\lambda$, which the paper cites from earlier work without proving; if that factorization is not valid for every $\lambda \in (0,1)$ and every $n$, those bounds do not follow. Additionally, the proof of the eigenvalue bound uses the numerical-radius relation $\omega(I+A) = 1+\omega(A)$ as a general fact, although it holds only when $A$ is nonnegative and symmetric, a condition that is met here by $A = L+L^T$ but is not stated.

Editorial extensions

If this is right

  • The sharper bound for $\|Q\|_1$ gives a tighter worst-case cost estimate for the matrix-vector product $Qv$ used in Sinkhorn iterations for optimal transport.
  • The condition-number bound $(\mathrm{ix})$ implies that linear systems $Qx = b$ are better behaved than earlier bounds suggested, especially for large $n$.
  • The eigenvalue and numerical-range inclusions can be used to design preconditioners or to certify convergence of iterative solvers for Wasserstein metric matrices.
  • In the two-dimensional case, the bounds scale as products of one-dimensional bounds, so sharpness carries over from $n$ to $nm$ dimensions.
  • The Hadamard decomposition $Q = A \circ A^T$ yields a cheap upper bound for the spectral norm, and exact formulas for the Hadamard inverse give closed-form norms of the entrywise reciprocal matrix.

Reading between the lines

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

  • The upper bound in Theorem 3.1(v) is not achieved by any row of $Q$; the exact 1-norm is the maximum of the row sums, which is smaller than the geometric sum over all off-diagonal distances. A natural tightening would replace the sum by the middle-row sum.
  • The inverse-norm inequalities inherit any hidden assumptions in the factorization cited from earlier work; a direct proof of $(I - \lambda N^T)Q(I - \lambda N) = \hat{D}_\lambda$ for all $\lambda \in (0,1)$ and all $n$ would put the conditioning bounds on a self-contained footing.
  • The identity $\omega(I+A) = 1+\omega(A)$ is applied without stating its conditions; it holds for nonnegative symmetric $A$, which is true for $A = L+L^T$ here, but the proof should say so explicitly to avoid an invalid general inference.
  • The Hadamard-inverse results suggest that the entrywise reciprocal of $Q$, which is cheap to compute, has norms comparable to the inverse in closed form; this could be useful in kernel methods requiring entrywise operations.
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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 / 6 minor

Summary. The paper studies the one- and two-dimensional Wasserstein-1 metric matrices Q defined by q_ij = λ^{|i-j|} (and their Kronecker-product analogues). It claims lower and upper bounds for the 1-, 2-, and ∞-norms of Q, Q^{-1}, and the associated condition numbers, asserting that these bounds are much sharper than those of Bai [2,3]. It also gives eigenvalue and numerical-range inclusion regions, determinant formulas, Hadamard-product decompositions, and numerical-radius-based condition-number estimates. The main results are Theorem 3.1 (one-dimensional case) and Theorem 3.9 (two-dimensional case), with supporting computations in Theorems 3.3 and 3.7 and numerical tables.

Significance. If the stated bounds are correct, they would provide a useful refinement of Bai's norm and conditioning estimates for a matrix class that arises in entropy-regularized optimal transport. A strength of the paper is that the bounds are explicit functions of the regularization parameter λ and the dimension n, with no fitted parameters, and several identities (determinant, Hadamard-inverse norms) are exact. However, the significance is currently undermined by the absence of proofs for the main norm and condition-number bounds in Theorem 3.1(v)-(x), by a concrete counterexample to part (vii) in the n=1 case, and by the use of an invalid general numerical-radius identity. These issues must be corrected before the claims can be accepted; they appear fixable within the scope of the manuscript.

major comments (4)
  1. [Theorem 3.1] The proof of Theorem 3.1 stops after part (iv); parts (v)-(x), which are the paper's headline norm and condition-number bounds, are asserted without any proof. Since these bounds are the main contribution, the authors must supply a complete derivation for each part, or at least reduce each bound explicitly to the factorization (4)-(6) from Bai [3] with all hypotheses stated. The two-dimensional claims in Theorem 3.9(v)-(x) inherit this gap, since their proofs appeal directly to the corresponding unproved parts of Theorem 3.1.
  2. [Theorem 3.1(vii)] Theorem 3.1(vii) is false as stated for n=1. For n=1, Q=[1], so ||Q^{-1}||_1=1, while the claimed lower bound is (1-λ)/((1+λ)(1-λ)^2)=1/(1-λ^2)>1 for every λ∈(0,1). Parts (iv) and (vi) also fail for n=1: for Q=[1] the claimed numerical-range upper bound becomes 0 and the claimed 2-norm upper bound becomes 0. The theorem must include an explicit n≥2 hypothesis, and Theorem 3.9 must include an analogous n,m≥2 hypothesis.
  3. [Theorem 3.1(ii), Section 2 (P2)] The proof of Theorem 3.1(ii) uses the identity ω(I+A)=1+ω(A), listed as property (P2) in Section 2. This identity is false in general (for example A=-I or A=[[0,1],[1,0]]). It is valid for A=L+L^T because this matrix is nonnegative and symmetric, but the manuscript does not state or justify that condition. The background property (P2) should be corrected to a qualified statement, and the proof of (ii) should mention the applicable hypothesis.
  4. [Theorem 3.1(v)] The upper bound in Theorem 3.1(v) is called 'tight' in the text preceding the theorem, but it is not tight in the natural sense. For n=5, λ=0.5, the exact ∞-norm of Q is the largest row sum, which equals 2.5, while the claimed bound gives 1+2λ(1-λ^{n-1})/(1-λ)=2.875. The exact row-sum value is easy to compute, so the paper should either give the exact maximum row sum or substantially soften the claim of tightness; as it stands, the title's 'tight bounds' claim is unsupported.
minor comments (6)
  1. [Section 2, property (P2)] Property (P2) should be replaced by a correct statement, for example that ω(I+A)=1+ω(A) holds when A is Hermitian and positive semidefinite, or more generally when A is nonnegative and symmetric; this prevents the reader from relying on a false general identity elsewhere.
  2. [Theorem 3.9 proof] In the proof of Theorem 3.9, part (i) says 'Similar to (1)', which should read 'Similar to (i)'; this is a typographical error that should be corrected.
  3. [Reference [9]] Reference [9] is incomplete: 'Application of the Wasserstein metric seismic signal, (2013), arXiv:1311.4581v1' lacks the authors' initials and a proper venue; it should be completed.
  4. [General] There are several typographical issues, including 'Wassetrstein matrix' in the sentence before Theorem 3.3. A careful proofreading pass is needed.
  5. [Theorem 3.3(i)] The bound ||Q||_2 ≤ n from the Hadamard-product argument is much weaker than the bounds already given in Theorem 3.1(vi) and is not used later; consider removing it or noting its limited role.
  6. [Remark 3.5(2)] The statement that the Cayley transform C(Q) can never be positive definite should explicitly mention the known criterion that the Cayley transform of a positive definite matrix is positive definite if and only if all eigenvalues lie in (0,1); the current wording leaves the reasoning implicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounds are explicit functions of the external parameters λ and n, and the key factorization is cited from independent work by Bai; proof gaps are correctness issues, not circular reductions.

full rationale

The paper derives norm, inverse-norm, and condition-number bounds for the Wasserstein metric matrix Q(λ) as explicit closed-form functions of the external regularization parameter λ and dimension n; no parameter is fitted to data and then renamed a prediction. The inverse-norm lower bounds in Theorem 3.1(vii)-(x) are obtained from Bai's factorization (I−λN^T)Q(I−λN)=D_hat_λ, quoted as Eqs. (3)-(6) from Bai [3]. Because Bai is a different research group and the factorization is an explicit algebraic identity, using it is ordinary external evidence, not a self-citation or a by-construction equivalence. The present authors appear only in References [25,26], which are used in the optional Remark 3.5 about Cayley transforms; that citation is not load-bearing for any main theorem. The criticisms in the skeptical review concern proof completeness and correctness rather than circularity: the proof of Theorem 3.1 stops after part (iv), so parts (v)-(x) are asserted rather than derived, and part (vii) fails for n=1. A missing proof or edge-case error does not make the claim circular, because the asserted bounds do not reduce by construction to any fitted input or self-citation chain. Therefore the circularity score is 0.

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

No free parameters or invented entities. The regularization parameter lambda is an external input from the Wasserstein kernel, and n is the matrix size. The bounds rely on Bai's factorization (external prior work), on standard matrix-analysis theorems, and on one mis-stated numerical-radius property that happens to hold in the specific application.

assumptions (7)
  • domain assumption Bai's factorization (I-lambda*N^T) * Q * (I-lambda*N) = D_hat_lambda, Eq. (4)
    Cited from Bai [3] and used for the inverse bounds in Theorem 3.1(vii)-(x). It is not re-derived; if false, those bounds do not follow.
  • ad hoc to paper Numerical radius identity omega(I+A)=1+omega(A) for A=L+L^T
    Stated in Section 2 as a general property, it fails for general A (e.g., A=-I). It is used in Theorem 3.1(ii) and holds only because L+L^T is nonnegative and symmetric; the restriction is not stated.
  • standard math Kronecker product norm multiplicativity ||A⊗B|| = ||A|| ||B||
    Used in Theorem 3.9 to reduce two-dimensional bounds to products of one-dimensional bounds.
  • standard math Numerical range formula W(A⊗B)=Co(W(A)W(B)) for normal A,B
    Used in Theorem 3.9(iv) to bound the numerical range of the two-dimensional matrix.
  • standard math Hadamard product norm bound ||A∘B||_2 ≤ r_1(A) c_1(B)
    Used in Theorem 3.3 and Theorem 3.7 for 2-norm bounds.
  • standard math Haagerup-de la Harpe bound omega(A) ≤ ||A||_2 cos(pi/(m+1)) for A^m=O
    Used in Theorem 3.1(ii) to bound the numerical radius of the nilpotent matrix L.
  • standard math Gershgorin disc theorem and Perron-Frobenius theorems for nonnegative matrices
    Used in Theorem 3.1(i) and (iii) for eigenvalue inclusion and spectral radius bounds.

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Cite this review

Pith. "Pith review of Tight bounds of norms of Wasserstein metric matrix." pith.science (2026). https://pith.science/paper/IFNSWAUI

@misc{pith2026250615326,
  author       = {Pith},
  title        = {Pith review of: Tight bounds of norms of Wasserstein metric matrix},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IFNSWAUI}},
  note         = {Machine review of arXiv:2506.15326}
}
abstract

Very recently, Bai [Linear Algebra Appl., 681:150-186, 2024 \& Appl. Math. Lett., 166:109510, 2025] studied some concrete structures, and obtained essential algebraic and computational properties of the one-dimensional, two-dimensional and generalized Wasserstein-1 metric matrices. This article further studies some algebraic and computational properties of these classes of matrices. Specifically, it provides lower and upper bounds on the $1,2,\infty$-norms of these matrices, their inverses, and their condition numbers. For the $1$ and $\infty$-norms, these upper bounds are much sharper than the existing ones established in the above-mentioned articles. These results are also illustrated using graphs, and the computation of the bounds is presented in tables for various matrix sizes. It also finds regions for the inclusion of the eigenvalues and numerical ranges of these matrices. A new decomposition of the Wasserstein matrix in terms of the Hadamard product can also be seen. Also, a decomposition of the Hadamard inverse of the Wasserstein matrix is obtained. Finally, a few bounds on the condition number are obtained using numerical radius inequalities.

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Works this paper leans on

26 extracted references · 25 canonical work pages

  1. [1]

    Arjovsky, M.; Chintala, S.; Bottou, L., Wasserstein generative adversarial networks, Proceedings of the 34th International Conference on Machine Learning, in: Proceedings of Machine Learning Research, 70 (2017), 214-223

    work page 2017

  2. [3]

    Bai, Z.-Z., On bounds for norms and conditioning of Wasserstein metric matrix , Appl. Math. Lett., 166 (2025), 109510

    work page 2025

  3. [2]

    Bai, Z.-Z., The Wasserstein metric matrix and its computational property , Linear Algebra Appl., 681 (2024), 150-186

    work page 2024

  4. [4]

    Balanchet, J.; Kang, Y.; Murthy, K., Robust Wasserstein profile inference and applications to machine learning, 56 (2019), 830-857

    work page 2019

  5. [5]

    Math., 37 (2019), 165-191

    Bhatia, R.; Jain, T.; Yangdo, L., On the Bures–Wasserstein distance between positive definite matrices, Expo. Math., 37 (2019), 165-191

    work page 2019

  6. [6]

    Bhatia, R.; Jain, T.; Yangdo, L., Inequalities for the Wasserstein mean of positive definite matrices , Linear Algebra Appl., 576 (2019), 108-123

    work page 2019

  7. [7]

    T., Numerical range of Moore-Penrose inverse matrices , Mathematics, 8 (2020), 830

    Chien, M. T., Numerical range of Moore-Penrose inverse matrices , Mathematics, 8 (2020), 830

    work page 2020

  8. [8]

    Neu- ral Inf

    Cuturi, M., Sinkhorn distances: Lightspeed computation of optimal transportation distances, Adv. Neu- ral Inf. Process. Syst., 26 (2013), 2292-2300

    work page 2013

Show all 26 references
  1. [9]

    D., Application of the Wasserstein metric seismic signal , (2013), arXiv:1311.4581v1

    Engquist, B.; Froese, B. D., Application of the Wasserstein metric seismic signal , (2013), arXiv:1311.4581v1

    2013 arXiv

  2. [10]

    E.; Rao, D

    Gustafson, K. E.; Rao, D. K. M., Numerical range: The field of values of linear operators and matrices , Springer-Verlag, New York, 1997

    1997

  3. [11]

    Haagerup, U.; Harpe, P. D. L., The numerical radius of a nilpotent operator on a Hilbert space , Proc. Amer. Math. Soc., 115 (1992), 371-379. 18

    1992

  4. [12]

    Hayoung, C.; Sejong, K.; Yongdo, L., Linearity of Cartan and Wasserstein means , Linear Algebra Appl., 681 (2024), 66-88

    2024

  5. [13]

    A.; Johnson, C

    Horn, R. A.; Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, 1985

    1985

  6. [14]

    A.; Johnson, C

    Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991

    1991

  7. [15]

    R.; Rohde, G

    Kolouri, S.; Park, S. R.; Rohde, G. K., The Radon cumulative distribution transform and its application to image classification , IEEE Trans. Image Process., 25 (2016), 920-934

    2016

  8. [16]

    R.; Thorpe, M.; Slepcev, D.; Rohde, G

    Kolouri, S.; Park, S. R.; Thorpe, M.; Slepcev, D.; Rohde, G. K., Optimal mass transport: Signal processing and machine-learning applications, IEEE Signal Process. Mag., 34 (2017), 43-59

    2017

  9. [17]

    Liao, Q.-C.; Chen, J.; Wang, Z.-H.; Bai, B., Jin, S., Wu, H., Fast Sinkhorn I: An O(N) algorithm for the Wasserstein-1 metric , Commun. Math. Sci., 20 (2022), 2053-2067

    2022

  10. [18]

    Liao, Q.-C.; Wang, Z.-H.; Chen, J.; Bai, B., Jin, S., Wu, H., Fast Sinkhorn II: Collinear triangular matrix and linear time accurate computation of optimal transport , J. Sci. Comput., 98 (2024), 1-19

    2024

  11. [19]

    Ling, H.; Okada, K., An efficient earth mover’s distance algorithm for robust histogram comparison , 29 (2007), 840-853

    2007

  12. [20]

    C.; Tsatsomeros, M

    Mondal, S.; Sivakumar, K. C.; Tsatsomeros, M. J., The Cayley transform of prevalent matrix classes, Linear Algebra Appl., 681 (2024), 1-20

    2024

  13. [21]

    J., The earth mover’s distance as a metric for image retrieval , J

    Rubner, Y.; Tomasi, C.; Guibas, L. J., The earth mover’s distance as a metric for image retrieval , J. Comput. Vis., 40 (2000), 99-121

    2000

  14. [22]

    Sinkhorn, R., Diagonal equivalence to matrices with prescribed row and column sums, Am. Math. Mon., 74 (1967), 402-405

    1967

  15. [23]

    II , Proc

    Sinkhorn, R., Diagonal equivalence to matrices with prescribed row and column sums. II , Proc. Am. Math. Soc., 45 (1974), 195-198

    1974

  16. [24]

    Graph., 34 (2015), 1-11

    Solomon, J.; De Goes, F.; Peyr´ e, G.; Cuturi, M.; Butscher, A.; Nguyen, A.; Du, T.; Guibas, L., Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains , ACM Trans. Graph., 34 (2015), 1-11

    2015

  17. [25]

    Verma, T.; Mishra, D.; Tsatsomeros, M., Cayley transform for Toeplitz and dual matrices, Linear Algebra Appl., 703 (2024), 627-644

    2024

  18. [26]

    Verma, T.; Mishra, D.; Tsatsomeros, M., Further results on Cayley transform of matrix classes , Preprint. 19

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