Tensor product and Hadamard product for the Wasserstein means

Jinmi Hwang; Sejong Kim

REVIEW 2 major objections 4 minor 18 references

Tensor product and Hadamard product for the Wasserstein means

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Wasserstein mean of positive definite matrices satisfies an exact tensor-product identity: the mean of a tensor-product family is the tensor product of the means of the factor families.

desk verdict Main tensor identity is correct and cleanly proved; the real gap is an unpublished cited bound (Theorem 3.1) that some later inequalities depend on. read the letter →

arxiv 1908.09261 v1 pith:BPC6XACR submitted 2019-08-25 math.FA

Jinmi Hwang , Sejong Kim This is my paper · ORCID

classification math.FA MSC 15B4815A69

keywords WassersteinmeanBuresdistancepositivedefinitematricestensorproductHadamardlinearmapleastsquaresdeterminantalinequality

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 Wasserstein (Bures) mean of positive definite Hermitian matrices, defined as the unique least-squares barycenter for the Wasserstein distance between Gaussian covariances. It establishes that this mean obeys an exact composition rule: the mean of a tensor-product family is the tensor product of the means of the factor families. It also proves a determinantal inequality with a sharp equality condition, lower bounds for the mean under strictly positive unital linear maps, and an upper bound for the Hadamard product of two Wasserstein means in terms of the weighted arithmetic mean of entrywise products. These results matter because they let Wasserstein barycenters of product systems be computed by averaging subsystems separately.

What carries the argument

The load-bearing object is the nonlinear fixed-point characterization of the Wasserstein mean: $\Omega(\omega;A)$ is the unique positive definite $X$ satisfying $I = \sum_j w_j (A_j \# X^{-1})$, equivalently $X = \sum_j w_j (X^{1/2} A_j X^{1/2})^{1/2}$, where $\#$ is the two-variable geometric mean $A\#B = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$. This equation turns every mean identity into an algebraic substitution: the tensor identity follows by substituting the two factor equations and using $(A\otimes B)^t = A^t\otimes B^t$; the linear-map inequalities follow by applying $\Phi$ to the same equation and bounding the geometric mean by the arithmetic mean; the Hadamard result follows by passing the tensor equation through a linear map that sends $A\otimes B$ to $A\circ B$.

What would settle it

Compare the two sides of the tensor identity numerically for non-commuting $2\times2$ positive definite matrices and unequal weights by solving $X=\sum_j w_j(X^{1/2}A_jX^{1/2})^{1/2}$ for each side; any positive definite instance where the two matrices differ would refute the paper's central claim.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that for positive definite matrices $A_i, B_j$ and probability weights $\omega, \mu$, the identity $\Omega(\omega;A) \otimes \Omega(\mu;B) = \Omega(\omega\otimes\mu; A_1\otimes B_1, \ldots, A_n\otimes B_n)$ holds. In words, averaging the two factor families separately and then forming a tensor product gives the same matrix as averaging all pairwise tensor products against the product weights. The same mechanism yields, for the Hadamard product, the inequality $\Omega(\omega;A) \circ \Omega(\mu;B) \le \sum_{i,j} \omega_i\mu_j (A_i\circ B_j)$, and for any strictly positive unital linear map $\Phi$ the lower bounds $\Phi(\Omega(\omega;A)) \ge 2I - \sum_j w_j \Phi(A_j^{-1})$ and $\Phi(\Omega(\omega;A)^{-1}) \ge 2I - \sum_j w_j \Phi(A_j)$.

Load-bearing premise

Everything in Sections 3 and 4 rests on the theorem that the Wasserstein mean is the unique positive definite matrix solving $I=\sum_j w_j(A_j\#X^{-1})$; if that characterization were wrong, the tensor identity and the inequalities would not follow.

Editorial extensions

If this is right

For product systems, the Wasserstein mean of all pairwise tensor products collapses to the tensor product of the two marginal means, so the barycenter of a joint Gaussian model can be built from separately averaged subsystems.
The determinantal inequality $\det\Omega(\omega;A)\ge\prod_j(\det A_j)^{w_j}$ gives a volume lower bound for the mean covariance, with equality exactly when all input covariances coincide.
The linear-map inequalities give two-sided bounds on $\Phi(\Omega(\omega;A))$ that depend only on arithmetic means of the transformed inputs, so the mean's image under coarse graining is controlled without solving the fixed-point equation.
The Hadamard inequality $\Omega(\omega;A)\circ\Omega(\mu;B)\le\sum_{i,j}\omega_i\mu_j(A_i\circ B_j)$ gives a computable entrywise upper bound for the product of two means.
Under the spectral conditions and contraction assumptions, the paper's final result turns that bound around into a lower bound on the weighted average of $(A_i\circ B_j)^{1/2}$.

Reading between the lines

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

The tensor identity is the kind of law that would let a Wasserstein barycenter of a joint Gaussian model be assembled from separately averaged subsystems; the paper proves it for Gaussian covariance matrices, and the same question is natural for Wasserstein barycenters of arbitrary measures.
The proof route suggests that any mean satisfying a fixed-point equation $X=\sum_j w_j f(X,A_j)$ with $f$ compatible with tensor powers will inherit a tensor identity; testing this family of means could transfer the result beyond the Wasserstein case.
The paper leaves open whether $\Phi(\Omega(\omega;A))\le\Omega(\omega;\Phi(A_1),\ldots,\Phi(A_n))$ for positive unital maps; this could be probed numerically on random $2\times2$ examples with maps such as $\Phi(A)=A\circ I$ or unitary conjugation followed by compression to a subsystem.

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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

Here is my take on arXiv:1908.09261. The thing to know first: the tensor product identity (Theorem 4.2) is sound. The proof is short and correct: assuming the fixed-point characterization from Bhatia–Jain–Lim, you substitute X⊗Y into the equation for the product list, the algebra works because of the mixed-product rule and (A⊗B)^{1/2}=A^{1/2}⊗B^{1/2}, and uniqueness gives the result. I checked the line and it is not a gap. The determinant inequality in Theorem 2.4 is also worth a second look; the proof from concavity of log-det is neat, and the equality condition is new, even though the inequality itself is weaker than the log-majorization already known. The paper is honest about that.

The soft spot is real but localized: Theorem 3.1 asserts a lower bound 2I - Σ w_j A_j^{-1} ≤ Ω(ω;A) and cites it to the authors' own unpublished manuscript [10] ('in preparation'). No proof is given. Theorem 3.3 and Remark 3.4 lean directly on that bound. Proposition 4.9, one of the Hadamard results, also invokes Lemma 2.4 from [15] (a 'to appear' paper), which is less alarming but still a self-citation to an inaccessible source. The central tensor result does not depend on these; the Hadamard-product estimates partly do. So the right fix is to make the missing material available: either prove Theorem 3.1 in this paper or replace the citation.

Citation pattern: the authors cite their own work several times, but that only becomes a problem when the cited result is unpublished. The rest of the references look standard.

Who this is for: anyone working on Bures–Wasserstein means of positive definite matrices, especially in quantum information or optimal transport, will find the tensor identity a convenient tool. It is a modest but useful paper, not a breakthrough. It deserves a serious referee: the main identity is correct, the supporting material is partially inaccessible, and a referee can pin down exactly what needs to be supplied.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the Bures–Wasserstein mean Ω(ω; A) of positive definite Hermitian matrices, defined as the minimizer of weighted squared Wasserstein distances. It presents four groups of results: (i) a determinantal inequality det Ω(ω;A) ≥ ∏ (det A_j)^{w_j} with an equality condition (Theorem 2.4); (ii) inequalities involving strictly positive unital linear maps (Theorem 3.3 and Remark 3.4) that are derived from a two-sided bound (Theorem 3.1) attributed to the authors' unpublished work [10]; (iii) the tensor-product identity Ω(ω;A) ⊗ Ω(μ;B) = Ω(ω⊗μ; A_i⊗B_j) (Theorem 4.2), proved by substituting the fixed-point equation of Theorem 2.1; and (iv) Hadamard-product estimates (Theorem 4.5, Proposition 4.6, Proposition 4.9, Theorem 4.11) obtained by combining the tensor results with the Schur-product map and several lemmas from [14], [15], and [18].

Significance. The tensor-product identity is the strongest and most attractive result of the paper. Its proof is short and correct assuming Theorem 2.1: the authors verify that X ⊗ Y satisfies the defining fixed-point equation for the tensor list with weights ω⊗μ, and then use uniqueness of the solution. The determinantal inequality proof is also self-contained and gives a clean equality condition. Notably, Theorem 4.2 does not depend on the disputed Theorem 3.1; it follows entirely from Theorem 2.1 and Lemma 4.1. If Theorem 3.1's lower bound can be established or properly referenced, the paper would provide a coherent set of composition rules and order inequalities for the Wasserstein mean. The main caveat is external dependence: Section 3 and part of Section 4 rely on [10] (in preparation) and [15] (to appear), so the manuscript is not fully verifiable without those sources.

major comments (2)

[§3, Theorem 3.1] The lower bound 2I − Σ_{j=1}^n w_j A_j^{-1} ≤ Ω(ω;A) is stated as Theorem 3.1 and cited only to [10], which the reference list describes as 'in preparation'. This bound is load-bearing: the first inequality in Theorem 3.3 is obtained by applying it under a positive unital linear map, and Remark 3.4 relies on it as well. Because no proof is supplied, these Section 3 inequalities are not verifiable from the manuscript. Please include a proof of the lower bound, or cite a published or otherwise publicly available source; if this is not possible, the affected results should be presented as conditional on [10].
[§4, Proposition 4.9] The proof of Proposition 4.9 invokes Lemma 2.4 of [15] to pass from α_i I ≤ A_i ≤ β_i I to α I ≤ X ≤ β I (and similarly for Y), and it invokes Lemma 3.1 of [14] for the Hadamard-product inequality for the geometric mean. Reference [15] is listed as 'to appear', and Lemma 2.4 is not stated anywhere in the present paper. The reader therefore cannot check the two main inequalities in the proof. Please state these lemmas with proofs, or verify that both references are in print and accessible.

minor comments (4)

[§2, Theorem 2.4] The equality analysis jumps from A_i#X^{-1} = A_j#X^{-1} to A_i = A_j; this uses the injectivity of the map A ↦ A#B for fixed B and should be justified explicitly.
[§4, Theorem 4.2] The displayed list of tensor products contains typesetting artifacts (the 'bracehtip' tokens); writing the list cleanly as (A_i ⊗ B_j)_{1≤i,j≤n} would remove ambiguity.
[§4, Proposition 4.6] The sentence beginning 'It reduces to' is algebraically opaque, and the displayed inequality appears to involve a different factor from the preceding bound; the simplification should be expanded or corrected.
[References] References [5], [6], and [15] are marked 'to appear'; please update them to final publication data if available.

Circularity Check

2 steps flagged · score 4.0 of 10

The tensor-product identity is independently derived from the fixed-point equation; Section 3 and Hadamard results depend on unpublished self-citations.

self citation load bearing [Section 3, Theorem 3.1 and proof of Theorem 3.3; reference [10]]
"Theorem 3.1. [10] The Wasserstein mean Ω(ω;A) satisfies the following inequalities: 2I − Σ w_j A_j^{-1} ≤ Ω(ω;A) ≤ Σ w_j A_j. ... By Theorem 3.1 and the positive unital linear map Φ, ..."

The first advertised linear-map inequality is just Φ applied to the lower bound of Theorem 3.1. Theorem 3.1 is not proved in this paper; it is cited to [10], an 'in preparation' manuscript by the same two authors. Consequently the Section 3 inequalities, Corollary 4.3, and the Hadamard inequality in Theorem 4.5 are supported by a self-citation to an unavailable source rather than by an independent proof.
self citation load bearing [Section 4, proof of Proposition 4.9; references [14] and [15]]
"The second equality follows from the linearity of Hadamard product, and the first inequality follows from Lemma 3.1 in [14]. ... Indeed, α_iI ≤ A_i ≤ β_iI implies αI ≤ X ≤ βI by Lemma 2.4 in [15]."

The Hadamard estimates in Proposition 4.9 and Theorem 4.11 rely on Lemma 3.1 of [14] and Lemma 2.4 of [15], both by H. Lee and S. Kim, the current second author. These lemmas are quoted without proof, and the results depending on them are not independently established in this paper. This raises the self-citation burden, though it does not infect Theorem 4.2.

full rationale

The paper's main tensor identity, Theorem 4.2, is not circular. The proof sets X=Ω(ω;A), Y=Ω(μ;B), substitutes the fixed-point characterization of Theorem 2.1 (which is taken from [6], an external source), and uses the tensor-product rules (A⊗B)(C⊗D)=AC⊗BD and (A⊗B)^t=A^t⊗B^t to show X⊗Y satisfies the characterizing equation for the tensor list with weights ω⊗μ; uniqueness then gives the identity. This is a direct derivation, not a definitional identity or a fitted-parameter prediction. Theorem 2.4's determinant inequality is likewise proved from the fixed-point equation and concavity of log det, with no self-citation load. The circularity concerns are confined to Theorem 3.1, which is cited to the authors' own in-preparation manuscript [10], and to Proposition 4.9, which imports lemmas from [14] and [15] by the same second author. These are load-bearing for the linear-map inequalities and the Hadamard estimates, but the central tensor-product result and its proof remain independent. Therefore the paper does not involve construction-forcing circularity in its main theorem; the self-citation burden warrants a moderate score rather than zero.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. It relies on standard matrix analysis background and on several cited results, two of which are by the same authors and one of which is unpublished (the lower bound in Theorem 3.1).

assumptions (6)

domain assumption The Wasserstein mean Ω(ω;A) is the unique positive definite solution of I = Σ w_j (A_j # X^{-1}) (Theorem 2.1 of [6]).
Used in every proof in Sections 3 and 4; the tensor identity follows by substituting this equation.
standard math Strict concavity of log det on positive definite matrices.
Used in the proof of Theorem 2.4 for the determinantal inequality and its equality condition.
standard math Properties of the two-variable geometric mean (G1)-(G7), including monotonicity and joint homogeneity.
Used throughout, e.g., in Theorem 3.3 and Proposition 4.9.
domain assumption Ando's inequality Φ(A#B) ≤ Φ(A)#Φ(B) for positive linear maps (Lemma 3.2 from [4]).
Used in the proof of the second inequality in Theorem 3.3.
domain assumption The bounds 2I − Σ w_j A_j^{-1} ≤ Ω(ω;A) ≤ Σ w_j A_j (Theorem 3.1), with the lower bound cited to the unpublished paper [10].
Used for the first inequality in Theorem 3.3 and for Corollary 4.3 (only the upper bound is needed there, which is known from [6]).
domain assumption The Hadamard product inverse inequality of Lemma 4.7 from [18] and Lemma 3.1 of [14].
Used in the proofs of Proposition 4.9 and Theorem 4.11.

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Cite this review

Pith. "Pith review of Tensor product and Hadamard product for the Wasserstein means." pith.science (2026). https://pith.science/paper/BPC6XACR

@misc{pith2026190809261,
  author       = {Pith},
  title        = {Pith review of: Tensor product and Hadamard product for the Wasserstein means},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPC6XACR}},
  note         = {Machine review of arXiv:1908.09261}
}

read the original abstract

As one of the least squares mean, we consider the Wasserstein mean of positive definite Hermitian matrices. We verify in this paper the inequalities of the Wasserstein mean related with a strictly positive and unital linear map, the identity of the Wasserstein mean for tensor product, and some inequalities of the Wasserstein mean for Hadamard product.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[10]

Hwang and S

J. Hwang and S. Kim, Bounds for the Wasserstein mean with applications to the Lie-Trotter mean, in preparation

work page
[15]

Lee and S

H. Lee and S. Kim, Inequalities of the Wasserstein mean w ith other matrix means, Annals of Functional Analysis, to appear

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Lee and S

H. Lee and S. Kim, The Hadamard product for the weighted K archer means, Linear Algebra Appl. 501 (2016), 290-303

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[18]

Zhang, Matrix Theory: Basic Results and Techniques, 2nd edition, Springer, 2011

F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd edition, Springer, 2011. 14 JINMI HW ANG AND SEJONG KIM Jinmi Hw ang, Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea E-mail address : jinmi0401@chungbuk.ac.kr Sejong Kim, Department of Mathematics, Chungbuk National U niversity, Cheongju 28644, Korea E-mail address ...

work page 2011
[1]

Agueh and G

M. Agueh and G. Carlier, Barycenters in the Wasserstein s pace, SIAM J. Math. Anal. Appl. 43 (2011), 904-924

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Hwang and S

J. Hwang and S. Kim, Bounds for the Wasserstein mean with applications to the Lie-Trotter mean, in preparation

work page

[2] [15]

Lee and S

H. Lee and S. Kim, Inequalities of the Wasserstein mean w ith other matrix means, Annals of Functional Analysis, to appear

work page

[3] [14]

Lee and S

H. Lee and S. Kim, The Hadamard product for the weighted K archer means, Linear Algebra Appl. 501 (2016), 290-303

work page 2016

[4] [18]

Zhang, Matrix Theory: Basic Results and Techniques, 2nd edition, Springer, 2011

F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd edition, Springer, 2011. 14 JINMI HW ANG AND SEJONG KIM Jinmi Hw ang, Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea E-mail address : jinmi0401@chungbuk.ac.kr Sejong Kim, Department of Mathematics, Chungbuk National U niversity, Cheongju 28644, Korea E-mail address ...

work page 2011

[5] [1]

Agueh and G

M. Agueh and G. Carlier, Barycenters in the Wasserstein s pace, SIAM J. Math. Anal. Appl. 43 (2011), 904-924

work page 2011

[6] [2]

P. C. Alvarez-Esteban, E. del Barrio, J. A. Cuesta-Alber tos and C. Matran, A ﬁxed point approach to barycenters in Wasserstein spaces, J. Math. Anal. Appl. 441 (2016), 744-762

work page 2016

[7] [3]

Ando, Concavity of certain maps on positive deﬁnite ma trices and applications to Hadamard prod- ucts, Linear Algebra Appl

T. Ando, Concavity of certain maps on positive deﬁnite ma trices and applications to Hadamard prod- ucts, Linear Algebra Appl. 26 (1979), 203-241

work page 1979

[8] [4]

Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, 2007

R. Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, 2007

work page 2007