bibkey: hwangkim2024nearorder
authors: Jinmi Hwang; Sejong Kim
year: 2024
title: Near-order relation of power means
doi: 10.1007/s11117-024-01099-4
url: https://arxiv.org/abs/2407.07438v1
claim: "The final unnumbered question in Section 6 asks whether LE(ω; 𝔸) ≼ Ω(ω; 𝔸) for positive definite matrices and strictly positive probability weights."
strata_touched:
  - D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation
license: citation-only
triage: anchor

Near-order relation of power means

Verified locator

DOI: 10.1007/s11117-024-01099-4

Source: https://arxiv.org/abs/2407.07438v1

The source version is arXiv:2407.07438v1. The DOI identifies the journal publication in Positivity 29, article 3. Agreement of the journal body with the arXiv question is ASSUMED-UNVERIFIED; the statement below is read from v1.

Source statement

Page 17, Section 6, final paragraph:

It would be an interesting problem to study whether the weak log-majorization relation generalizes to the near-order or entrywise eigenvalue relation. Note that (6.21) is a generalization of (4.14). From this point of view, we give an open question:

$$\mathrm{LE}(\omega ;\mathbb{A})⪯\Omega (\omega ;\mathbb{A}).$$

This display is unnumbered. Equation (6.21) is the harmonic–log-Euclidean–arithmetic comparison. The near-order question is distinct from the entrywise eigenvalue question in the preceding sentence.

Page 5, equation (3.5): “For $A,B\in \mathbb{P}$,” followed by “$A⪯B$ if and only if $A^{-1}\#B\ge I$.” Page 9, equation (4.10), defines $\exp ({\sum }_{j=1}^n{w}_j\log A_j)=:\mathrm{LE}(\omega ;\mathbb{A})$. The inputs are complex positive definite matrices and the weights are strictly positive and sum to one. Page 4, Section 2: “where

$$d_W(A,B)={\left[\mathrm{tr}(A+B)-2\mathrm{tr}(A^{1/2}B{A}^{1/2}{)}^{1/2}\right]}^{1/2}.$$

is the Bures-Wasserstein distance of $A,B\in {\mathbb{P}}_m$.”

The multivariable least-squares definition used for $\Omega$ is equation (2) of Bhatia, Jain and Lim, Inequalities for the Wasserstein mean of positive definite matrices, https://arxiv.org/abs/1803.03357. The positive definite minimizer is unique for admissible inputs; the Lean definition uses Classical.epsilon on the literal minimizer predicate. On page 1 the source states:

Let $A_1,\dots ,A_m$ be given positive definite matrices and let $w=(w_1,\dots ,w_m)$ be a vector of weights; i.e., $w_j\ge 0$ and ${\sum }_{j=1}^m{w}_j=1.$ Then the (weighted) Wasserstein mean, or the Wasserstein barycentre of $A_1,\dots ,A_m$ is defined as

$$\Omega (w;A_1,\dots ,A_m)=\underset{X\in \mathbb{P}}{\mathrm{argmin}}\sum _{j=1}^m{w}_j{d}^2(X,A_j).$$

The Lean encoding uses $n$ for the number of inputs, $m$ for the matrix dimension, and Fin n for the input indices.

Scope

The refutation uses two positive definite $2\times 2$ matrices and equal weights. It disproves the universal near-order comparison. It does not refute weak log-majorization or settle the separate entrywise-eigenvalue question. Completeness of the literature screen and priority beyond the sources read on issue #21282 are ASSUMED-UNVERIFIED.