The log-Euclidean mean and Wasserstein near order

Abstract

Two positive definite complex matrices refute the proposed near-order comparison of the log-Euclidean and Wasserstein means.

Definition 1.1 (Near order).

$$\forall m:\mathbb{N},\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),\forall B:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),(\mathrm{NearOrder}\left(A,B\right))\iff (1\le \mathrm{geometricMean}\left(\frac{1}{2},A^{-1},B\right))$$

Formalization. D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.NearOrder (✓ std3).

Citation. Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Commentary.

Hwang and Kim, page 5, equation (3.5): “For A, B ∈ ℙ,” followed by “A ≼ B if and only if A⁻¹ # B ≥ I.” Here matrices have indices Fin m, inverse means Matrix nonsingular inverse, and geometricMean has weight 1/2. MatrixOrder is the positive semidefinite order.

Definition 1.2 (The log-Euclidean mean).

$$\forall m:\mathbb{N},\forall n:\mathbb{N},\forall w:\mathrm{Fin}\left(n\right)\to \mathbb{R},\forall A:\mathrm{Fin}\left(n\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),\mathrm{logEuclidean}\left(w,A\right)=\mathrm{cfc}\left(\mathrm{Real}.\exp ,\sum _{j:\mathrm{Fin}\left(n\right)}(w(j)\cdot \mathrm{cfc}\left(\mathrm{Real}.\log ,A(j)\right))\right)$$

Formalization. D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.logEuclidean (✓ std3).

Citation. Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Commentary.

Page 9, equation (4.10), defines “exp [∑ⁿⱼ₌₁ wⱼ log Aⱼ] =: LE(ω; 𝔸).” The finite index Fin n enumerates the n inputs; cfc applies Real.log and Real.exp to the matrices.

Definition 1.3 (The Bures–Wasserstein distance).

$$\forall m:\mathbb{N},\forall A:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),\forall B:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),\mathrm{buresWassersteinDistance}\left(A,B\right)=\mathrm{Real}.\mathrm{sqrt}(\mathrm{Complex}.\mathrm{re}(\mathrm{Matrix}.\mathrm{trace}(A+B))-2\cdot \mathrm{Complex}.\mathrm{re}(\mathrm{Matrix}.\mathrm{trace}(\mathrm{mpow}\left(\mathrm{mpow}\left(A,\frac{1}{2}\right)\cdot B\cdot \mathrm{mpow}\left(A,\frac{1}{2}\right),\frac{1}{2}\right))))$$

Formalization. D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.buresWassersteinDistance (✓ std3).

Citation. Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Commentary.

Page 4, Section 2: “where dW(A, B) = [tr(A + B) − 2 tr(A¹ᐟ² B A¹ᐟ²)¹ᐟ²]¹ᐟ² is the Bures-Wasserstein distance of A, B ∈ ℙₘ.” SpectralPower.mpow is the real spectral power. Complex traces are read through Complex.re.

Definition 1.4 (The Wasserstein barycenter).

$$\forall m:\mathbb{N},\forall n:\mathbb{N},\forall w:\mathrm{Fin}\left(n\right)\to \mathbb{R},\forall A:\mathrm{Fin}\left(n\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),\mathrm{wassersteinBarycenter}\left(w,A\right)=\mathrm{Classical}.\mathrm{epsilon}(\lambda X:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),(\mathrm{Matrix}.\mathrm{PosDef}(X))\wedge (\forall Y:\mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),(\mathrm{Matrix}.\mathrm{PosDef}(Y))\Rightarrow (\sum _{j:\mathrm{Fin}\left(n\right)}(w(j)\cdot \mathrm{buresWassersteinDistance}{\left(X,A(j)\right)}^2)\le \sum _{j:\mathrm{Fin}\left(n\right)}(w(j)\cdot \mathrm{buresWassersteinDistance}{\left(Y,A(j)\right)}^2))))$$

Formalization. D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.wassersteinBarycenter (✓ std3).

Citation. Rajendra Bhatia; Tanvi Jain; Yongdo Lim (2018). Inequalities for the Wasserstein mean of positive definite matrices. URL: https://arxiv.org/abs/1803.03357.

Commentary.

Bhatia, Jain and Lim, arXiv:1803.03357, page 1, equation (2): “Let A₁, …, Aₘ be given positive definite matrices and let w = (w₁, …, wₘ) be a vector of weights; i.e., wⱼ ≥ 0 and ∑ⱼ₌₁ᵐ wⱼ = 1. Then the (weighted) Wasserstein mean, or the Wasserstein barycentre of A₁, …, Aₘ is defined as”. Equation (2) displays Ω as the argmin of the weighted squared distances. Here n counts the inputs and m is the matrix dimension; Fin n enumerates the inputs. Classical.epsilon selects that minimizer. The formula below includes both positive definiteness and comparison with every positive definite matrix.

Definition 1.5 (The final open inequality).

$$(claim)\iff (\forall m:\mathbb{N},(1\le m)\Rightarrow (\forall n:\mathbb{N},(2\le n)\Rightarrow (\forall w:\mathrm{Fin}\left(n\right)\to \mathbb{R},(\forall j:\mathrm{Fin}\left(n\right),0<w(j))\Rightarrow ((\sum _{j:\mathrm{Fin}\left(n\right)}(w(j))=1)\Rightarrow (\forall A:\mathrm{Fin}\left(n\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),(\forall j:\mathrm{Fin}\left(n\right),\mathrm{Matrix}.\mathrm{PosDef}(A(j)))\Rightarrow (\mathrm{NearOrder}\left(\mathrm{logEuclidean}\left(w,A\right),\mathrm{wassersteinBarycenter}\left(w,A\right)\right)))))))$$

Formalization. D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.claim (✓ std3).

Citation. Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Commentary.

Hwang and Kim, arXiv:2407.07438v1, page 17, §6: “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:” The following display is “LE(ω; 𝔸) ≼ Ω(ω; 𝔸).” Here m is the complex matrix dimension, n is the number of inputs, w gives strictly positive weights summing to one, and every A j is positive definite. Fin n replaces the source indices 1,…,n. The assertion retains all dimensions m ≥ 1 and all input counts n ≥ 2.

Theorem 1.6 (A counterexample in dimension two).

$$\neg (\forall m:\mathbb{N},(1\le m)\Rightarrow (\forall n:\mathbb{N},(2\le n)\Rightarrow (\forall w:\mathrm{Fin}\left(n\right)\to \mathbb{R},(\forall j:\mathrm{Fin}\left(n\right),0<w(j))\Rightarrow ((\sum _{j:\mathrm{Fin}\left(n\right)}(w(j))=1)\Rightarrow (\forall A:\mathrm{Fin}\left(n\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(m\right),\mathrm{Fin}\left(m\right),\mathbb{C}\right),(\forall j:\mathrm{Fin}\left(n\right),\mathrm{Matrix}.\mathrm{PosDef}(A(j)))\Rightarrow (\mathrm{NearOrder}\left(\mathrm{logEuclidean}\left(w,A\right),\mathrm{wassersteinBarycenter}\left(w,A\right)\right)))))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.result (✓ std3). ∎

Resolves. Problems/hwang-kim-2024-near-order-log-euclidean-wasserstein (refuted) by D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.result.

Source. Repository-derived.

Acknowledgement. Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Commentary.

Take A = diag(4,1) and B = [[19,−3√15],[−3√15,61]]/64, at weights (1/2,1/2). Both matrices are positive definite. Let L be their log-Euclidean mean and W their Wasserstein barycenter. Their explicit formulas give det L = 1 and det W = d² with d > 0. For C = geometricMean(1/2,L⁻¹,W), positivity and CLC = W imply det C = d. If I ≤ C, positive semidefiniteness of C−I forces tr C ≤ 1+d. Cayley–Hamilton trace identities then force (tr W+d tr L)² ≤ (1+d)²(tr(LW)+2d), whereas rational bounds for √√2 and √265 make the difference greater than 3/2500. The barycenter identity follows by minimizing the actual sum of squared distances, using a trace-norm inequality with its equality condition. The separate entrywise-eigenvalue question is not answered by this theorem.

References

• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.NearOrder
• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.buresWassersteinDistance
• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.claim
• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.logEuclidean
• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.result
• Truth anchor: D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation.wassersteinBarycenter
• Dependency: D5/S0/Observation/MatrixTracePowerSum
• Dependency: D5/S3/Quantum/Dynamics/EntropyProductionCoherenceDeletionIdentity
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow
• Dependency: D5/S3/Quantum/Foundation/FiniteTraceDistance
• Dependency: D5/S3/Quantum/Hiai/PsiMeanTraceConcavity
• Dependency: D5/S3/Quantum/Matrix/AlternativeMeanLinearization
• Dependency: D5/S3/Quantum/Matrix/GanKimConjectureRefutation
• Dependency: D5/S3/Quantum/Matrix/SpectralPower