bibkey: bhatiajainlim2018wasserstein
authors: Rajendra Bhatia; Tanvi Jain; Yongdo Lim
year: 2018
title: Inequalities for the Wasserstein mean of positive definite matrices
doi: null
url: https://arxiv.org/abs/1803.03357
claim: "Equation (2) defines the weighted Wasserstein mean as the positive definite minimizer of the weighted sum of squared Bures–Wasserstein distances."
strata_touched:
  - D5/S3/Quantum/Matrix/HwangKimNearOrderRefutation
license: citation-only
triage: anchor

Inequalities for the Wasserstein mean of positive definite matrices

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Source: https://arxiv.org/abs/1803.03357

Source definition

Page 1, equation (2):

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

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

For admissible inputs the positive definite minimizer is unique. The Lean definition uses Classical.epsilon on the minimizer predicate, with n counting the inputs, m the matrix dimension, and Fin n indexing the inputs.