# On the Bures-Wasserstein distance between positive definite matrices

@article{Bhatia2017OnTB,
title={On the Bures-Wasserstein distance between positive definite matrices},
author={R. Bhatia and Tanvi Jain and Y. Lim},
journal={arXiv: Functional Analysis},
year={2017}
}
• Published 2017
• Mathematics
• arXiv: Functional Analysis
The metric $d(A,B)=\left[ \tr\, A+\tr\, B-2\tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several… Expand
99 Citations

#### Figures from this paper

Bures-Wasserstein Geometry.
The Bures-Wasserstein distance is a Riemannian distance on the space of positive definite Hermitian matrices and is given by: $d(\Sigma,T) = \left[\text{tr}(\Sigma) + \text{tr}(T) - 2 \text{tr}Expand On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry • Computer Science, Mathematics • ArXiv • 2021 This study comparatively analyzes the Bures-Wasserstein geometry with the popular Affine-Invariant geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold to show that the BW metric is a more suitable and robust choice for several RiemANNian optimization problems over ill-conditioned SPD matrices. Expand QUOTIENT GEOMETRY WITH SIMPLE GEODESICS FOR THE MANIFOLD OF FIXED-RANK POSITIVE-SEMIDEFINITE MATRICES\ast • 2020 This paper explores the well-known identification of the manifold of rank p positivesemidefinite matrices of size n with the quotient of the set of full-rank n-by-p matrices by the orthogonal groupExpand Some Geometric Properties of Matrix Means with respect to Different Distance Function. • Mathematics • 2019 In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures-Wasserstein, Hellinger and Log-Determinant metrics. More precisely, we show thatExpand Matrix versions of the Hellinger distance • Physics, Mathematics • 2019 On the space of positive definite matrices, we consider distance functions of the form$$d(A,B)=\left[ \mathrm{tr}\mathcal {A}(A,B)-\mathrm{tr}\mathcal {G}(A,B)\right]Expand Statistical inference for Bures–Wasserstein barycenters • Mathematics • 2019 In this work we introduce the concept of Bures-Wasserstein barycenter$Q_*$, that is essentially a Frechet mean of some distribution$\mathbb{P}$supported on a subspace of positive semi-definiteExpand Some geometric properties of matrix means with respect to different metrics • Mathematics • 2020 In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures–Wasserstein, Hellinger and log-determinant metrics. More precisely, we show thatExpand Wasserstein Riemannian Geometry of Positive Definite Matrices • Mathematics • 2018 The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we derive an explicit formExpand Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent • Computer Science, Mathematics • ArXiv • 2021 This work proves new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate and enables the analysis of two related notions of averaging, the entropicallyregularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems. Expand Optimal transport natural gradient for statistical manifolds with continuous sample space • Mathematics, Computer Science • 2018 In optimization problems, it is observed that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function, and it is proved that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. Expand #### References SHOWING 1-10 OF 34 REFERENCES A fixed-point approach to barycenters in Wasserstein space • Mathematics • 2015 Let$\mathcal{P}_{2,ac}$be the set of Borel probabilities on$\mathbb{R}^d\$ with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of findingExpand
The Riemannian Mean of Positive Matrices
Recent work in the study of the geometric mean of positive definite matrices has seen the coming together of several subjects: matrix analysis, operator theory, differential geometry (Riemannian andExpand
Riemannian geometry and matrix geometric means
• Mathematics
• 2006
The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authorsExpand
Density operators as an arena for differential geometry
What I am going to describe may be called an interplay between concepts of differential geometry and the superposition principle of quantum physics. In particular, it concerns a metrical distanceExpand
Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transportExpand
Wasserstein geometry of Gaussian measures
This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2 -Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allowsExpand
Matrix power means and the Karcher mean
• Mathematics
• 2012
We define a new family of matrix means {Pt(ω;A)}t∈[−1,1], where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these means exceptExpand
Monotonicity of the matrix geometric mean
• Mathematics
• 2012
An attractive candidate for the geometric mean of m positive definite matrices A1, . . . , Am is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in theExpand
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss'sExpand
Positive Definite Matrices
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive realExpand