# On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry

@article{Han2021OnRO, title={On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry}, author={Andi Han and Bamdev Mishra and Pratik Jawanpuria and Junbin Gao}, journal={ArXiv}, year={2021}, volume={abs/2106.00286} }

In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill…

## Figures and Tables from this paper

## 2 Citations

Generalized Bures-Wasserstein Geometry for Positive Definite Matrices

- Mathematics
- 2021

This paper proposes a generalized Bures-Wasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three…

Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses

- Mathematics
- 2021

This paper proposes a novel general framework of Riemannian conjugate gradient methods, that is, conjugate gradient methods on Riemannian manifolds. The conjugate gradient methods are important…

## References

SHOWING 1-10 OF 88 REFERENCES

Generalized Bures-Wasserstein Geometry for Positive Definite Matrices

- Mathematics
- 2021

This paper proposes a generalized Bures-Wasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three…

Conic Geometric Optimization on the Manifold of Positive Definite Matrices

- Mathematics, Computer ScienceSIAM J. Optim.
- 2015

This work develops theoretical tools to recognize and generate g-convex functions as well as cone theoretic fixed-point optimization algorithms and illustrates their techniques by applying them to maximum-likelihood parameter estimation for elliptically contoured distributions.

On the Bures–Wasserstein distance between positive definite matrices

- MathematicsExpositiones Mathematicae
- 2019

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…

Wasserstein Riemannian geometry of Gaussian densities

- MathematicsInformation Geometry
- 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 present an explicit…

Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds

- Computer Science, MathematicsNIPS
- 2016

This paper introduces Riemannian SVRG (RSVRG), a new variance reduced RiemANNian optimization method, and presents the first non-asymptotic complexity analysis (novel even for the batch setting) for nonconvex Riem Mannian optimization.

Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

- Computer Science, Mathematics2013 IEEE Conference on Computer Vision and Pattern Recognition
- 2013

To encode the geometry of the manifold in the mapping, a family of provably positive definite kernels on the Riemannian manifold of SPD matrices is introduced, derived from the Gaussian kernel, but exploit different metrics on the manifold.

A new metric on the manifold of kernel matrices with application to matrix geometric means

- Mathematics, Computer ScienceNIPS
- 2012

A new metric on spd matrices is introduced, which not only respects non-Euclidean geometry but also offers faster computation than δR while being less complicated to use.

Manifold-valued image processing with SPD matrices

- Mathematics
- 2020

Abstract Symmetric positive definite (SPD) matrices are geometric data that appear in many applications. They are used in particular in Diffusion Tensor Imaging (DTI) as a simple model of the…

A Riemannian Framework for Tensor Computing

- Mathematics, Computer ScienceInternational Journal of Computer Vision
- 2005

This paper proposes to endow the tensor space with an affine-invariant Riemannian metric and demonstrates that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries, the geodesic between two tensors and the mean of a set of tensors are uniquely defined.

Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces

- Computer Science, MathematicsNIPS
- 2014

This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the…