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

arXiv: https://arxiv.org/abs/2106.00286 · HTML: https://mlsanity.com/paper/2106.00286/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 82 references that are in the map.
- First posted on arXiv 1 Jun 2021.

## Builds on (most cited first, 28 of 28)

- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Computationally Tractable Riemannian Manifolds for Graph Embeddings](https://mlsanity.com/paper/2002.08665/) (2020)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Learning Multilingual Word Embeddings in Latent Metric Space: A Geometric Approach](https://mlsanity.com/paper/1808.08773/) (2018)
- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization](https://mlsanity.com/paper/1706.03267/) (2017)
- [ROP: Matrix recovery via rank-one projections](https://mlsanity.com/paper/1310.5791/) (2013)
- [Adaptive regularization with cubics on manifolds](https://mlsanity.com/paper/1806.00065/) (2018)
- [McTorch, a manifold optimization library for deep learning](https://mlsanity.com/paper/1810.01811/) (2018)
- [Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity](https://mlsanity.com/paper/1806.06373/) (2018)
- [Wasserstein Exponential Kernels](https://mlsanity.com/paper/2002.01878/) (2020)
- [Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach](https://mlsanity.com/paper/1304.4344/) (2013)
- [Efficient Output Kernel Learning for Multiple Tasks](https://mlsanity.com/paper/1511.05706/) (2015)
- [Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition](https://mlsanity.com/paper/1608.04914/) (2016)
- [A dual framework for low-rank tensor completion](https://mlsanity.com/paper/1712.01193/) (2017)
- [A Unified Framework for Domain Adaptation using Metric Learning on Manifolds](https://mlsanity.com/paper/1804.10834/) (2018)
- [Learning Geometric Word Meta-Embeddings](https://mlsanity.com/paper/2004.09219/) (2020)
- [Efficient Robust Optimal Transport with Application to Multi-Label Classification](https://mlsanity.com/paper/2010.11852/) (2020)
- [Regularization-free estimation in trace regression with symmetric positive semidefinite matrices](https://mlsanity.com/paper/1504.06305/) (2015)
- [MixEst: An Estimation Toolbox for Mixture Models](https://mlsanity.com/paper/1507.06065/) (2015)
- [Riemannian optimization on the simplex of positive definite matrices](https://mlsanity.com/paper/1906.10436/) (2019)
- [Low-rank geometric mean metric learning](https://mlsanity.com/paper/1806.05454/) (2018)
- [Variance reduction for Riemannian non-convex optimization with batch size adaptation](https://mlsanity.com/paper/2007.01494/) (2020)
- [Learning Discriminative Stein Kernel for SPD Matrices and Its Applications](https://mlsanity.com/paper/1407.1974/) (2014)

## Built on by (most cited first, 30 of 32)

- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Convergence of policy gradient methods for finite-horizon exploratory linear-quadratic control problems](https://mlsanity.com/paper/2211.00617/) (2022)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning](https://mlsanity.com/paper/2302.09738/) (2023)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies](https://mlsanity.com/paper/2305.10411/) (2023)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [Riemannian Federated Learning via Averaging Gradient Streams](https://mlsanity.com/paper/2409.07223/) (2024)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [Manifold Gaussian Variational Bayes on the Precision Matrix](https://mlsanity.com/paper/2210.14598/) (2022)
- [Random matrix theory improved Fr'echet mean of symmetric positive definite matrices](https://mlsanity.com/paper/2405.06558/) (2024)
- [Metric Graph Kernels via the Tropical Torelli Map](https://mlsanity.com/paper/2505.12129/) (2025)
- [Gram-MMD: A Texture-Aware Metric for Image Realism Assessment](https://mlsanity.com/paper/2604.03064/) (2026)
- [Riemannian Optimization for Variance Estimation in Linear Mixed Models](https://mlsanity.com/paper/2212.09081/) (2022)
- [Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularization](https://mlsanity.com/paper/2302.11076/) (2023)
- [On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians](https://mlsanity.com/paper/2410.08117/) (2024)
- [Spectral-factorized Positive-definite Curvature Learning for NN Training](https://mlsanity.com/paper/2502.06268/) (2025)
- [Generalized infinite dimensional Alpha-Procrustes based geometries](https://mlsanity.com/paper/2511.09801/) (2025)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)

## Abstract (arXiv)

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-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.