mlsanityData to 7 Oct 2026

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

arXiv 2106.00286 ↗First posted 1 Jun 2021math.OC cs.LGcited by 32

Not on the map. Why?

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.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). Builds on 28 of its 82 references that are in the map. First posted on arXiv 1 Jun 2021.

Builds on

28 of 82 references.

References · 28

  1. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  2. 2013Manopt, a Matlab toolbox for optimization on manifoldscited by 112
  3. 2016Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiationcited by 93
  4. 2016First-order Methods for Geodesically Convex Optimizationcited by 92
  5. 2016Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifoldscited by 64
  6. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  7. 2014From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matricescited by 22
  8. 2018Learning Multilingual Word Embeddings in Latent Metric Space: A Geometric Approachcited by 22
  9. 2014Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matricescited by 20
  10. 2017An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimizationcited by 20
  11. 2013ROP: Matrix recovery via rank-one projectionscited by 18
  12. 2018Adaptive regularization with cubics on manifoldscited by 17
16 more
  1. 2018McTorch, a manifold optimization library for deep learningcited by 16
  2. 2018Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexitycited by 13
  3. 2020Wasserstein Exponential Kernelscited by 10
  4. 2013Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approachcited by 9
  5. 2015Efficient Output Kernel Learning for Multiple Taskscited by 9
  6. 2016Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognitioncited by 9
  7. 2017A dual framework for low-rank tensor completioncited by 9
  8. 2018A Unified Framework for Domain Adaptation using Metric Learning on Manifoldscited by 8
  9. 2020Learning Geometric Word Meta-Embeddingscited by 6
  10. 2020Efficient Robust Optimal Transport with Application to Multi-Label Classificationcited by 6
  11. 2015Regularization-free estimation in trace regression with symmetric positive semidefinite matricescited by 4
  12. 2019Riemannian optimization on the simplex of positive definite matricescited by 4
  13. 2015MixEst: An Estimation Toolbox for Mixture Modelscited by 3
  14. 2018Low-rank geometric mean metric learningcited by 2
  15. 2020Variance reduction for Riemannian non-convex optimization with batch size adaptationcited by 2
  16. 2014Learning Discriminative Stein Kernel for SPD Matrices and Its Applicationscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 9; 2023: 7; 2024: 6; 2025: 4; 2026: 5.

32 papers.

Sort:

  1. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  2. 2023Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Spacecited by 36
  3. 2023A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifoldcited by 21
  4. 2022Convergence of policy gradient methods for finite-horizon exploratory linear-quadratic control problemscited by 19
  5. 2022Riemannian Hamiltonian methods for min-max optimization on manifoldscited by 12
  6. 2022Differentially private Riemannian optimizationcited by 12
  7. 2023Riemannian Multinomial Logistics Regression for SPD Neural Networkscited by 11
  8. 2021Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometrycited by 10
  9. 2022First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spacescited by 10
  10. 2023Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learningcited by 10
  11. 2024Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometrycited by 9
  12. 2023Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policiescited by 8
20 more
  1. 2022Riemannian block SPD coupling manifold and its application to optimal transportcited by 7
  2. 2025Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometrycited by 7
  3. 2023The Fisher-Rao geometry of CES distributionscited by 5
  4. 2024Riemannian coordinate descent algorithms on matrix manifoldscited by 4
  5. 2024Riemannian Federated Learning via Averaging Gradient Streamscited by 4
  6. 2024A Framework for Bilevel Optimization on Riemannian Manifoldscited by 3
  7. 2022Rieoptax: Riemannian Optimization in JAXcited by 1
  8. 2022Manifold Gaussian Variational Bayes on the Precision Matrixcited by 1
  9. 2024Random matrix theory improved Fr'echet mean of symmetric positive definite matricescited by 1
  10. 2025Metric Graph Kernels via the Tropical Torelli Mapcited by 1
  11. 2022Riemannian Optimization for Variance Estimation in Linear Mixed Modelscited by 0
  12. 2023Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularizationcited by 0
  13. 2024On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussianscited by 0
  14. 2025Spectral-factorized Positive-definite Curvature Learning for NN Trainingcited by 0
  15. 2025Generalized infinite dimensional Alpha-Procrustes based geometriescited by 0
  16. 2026Gram-MMD: A Texture-Aware Metric for Image Realism Assessmentcited by 0
  17. 2026Fr'echet Regression on the Bures-Wasserstein Manifoldcited by 0
  18. 2026Towards Robust EEG Decoding Based on Riemannian Self-Attentioncited by 0
  19. 2026ITSPACE: Monotone Gaussian Optimal Transport Updatescited by 0
  20. 2026Optimization over covariance matrices with a parameterized metriccited by 0