mlsanityData to 7 Oct 2026

Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry

arXiv 2110.10464 ↗First posted 20 Oct 2021math.FA math.DG math.OCcited by 10

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Abstract (arXiv)

Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix M such that when M = I, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.

Key facts. Cited by 10 papers in the map (data through 7 Oct 2026). Builds on 17 of its 57 references that are in the map. First posted on arXiv 20 Oct 2021.

Builds on

17 of 57 references.

References · 17

  1. 2018Computational Optimal Transportcited by 765
  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. 2019Subspace Robust Wasserstein Distancescited by 89
  6. 2021On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometrycited by 32
  7. 2015Fast Optimal Transport Averaging of Neuroimaging Datacited by 26
  8. 2015Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matricescited by 24
  9. 2014From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matricescited by 22
  10. 2016Geometric Mean Metric Learningcited by 22
  11. 2017An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimizationcited by 20
  12. 2014Riemannian preconditioningcited by 19
5 more
  1. 2013Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximationcited by 11
  2. 2016Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognitioncited by 9
  3. 2018A Unified Framework for Domain Adaptation using Metric Learning on Manifoldscited by 8
  4. 2021Projection Robust Wasserstein Barycenterscited by 8
  5. 2014Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergencescited by 6

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 3; 2025: 1; 2026: 5.

10 papers.

  1. 2023A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifoldcited by 21
  2. 2024Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometrycited by 9
  3. 2025Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometrycited by 7
  4. 2024Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometrycited by 4
  5. 2024A Framework for Bilevel Optimization on Riemannian Manifoldscited by 3
  6. 2026Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learningcited by 1
  7. 2026Batch Normalization for Neural Networks on Complex Domainscited by 0
  8. 2026Towards Robust EEG Decoding Based on Riemannian Self-Attentioncited by 0
  9. 2026BrainLinear: A Linear Model for Brain Network Analysis in Sparse Tangent Subspacescited by 0
  10. 2026Optimization over covariance matrices with a parameterized metriccited by 0