mlsanityData to 7 Oct 2026

Riemannian block SPD coupling manifold and its application to optimal transport

arXiv 2201.12933 ↗First posted 30 Jan 2022math.FA math.OC stat.MLcited by 7

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Abstract (arXiv)

In this work, we study the optimal transport (OT) problem between symmetric positive definite (SPD) matrix-valued measures. We formulate the above as a generalized optimal transport problem where the cost, the marginals, and the coupling are represented as block matrices and each component block is a SPD matrix. The summation of row blocks and column blocks in the coupling matrix are constrained by the given block-SPD marginals. We endow the set of such block-coupling matrices with a novel Riemannian manifold structure. This allows to exploit the versatile Riemannian optimization framework to solve generic SPD matrix-valued OT problems. We illustrate the usefulness of the proposed approach in several applications.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 29 of its 80 references that are in the map. First posted on arXiv 30 Jan 2022.

Builds on

29 of 80 references.

References · 29

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  2. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  3. 2018Computational Optimal Transportcited by 765
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  11. 2019GOT: An Optimal Transport framework for Graph comparisoncited by 64
  12. 2019Hierarchical Optimal Transport for Document Representationcited by 53
17 more
  1. 2021On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometrycited by 32
  2. 2014From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matricescited by 22
  3. 2018Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptationcited by 20
  4. 2014Riemannian preconditioningcited by 19
  5. 2018Adaptive regularization with cubics on manifoldscited by 17
  6. 2020Geometry-aware Domain Adaptation for Unsupervised Alignment of Word Embeddingscited by 14
  7. 2021SPOT: A framework for selection of prototypes using optimal transportcited by 13
  8. 2018Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometrycited by 12
  9. 2015Heterogeneous Tensor Decomposition for Clustering via Manifold Optimizationcited by 10
  10. 2021Manifold optimization for non-linear optimal transport problemscited by 8
  11. 2016Quantum Optimal Transport for Tensor Field Processingcited by 7
  12. 2020Statistical Optimal Transport posed as Learning Kernel Embeddingcited by 7
  13. 2013Matrix-valued Monge-Kantorovich Optimal Mass Transportcited by 4
  14. 2016Matrix Optimal Mass Transport: A Quantum Mechanical Approachcited by 4
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  16. 2016Vector-Valued Optimal Mass Transportcited by 3
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 1; 2024: 3; 2025: 0; 2026: 1.

7 papers.

  1. 2023A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifoldcited by 21
  2. 2022Riemannian accelerated gradient methods via extrapolationcited by 5
  3. 2024Riemannian coordinate descent algorithms on matrix manifoldscited by 4
  4. 2024Submodular Framework for Structured-Sparse Optimal Transportcited by 4
  5. 2024A Riemannian Approach to Ground Metric Learning for Optimal Transportcited by 3
  6. 2022Rieoptax: Riemannian Optimization in JAXcited by 1
  7. 2026A Riemannian Approach to Low-Rank Optimal Transportcited by 1