# Riemannian block SPD coupling manifold and its application to optimal transport

arXiv: https://arxiv.org/abs/2201.12933 · HTML: https://mlsanity.com/paper/2201.12933/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 80 references that are in the map.
- First posted on arXiv 30 Jan 2022.

## Builds on (most cited first, 29 of 29)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Graph Optimal Transport for Cross-Domain Alignment](https://mlsanity.com/paper/2006.14744/) (2020)
- [GOT: An Optimal Transport framework for Graph comparison](https://mlsanity.com/paper/1906.02085/) (2019)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation](https://mlsanity.com/paper/1807.10479/) (2018)
- [Riemannian preconditioning](https://mlsanity.com/paper/1405.6055/) (2014)
- [Adaptive regularization with cubics on manifolds](https://mlsanity.com/paper/1806.00065/) (2018)
- [Geometry-aware Domain Adaptation for Unsupervised Alignment of Word Embeddings](https://mlsanity.com/paper/2004.08243/) (2020)
- [SPOT: A framework for selection of prototypes using optimal transport](https://mlsanity.com/paper/2103.10159/) (2021)
- [Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry](https://mlsanity.com/paper/1802.02628/) (2018)
- [Heterogeneous Tensor Decomposition for Clustering via Manifold Optimization](https://mlsanity.com/paper/1504.01777/) (2015)
- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [Quantum Optimal Transport for Tensor Field Processing](https://mlsanity.com/paper/1612.08731/) (2016)
- [Statistical Optimal Transport posed as Learning Kernel Embedding](https://mlsanity.com/paper/2002.03179/) (2020)
- [Matrix-valued Monge-Kantorovich Optimal Mass Transport](https://mlsanity.com/paper/1304.3931/) (2013)
- [Matrix Optimal Mass Transport: A Quantum Mechanical Approach](https://mlsanity.com/paper/1610.03041/) (2016)
- [Riemannian optimization on the simplex of positive definite matrices](https://mlsanity.com/paper/1906.10436/) (2019)
- [Vector-Valued Optimal Mass Transport](https://mlsanity.com/paper/1611.09946/) (2016)
- [Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications](https://mlsanity.com/paper/1712.10279/) (2017)

## Built on by (most cited first, 7 of 7)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Submodular Framework for Structured-Sparse Optimal Transport](https://mlsanity.com/paper/2406.04914/) (2024)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [A Riemannian Approach to Low-Rank Optimal Transport](https://mlsanity.com/paper/2606.12120/) (2026)

## 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.