mlsanityData to 14 Oct 2026

Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space

arXiv 2310.11256 ↗First posted 17 Oct 2023stat.ML cs.CV cs.LGcited by 4

Not on the map. Why?

Abstract (arXiv)

The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a novel Wasserstein distance specifically tailored for Gaussian mixture models (GMMs) and known as MW2 (mixture Wasserstein) has been introduced by several authors. In scenarios where data exhibit clustering, this approach simplifies to a small-scale discrete optimal transport problem, which complexity depends solely on the number of Gaussian components in the GMMs. This paper aims to incorporate invariance properties into MW2. This is done by introducing new Gromov-type distances, designed to be isometry-invariant in Euclidean spaces and applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance (MGW2), which can be viewed as a "Gromovized" version of MW2. This new distance has a straightforward discrete formulation, making it highly efficient for estimating distances between GMMs in practical applications. To facilitate the derivation of a transport plan between GMMs, we present a second distance, the Embedded Wasserstein distance (EW2). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that EW2 can be adapted to derive a distance as well as optimal transportation plans between GMMs. We demonstrate the efficiency of these newly proposed distances on medium to large-scale problems, including shape matching and hyperspectral image color transfer.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 34 of its 75 references that are in the map. First posted on arXiv 17 Oct 2023.

Builds on

34 of 75 references.

References · 34

  1. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  2. 2015Automatic differentiation in machine learning: a surveycited by 855
  3. 2015Optimal Transport for Domain Adaptationcited by 463
  4. 2017Wasserstein Auto-Encoderscited by 373
  5. 2017Learning Generative Models with Sinkhorn Divergencescited by 314
  6. 2019Generalized Sliced Wasserstein Distancescited by 208
  7. 2018Optimal Transport for structured data with application on graphscited by 172
  8. 2017Large-Scale Optimal Transport and Mapping Estimationcited by 123
  9. 2018Gromov-Wasserstein Alignment of Word Embedding Spacescited by 122
  10. 2019Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matchingcited by 103
  11. 2022Variational inference via Wasserstein gradient flowscited by 85
  12. 2019A Wasserstein-type distance in the space of Gaussian Mixture Modelscited by 75
22 more
  1. 2021Low-Rank Sinkhorn Factorizationcited by 67
  2. 2019Sliced Gromov-Wassersteincited by 66
  3. 2021Minibatch optimal transport distances; analysis and applicationscited by 65
  4. 2021Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costscited by 61
  5. 2017Sliced Wasserstein Distance for Learning Gaussian Mixture Modelscited by 59
  6. 2019Learning Generative Models across Incomparable Spacescited by 53
  7. 2017Optimal transport for Gaussian mixture modelscited by 45
  8. 2018Massively scalable Sinkhorn distances via the Nystr"om methodcited by 41
  9. 2018Statistical Optimal Transport via Factored Couplingscited by 38
  10. 2018Towards Optimal Transport with Global Invariancescited by 33
  11. 2017Learning Wasserstein Embeddingscited by 26
  12. 2022Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenterscited by 22
  13. 2017An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimizationcited by 20
  14. 2021Quantized Gromov-Wassersteincited by 19
  15. 2018Distilled Wasserstein Learning for Word Embedding and Topic Modelingcited by 18
  16. 2020Linear Time Sinkhorn Divergences using Positive Featurescited by 17
  17. 2021Evaluating generative networks using Gaussian mixtures of image featurescited by 11
  18. 2020MREC: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular datacited by 10
  19. 2023Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spacescited by 7
  20. 2018Approximating the Quadratic Transportation Metric in Near-Linear Timecited by 4
  21. 2021A Riemannian Newton Trust-Region Method for Fitting Gaussian Mixture Modelscited by 2
  22. 2022Stochastic First-Order Learning for Large-Scale Flexibly Tied Gaussian Mixture Modelcited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 1.

4 papers.

  1. 2025Slicing the Gaussian Mixture Wasserstein Distancecited by 7
  2. 2025Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalizationcited by 2
  3. 2025Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognitioncited by 0
  4. 2026Entropic Partial Optimal Transport and Partial Gromov--Wasserstein Distance between Gaussian Mixturescited by 0