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Supervised Gromov-Wasserstein Optimal Transport with Metric-Preserving Constraints.

Zixuan Cang1, Yaqi Wu2, Yanxiang Zhao2

  • 1Department of Mathematics, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695 USA.

SIAM Journal on Mathematics of Data Science
|September 2, 2025
PubMed
Summary

We introduce supervised Gromov-Wasserstein (sGW), a new optimal transport method that enforces distance preservation constraints. This approach enhances data alignment, particularly for partially overlapping datasets like single-cell RNA sequencing data.

Keywords:
28A3349Q2265K10minimal vertex covermirror-C descentnonconvex optimizationsupervised Gromov–Wassersteinsupervised optimal transport solver

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Area of Science:

  • Optimal transport theory
  • Computational geometry
  • Data science

Background:

  • Gromov-Wasserstein (GW) is a powerful tool for comparing metric spaces.
  • Existing GW methods may not adequately preserve pairwise distances under certain constraints.
  • Aligning datasets, especially single-cell RNA sequencing data, often requires robust distance preservation.

Purpose of the Study:

  • Introduce supervised Gromov-Wasserstein (sGW) optimal transport.
  • Incorporate infinity entries in the cost tensor to enforce distance preservation constraints.
  • Develop and validate a numerical solver for the sGW problem.

Main Methods:

  • Extended Gromov-Wasserstein by incorporating potential infinity entries in the cost tensor.
  • Transformed high-order constraints into coupling matrix constraints via minimal vertex cover.
  • Employed mirror-C descent iteration coupled with a supervised optimal transport solver.

Main Results:

  • Demonstrated the effectiveness of sGW through various numerical experiments.
  • Validated the framework on synthetic datasets and single-cell RNA sequencing data.
  • Showcased sGW's ability to control distance preservation and automatically estimate dataset overlaps.

Conclusions:

  • Supervised Gromov-Wasserstein (sGW) provides enhanced control over distance preservation in optimal transport.
  • sGW improves stability and flexibility in data-driven applications, including single-cell data alignment.
  • The method facilitates automatic estimation of overlapping portions in datasets.