# On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians

arXiv: https://arxiv.org/abs/2410.08117 · HTML: https://mlsanity.com/paper/2410.08117/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 19 of 19)

- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm](https://mlsanity.com/paper/2002.03293/) (2020)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Scaling Algorithms for Unbalanced Transport Problems](https://mlsanity.com/paper/1607.05816/) (2016)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Unbalanced Optimal Transport through Non-negative Penalized Linear Regression](https://mlsanity.com/paper/2106.04145/) (2021)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [Block Coordinate Descent on Smooth Manifolds: Convergence Theory and Twenty-One Examples](https://mlsanity.com/paper/2305.14744/) (2023)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Barycenters of Natural Images -- Constrained Wasserstein Barycenters for Image Morphing](https://mlsanity.com/paper/1912.11545/) (2019)
- [Dynamical Wasserstein Barycenters for Time-series Modeling](https://mlsanity.com/paper/2110.06741/) (2021)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [ExGra-Med: Extended Context Graph Alignment for Medical Vision-Language Models](https://mlsanity.com/paper/2410.02615/) (2024)
- [node2coords: Graph Representation Learning with Wasserstein Barycenters](https://mlsanity.com/paper/2007.16056/) (2020)
- [Graph Signal Representation with Wasserstein Barycenters](https://mlsanity.com/paper/1812.05517/) (2018)

## Abstract (arXiv)

We explore a robust version of the barycenter problem among n centered Gaussian probability measures, termed Semi-Unbalanced Optimal Transport (SUOT)-based Barycenter, wherein the barycenter remains fixed while the others are relaxed using Kullback-Leibler divergence. We develop optimization algorithms on Bures-Wasserstein manifold, named the Exact Geodesic Gradient Descent and Hybrid Gradient Descent algorithms. While the Exact Geodesic Gradient Descent method is based on computing the exact closed form of the first-order derivative of the objective function of the barycenter along a geodesic on the Bures manifold, the Hybrid Gradient Descent method utilizes optimizer components when solving the SUOT problem to replace outlier measures before applying the Riemannian Gradient Descent. We establish the theoretical convergence guarantees for both methods and demonstrate that the Exact Geodesic Gradient Descent algorithm attains a dimension-free convergence rate. Finally, we conduct experiments to compare the normal Wasserstein Barycenter with ours and perform an ablation study.