Summary of On Barycenter Computation: Semi-unbalanced Optimal Transport-based Method on Gaussians, by Ngoc-hai Nguyen et al.
On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians
by Ngoc-Hai Nguyen, Dung Le, Hoang-Phi Nguyen, Tung Pham, Nhat Ho
First submitted to arxiv on: 10 Oct 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: None
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The proposed Semi-Unbalanced Optimal Transport (SUOT)-based Barycenter method tackles the barycenter problem among n centered Gaussian probability measures while fixing the barycenter. The SUOT-based Barycenter relies on Kullback-Leibler divergence to relax other measures. To optimize the problem, two algorithms are developed: Exact Geodesic Gradient Descent and Hybrid Gradient Descent. While Exact Geodesic Gradient Descent uses a closed-form derivative along a geodesic on the Bures-Wasserstein manifold, Hybrid Gradient Descent replaces outlier measures using optimizer components before applying Riemannian Gradient Descent. Theoretical convergence guarantees are established for both methods, with Exact Geodesic Gradient Descent achieving a dimension-free convergence rate. Experimental results compare normal Wasserstein Barycenter with SUOT-based Barycenter and perform an ablation study. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper explores a new way to find the average of multiple probability distributions while keeping one distribution fixed. The method, called Semi-Unbalanced Optimal Transport (SUOT)-based Barycenter, uses a special kind of distance between probability distributions to make the calculation more efficient. To do this, two different algorithms are developed: Exact Geodesic Gradient Descent and Hybrid Gradient Descent. These algorithms help find the optimal solution by adjusting the way they treat outlier data points. The paper shows that one algorithm works better than the other in certain situations and provides examples of how to use it. |
Keywords
» Artificial intelligence » Gradient descent » Probability
