提出一种更鲁棒的高斯分布中位数计算方法，能有效处理异常数据。
On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians
- 基于半不平衡最优传输框架，在Bures-Wasserstein流形上设计优化算法。
- 精确测地线梯度下降法实现无维度收敛率，优于传统方法。
- 适合处理含异常值的高斯分布聚类与统计分析任务。
本文研究中心高斯概率测度之间的一种稳健型中位数问题，提出基于半不平衡最优传输（SUOT）的中位数方法，其中中位数固定，其余分布通过KL散度松弛。在Bures-Wasserstein流形上构建了精确测地线梯度下降和混合梯度下降算法。前者基于目标函数沿测地线的一阶导数闭式解；后者在求解过程中引入优化组件，先替换异常测度再执行黎曼梯度下降。两种方法均具备理论收敛性保证，且精确测地线梯度下降实现无维度收敛率。实验对比了标准Wasserstein中位数与本方法，并进行了消融研究。
原文摘要 · Abstract (English)
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.
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