用于鲁棒分布值数据的Huber-Wasserstein重心
Huber-Wasserstein barycenters for robust distribution-valued data
AI总结：
该研究提出直接融入Huber损失的Huber-Wasserstein重心，兼具Wasserstein均值与L¹型Wasserstein中位数特性，具鲁棒性，适用于分布值数据，实验验证其优势。
AI中文摘要：
我们通过将Huber损失直接融入最优传输代价，提出了一种针对分布值数据的鲁棒重心。与在优化后将Huber损失应用于Wasserstein距离的度量空间Huber均值不同，我们的构造作用于单个传输位移，在局部保留二次行为的同时限制大位移的影响。由此得到的Huber-Wasserstein重心构成了Wasserstein均值与L¹型Wasserstein中位数之间的自然插值。我们建立了该构造的分析与统计基础：对于带Huber损失的最优传输，我们证明了对偶势的正则性与唯一性、最优传输映射的存在性，以及Huber参数变化时的稳定性；对于相关的重心问题，我们证明了存在性与刻画结果、经验插件估计量的一致性，以及本质上等于1/2的有限样本崩溃点。在一维情形下，我们进一步推导了逐点影响函数与渐近分布，量化了相关的鲁棒性-效率权衡，并表明在局部形状污染下，逐位移Huber化可保留基于距离的Huber化所丢失的一阶信息。对受污染的分布值数据的数值实验验证了所提出重心的鲁棒性，并展示了其在类均值与类中位数行为之间的插值特性。
英文摘要：
We propose a robust barycenter for distribution-valued data by incorporating the Huber loss directly into the optimal transport cost. In contrast to metric-space Huber means, which apply the Huber loss to the Wasserstein distance after optimization, our construction acts on individual transport displacements, preserving quadratic behavior locally while limiting the influence of large displacements. The resulting Huber-Wasserstein barycenters form a natural interpolation between Wasserstein means and $L^1$-type Wasserstein medians. We establish the analytical and statistical foundations of this construction. For optimal transport with Huber loss, we prove regularity and uniqueness properties of dual potentials, existence of optimal transport maps, and stability as the Huber parameter varies. For the associated barycenter problem, we prove existence and characterization results, consistency of empirical plug-in estimators, and a finite-sample breakdown point essentially equal to $1/2$. In dimension one, we further derive the pointwise influence function and asymptotic distribution, quantify the associated robustness-efficiency trade-off, and show that displacement-wise Huberization can retain first-order information that is lost by distance-based Huberization under localized shape contamination. Numerical experiments on contaminated distribution-valued data demonstrate the robustness of the proposed barycenters and illustrate their interpolation between mean- and median-like behavior.
