切片 Wasserstein 重心：Wasserstein 空间中重心编码模型内的分析途径

中文摘要 AI 辅助

我们从 Wasserstein 空间中的变分视角研究切片 Wasserstein 重心，重点关注重心编码模型中的分析问题：给定一个查询测度和一个有限概率测度字典，恢复单纯形约束的重心坐标。我们推导了切片 Wasserstein 重心泛函关于经典 2-Wasserstein 几何的显式一阶变分公式。所得梯度表示为球面上 1D 单调传输位移的平均值。平稳性方程产生一个 Gram 矩阵准则，用于通过二次规划估计重心坐标，而其不动点形式则导出一个合成迭代，以生成新的重心测度。对于高斯模板，我们证明每个全局切片 Wasserstein 重心都是高斯的；然而，该泛函可能允许非高斯临界点。我们还提供了基于 1D 传输势的全局最优性证书。数值实验展示了在合成查询上的准确坐标恢复，并说明了这些坐标在数据表示中的用途以及用于可靠性评估的平稳性残差。

英文摘要

We study sliced Wasserstein barycenters from a variational perspective in Wasserstein space, with emphasis on the analysis problem in the barycentric coding model: given a query measure and a finite dictionary of probability measures, recover simplex-constrained barycentric coordinates. We derive an explicit first-variation formula for the sliced Wasserstein barycenter functional with respect to the classical 2-Wasserstein geometry. The resulting gradient is expressed as an average of 1D monotone transport displacements over the sphere. The stationarity equation yields a Gram-matrix criterion for estimating barycentric coordinates through a quadratic program, while its fixed-point form leads to a synthesis iteration to new barycentric measures. For Gaussian templates, we show that every global sliced Wasserstein barycenter is Gaussian; however, the functional may admit non-Gaussian critical points. We also provide a certificate for global optimality built from the 1D transport potentials. Numerical experiments demonstrate accurate coordinate recovery on synthesized queries and illustrate the use of these coordinates for data representation and stationarity residuals for reliability assessment.

发表机构

• Florida State University（佛罗里达州立大学）
• Tufts University（塔夫茨大学）

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