Given a family of probability measures in P(X), the space of probability
measures on a Hilbert space X, our goal in this paper is to highlight one ore
more curves in P(X) that summarize efficiently that family. We propose to study
this problem under the optimal transport (Wasserstein) geometry, using curves
that are restricted to be geodesic segments under that metric. We show that
concepts that play a key role in Euclidean PCA, such as data centering or
orthogonality of principal directions, find a natural equivalent in the optimal
transport geometry, using Wasserstein means and differential geometry. The
implementation of these ideas is, however, computationally challenging. To
achieve scalable algorithms that can handle thousands of measures, we propose
to use a relaxed definition for geodesics and regularized optimal transport
distances. The interest of our approach is demonstrated on images seen either
as shapes or color histograms.

本文探讨了在概率测度空间 P（X）中，使用最优传输（Wasserstein）几何将被限制为该度量下的测地线段的曲线，以高效地总结该系列测度。我们展示了在最优传输几何中，重要概念可以通过使用 Wasserstein 平均值和微分几何，找到自然的等效物。然而，应用这些想法是具有挑战性的。为了处理数千个测度和实现可扩放的算法，我们建议使用放松的测地线定义和正则化的最优传输距离。本文的方法在将图像视为形状或颜色直方图方面具有重要意义。