Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric
Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric
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发表时间：
2015-06
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通讯作者：
Vivien Seguy;Marco Cuturi
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作者：
Vivien Seguy;Marco Cuturi
给定Hilbert空间X上的概率测度空间P（X）中的一个概率测度族，本文的目标是突出P（X）中的一条或多条曲线，这些曲线有效地概括了该概率测度族。我们建议研究这个问题下的最佳运输（Wasserstein）的几何形状，使用曲线，被限制为测地线段下的度量。我们发现，在欧几里得PCA中发挥关键作用的概念，如数据中心或主方向的正交性，找到一个自然的最佳运输几何等效，使用Wasserstein手段和微分几何。然而，这些想法的实现在计算上具有挑战性。为了实现可扩展的算法，可以处理成千上万的措施，我们建议使用一个宽松的定义测地线和正则化的最佳运输距离。我们的方法的兴趣被证明无论是作为形状或颜色直方图的图像。
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.