用沃瑟斯坦几何分析概率分布的主变化方向，找到最优测地线。
On the Wasserstein Geodesic Principal Component Analysis of probability measures
- 基于沃瑟斯坦几何构建测地线主成分分析方法
- 对高斯分布可简化为可逆线性映射计算，一般情形用神经网络参数化测地线
- 在真实数据集上验证效果，适合处理分布型数据的研究者
本文研究在奥托-沃瑟斯坦几何框架下对概率分布集合进行测地线主成分分析（GPCA），目标是识别能最好捕捉数据底层变化模式的测地线曲线。针对高斯分布集合，提出将计算映射到可逆线性变换空间的方法；对于更一般的绝对连续概率分布，采用新型神经网络参数化方式来表示沃瑟斯坦空间中的测地线。通过多个例子与经典切空间主成分分析对比，展示了该方法在真实数据集上的有效性。
原文摘要 · Abstract (English)
This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.
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