We consider the 2-Wasserstein space of probability measures supported on the unit-circle, and propose a framework for Principal Component Analysis (PCA) for data living in such a space. We build on a detailed investigation of the optimal transportation problem for measures on the unit-circle which might be of independent interest. In particular, we derive an expression for optimal transport maps in (almost) closed form and propose an alternative definition of the tangent space at an absolutely continuous probability measure, together with the associated exponential and logarithmic maps. PCA is performed by mapping data on the tangent space at the Wasserstein barycentre, which we approximate via an iterative scheme, and for which we establish a sufficient a posteriori condition to assess its convergence. Our methodology is illustrated on several simulated scenarios and a real data analysis of measurements of optical nerve thickness.

翻译：本文考虑单位圆上概率测度构成的2-Wasserstein空间，并针对该空间中的数据提出主成分分析（PCA）框架。我们基于对单位圆上测度最优传输问题的详细研究（该研究本身可能具有独立价值），特别是推导了（几乎）封闭形式的最优传输映射表达式，并提出了绝对连续概率测度处切空间的替代定义以及相应的指数映射和对数映射。PCA通过将数据映射到Wasserstein重心的切空间上实现，我们采用迭代算法近似该重心，并建立了评估其收敛性的充分后验条件。通过多个模拟场景以及视神经厚度测量的真实数据分析，验证了所提方法的有效性。