Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric
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.
In this paper, the authors proposed a novel algorithm to compute principal geodesics in the Wasserstein space. Experiments in simulated and real datasets show the promising performance. Essentially, the PCA is generalized into nonlinear PCA when such estimation is performed under the Wasserstein space, which is a high-dimensional nonlinear space. However, it is not easy to see such nonlinearity is well described through observing several experimental results reported in this paper. Specifically, 1) in Fig, 1, 2 and 3,I guess these principal Geodesics should be principal components whereas in my opinion, the real principal Geodesics should be principal curves in one-dimensional sense or principal manifolds in two or higher dimensional sense.Further, since the authors do not explain what the points are,I have no ideas on how the algorithms run for the data points and whether the results are the best ones. 2) Fig. 4 and Fig. 5 should be used to discover one intrinsic dimension or the first principal Geodesic. Therefore, if the intrinsic dimensions are greater than 3, maybe the algorithms will not work.I think the authors should discuss the limitation of their proposed algorithms related to the dimension of intrinsic variables. 3) Orthogonality: I agree that the authors are right that in supplementary material, Figure 1 indeed discovered two intrinsic variables: scale and translation. However, for other figures, it is not easy to find such a nonlinear orthogonality. Especially for other images, I have difficult to figure out the "orthogonality" from them. Therefore, I think that the authors should compare their proposed algorithm with not only PCA but also other nonlinear PCA algorithms so that we can see how much the algorithms improve the estimation of principal Geodesics based on different datasets.