Representing and Learning High Dimensional Data with the Optimal Transport Map from a Probabilistic Viewpoint

In this paper, we propose a generative model in the space of diffeomorphic deformation maps. More precisely, we utilize the Kantarovich-Wasserstein metric and accompanying geometry to represent an image as a deformation from templates. Moreover, we incorporate a probabilistic viewpoint by assuming that each image is locally generated from a reference image. We capture the local structure by modelling the tangent planes at reference images. Once basis vectors for each tangent plane are learned via probabilistic PCA, we can sample a local coordinate, that can be inverted back to image space exactly. With experiments using 4 different datasets, we show that the generative tangent plane model in the optimal transport (OT) manifold can be learned with small numbers of images and can be used to create infinitely many 'unseen' images. In addition, the Bayesian classification accompanied with the probabilist modeling of the tangent planes shows improved accuracy over that done in the image space. Combining the results of our experiments supports our claim that certain datasets can be better represented with the Kantarovich-Wasserstein metric. We envision that the proposed method could be a practical solution to learning and representing data that is generated with templates in situatons where only limited numbers of data points are available.