As a fundamental problem of natural language processing, it is important to measure the distance between different documents. Among the existing methods, the Word Mover's Distance (WMD) has shown remarkable success in document semantic matching for its clear physical insight as a parameter-free model. However, WMD is essentially based on the classical Wasserstein metric, thus it often fails to robustly represent the semantic similarity between texts of different lengths. In this paper, we apply the newly developed Wasserstein-Fisher-Rao (WFR) metric from unbalanced optimal transport theory to measure the distance between different documents. The proposed WFR document distance maintains the great interpretability and simplicity as WMD. We demonstrate that the WFR document distance has significant advantages when comparing the texts of different lengths. In addition, an accelerated Sinkhorn based algorithm with GPU implementation has been developed for the fast computation of WFR distances. The KNN classification results on eight datasets have shown its clear improvement over WMD.
 This paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing. This article introduces a new non-linear dictionary learning method for histograms in the probability simplex. The method leverages optimal transport theory, in the sense that our aim is to reconstruct histograms using so called displacement interpolations (a.k.a. Wasserstein barycenters) between dictionary atoms; such atoms are themselves synthetic histograms in the probability simplex. Our method simultaneously estimates such atoms, and, for each datapoint, the vector of weights that can optimally reconstruct it as an optimal transport barycenter of such atoms. Our method is computationally tractable thanks to the addition of an entropic regularization to the usual optimal transportation problem, leading to an approximation scheme that is efficient, parallel and simple to differentiate. Both atoms and weights are learned using a gradient-based descent method. Gradients are obtained by automatic differentiation of the generalized Sinkhorn iterations that yield barycenters with entropic smoothing. Because of its formulation relying on Wasserstein barycenters instead of the usual matrix product between dictionary and codes, our method allows for non-linear relationships between atoms and the reconstruction of input data. We illustrate its application in several different image processing settings. In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computational geometry is established. The elliptic Monge-Ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing, and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the Monge-Ampere equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to nondifferentiable. The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampere type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function. 