We address the problem of unsupervised domain adaptation (UDA) by learning a cross-domain agnostic embedding space, where the distance between the probability distributions of the two source and target visual domains is minimized. We use the output space of a shared cross-domain deep encoder to model the embedding space anduse the Sliced-Wasserstein Distance (SWD) to measure and minimize the distance between the embedded distributions of two source and target domains to enforce the embedding to be domain-agnostic.Additionally, we use the source domain labeled data to train a deep classifier from the embedding space to the label space to enforce the embedding space to be this http URL a result of this training scheme, we provide an effective solution to train the deep classification network on the source domain such that it will generalize well on the target domain, where only unlabeled training data is accessible. To mitigate the challenge of class matching, we also align corresponding classes in the embedding space by using high confidence pseudo-labels for the target domain, i.e. assigning the class for which the source classifier has a high prediction probability. We provide theoretical justification as well as experimental results on UDA benchmark tasks to demonstrate that our method is effective and leads to state-of-the-art performance.
As Wasserstein distance is finding more applications in deep learning, efficient computation of Wasserstein distance has become an active area of research. The reason is that Wasserstein distance is defined in form of a linear programming optimization and solving this optimization problem is computationally expensive for high-dimensional data. Although computationally efficient variations and approximations of the Wasserstein distance have been recently proposed @cite_34 @cite_39 @cite_3 , these variations still require an additional optimization in each iteration of the stochastic gradient descent (SGD) steps to match distributions. @cite_27 used a regularized version of the optimal transport for domain adaptation. @cite_13 used a dual stochastic gradient algorithm for solving the regularized optimal transport problem. Alternatively, we propose to address the above challenges using Sliced Wasserstein Distance (SWD). Definition of SWD is motivated by the fact that in contrast to higher dimensions, the Wasserstein distance for one-dimensional distributions has a closed form solution which can be computed efficiently. This fact is used to approximate Wasserstein distance by SWD, which is a computationally efficient approximation and has recently drawn interest from the machine learning and computer vision communities @cite_28 @cite_35 @cite_32 @cite_41 @cite_42 .
@cite_35 @cite_28 @cite_41 @cite_42 @cite_32 @cite_3 @cite_39 @cite_27 @cite_34 @cite_13
2019106840 2963398989 2963506208 2253327477 2009172320 1594039573 2158131535 2964252913
Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by G. Carlier, A. Oberman, and E. Oudet [ESAIM Math. Model. Numer. Anal., 6 (2015), pp. 1621--1642] can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals. (A correction is attached.) We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem. Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm. We study unsupervised generative modeling in terms of the optimal transport (OT) problem between true (but unknown) data distribution @math and the latent variable model distribution @math . We show that the OT problem can be equivalently written in terms of probabilistic encoders, which are constrained to match the posterior and prior distributions over the latent space. When relaxed, this constrained optimization problem leads to a penalized optimal transport (POT) objective, which can be efficiently minimized using stochastic gradient descent by sampling from @math and @math . We show that POT for the 2-Wasserstein distance coincides with the objective heuristically employed in adversarial auto-encoders (AAE) (, 2016), which provides the first theoretical justification for AAEs known to the authors. We also compare POT to other popular techniques like variational auto-encoders (VAE) (Kingma and Welling, 2014). Our theoretical results include (a) a better understanding of the commonly observed blurriness of images generated by VAEs, and (b) establishing duality between Wasserstein GAN (Arjovsky and Bottou, 2017) and POT for the 1-Wasserstein distance. Sliced Wasserstein metrics between probability measures solve the optimal transport (OT) problem on univariate projections, and average such maps across projections. The recent interest for the SW distance shows that much can be gained by looking at optimal maps between measures in smaller subspaces, as opposed to the curse-of-dimensionality price one has to pay in higher dimensions. Any transport estimated in a subspace remains, however, an object that can only be used in that subspace. We propose in this work two methods to extrapolate, from an transport map that is optimal on a subspace, one that is nearly optimal in the entire space. We prove that the best optimal transport plan that takes such "subspace detours" is a generalization of the Knothe-Rosenblatt transport. We show that these plans can be explicitly formulated when comparing Gaussians measures (between which the Wasserstein distance is usually referred to as the Bures or Fr 'echet distance). Building from there, we provide an algorithm to select optimal subspaces given pairs of Gaussian measures, and study scenarios in which that mediating subspace can be selected using prior information. We consider applications to NLP and evaluation of image quality (FID scores). This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space. To overcome the time complexity involved by the numerical solving of such problem, the original Wasserstein metric is replaced by a sliced approximation over 1D distributions. This enables us to introduce a new fast gradient descent algorithm to compute Wasserstein barycenters of point clouds. This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multi-scale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures. The notion of Sinkhorn divergence has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive central limit theorems for empirical Sinkhorn divergences. We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. The strategy of proof uses the notions of directional Hadamard differentiability and delta-method in this setting. It is inspired by the results in the work of Sommerfeld and Munk (2016) on the asymptotic distribution of empirical Wasserstein distance on finite space using un-regularized transportation costs. Simulated and real datasets are used to illustrate our approach. A comparison with existing methods to measure the discrepancy between multivariate distributions is also proposed. We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator. 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. Domain adaptation aims at generalizing a high-performance learner on a target domain via utilizing the knowledge distilled from a source domain which has a different but related data distribution. One solution to domain adaptation is to learn domain invariant feature representations while the learned representations should also be discriminative in prediction. To learn such representations, domain adaptation frameworks usually include a domain invariant representation learning approach to measure and reduce the domain discrepancy, as well as a discriminator for classification. Inspired by Wasserstein GAN, in this paper we propose a novel approach to learn domain invariant feature representations, namely Wasserstein Distance Guided Representation Learning (WDGRL). WDGRL utilizes a neural network, denoted by the domain critic, to estimate empirical Wasserstein distance between the source and target samples and optimizes the feature extractor network to minimize the estimated Wasserstein distance in an adversarial manner. The theoretical advantages of Wasserstein distance for domain adaptation lie in its gradient property and promising generalization bound. Empirical studies on common sentiment and image classification adaptation datasets demonstrate that our proposed WDGRL outperforms the state-of-the-art domain invariant representation learning approaches.