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Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on alternative formulations of the optimal transport have allowed for faster solutions to the problem and has revamped its practical applications in machine learning. In this paper, we exploit the widely used kernel methods and provide a family of provably positive definite kernels based on the Sliced Wasserstein distance and demonstrate the benefits of these kernels in a variety of learning tasks. Our work provides a new perspective on the application of optimal transport flavored distances through kernel methods in machine learning tasks.
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The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We also show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed distances. Finally, we compare the numerical performance of the proposed distances on several generative modeling tasks, including SW flows and SW auto-encoders.
Developing machine learning methods that are privacy preserving is today a central topic of research, with huge practical impacts. Among the numerous ways to address privacy-preserving learning, we here take the perspective of computing the divergences between distributions under the Differential Privacy (DP) framework -- being able to compute divergences between distributions is pivotal for many machine learning problems, such as learning generative models or domain adaptation problems. Instead of resorting to the popular gradient-based sanitization method for DP, we tackle the problem at its roots by focusing on the Sliced Wasserstein Distance and seamlessly making it differentially private. Our main contribution is as follows: we analyze the property of adding a Gaussian perturbation to the intrinsic randomized mechanism of the Sliced Wasserstein Distance, and we establish the sensitivityof the resulting differentially private mechanism. One of our important findings is that this DP mechanism transforms the Sliced Wasserstein distance into another distance, that we call the Smoothed Sliced Wasserstein Distance. This new differentially private distribution distance can be plugged into generative models and domain adaptation algorithms in a transparent way, and we empirically show that it yields highly competitive performance compared with gradient-based DP approaches from the literature, with almost no loss in accuracy for the domain adaptation problems that we consider.
In this paper we study generative modeling via autoencoders while using the elegant geometric properties of the optimal transport (OT) problem and the Wasserstein distances. We introduce Sliced-Wasserstein Autoencoders (SWAE), which are generative models that enable one to shape the distribution of the latent space into any samplable probability distribution without the need for training an adversarial network or defining a closed-form for the distribution. In short, we regularize the autoencoder loss with the sliced-Wasserstein distance between the distribution of the encoded training samples and a predefined samplable distribution. We show that the proposed formulation has an efficient numerical solution that provides similar capabilities to Wasserstein Autoencoders (WAE) and Variational Autoencoders (VAE), while benefiting from an embarrassingly simple implementation.
Probability metrics have become an indispensable part of modern statistics and machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a practical setting, the convergence behavior of the algorithms built upon these distances have not been well established, except for a few specific cases. In this paper, we introduce a broad family of probability metrics, coined as Generalized Sliced Probability Metrics (GSPMs), that are deeply rooted in the generalized Radon transform. We first verify that GSPMs are metrics. Then, we identify a subset of GSPMs that are equivalent to maximum mean discrepancy (MMD) with novel positive definite kernels, which come with a unique geometric interpretation. Finally, by exploiting this connection, we consider GSPM-based gradient flows for generative modeling applications and show that under mild assumptions, the gradient flow converges to the global optimum. We illustrate the utility of our approach on both real and synthetic problems.
While variational autoencoders have been successful generative models for a variety of tasks, the use of conventional Gaussian or Gaussian mixture priors are limited in their ability to capture topological or geometric properties of data in the latent representation. In this work, we introduce an Encoded Prior Sliced Wasserstein AutoEncoder (EPSWAE) wherein an additional prior-encoder network learns an unconstrained prior to match the encoded data manifold. The autoencoder and prior-encoder networks are iteratively trained using the Sliced Wasserstein Distance (SWD), which efficiently measures the distance between two $textit{arbitrary}$ sampleable distributions without being constrained to a specific form as in the KL divergence, and without requiring expensive adversarial training. Additionally, we enhance the conventional SWD by introducing a nonlinear shearing, i.e., averaging over random $textit{nonlinear}$ transformations, to better capture differences between two distributions. The prior is further encouraged to encode the data manifold by use of a structural consistency term that encourages isometry between feature space and latent space. Lastly, interpolation along $textit{geodesics}$ on the latent space representation of the data manifold generates samples that lie on the manifold and hence is advantageous compared with standard Euclidean interpolation. To this end, we introduce a graph-based algorithm for identifying network-geodesics in latent space from samples of the prior that maximize the density of samples along the path while minimizing total energy. We apply our framework to 3D-spiral, MNIST, and CelebA datasets, and show that its latent representations and interpolations are comparable to the state of the art on equivalent architectures.
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