Optimal transport (OT) barycenters are a mathematically grounded way of averaging probability distributions while capturing their geometric properties. In short, the barycenter task is to take the average of a collection of probability distributions w.r.t. given OT discrepancies. We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions. Our approach is built upon the dual reformulation of the EOT problem based on weak OT, which has recently gained the attention of the ML community. Beyond its novelty, our method enjoys several advantageous properties: (i) we establish quality bounds for the recovered solution; (ii) this approach seemlessly interconnects with the Energy-Based Models (EBMs) learning procedure enabling the use of well-tuned algorithms for the problem of interest; (iii) it provides an intuitive optimization scheme avoiding min-max, reinforce and other intricate technical tricks. For validation, we consider several low-dimensional scenarios and image-space setups, including non-Euclidean cost functions. Furthermore, we investigate the practical task of learning the barycenter on an image manifold generated by a pretrained generative model, opening up new directions for real-world applications.
We propose a novel neural method to compute partial optimal transport (OT) maps, i.e., OT maps between parts of measures of the specified masses. We test our partial neural optimal transport algorithm on synthetic examples.
We propose the extremal transport (ET) which is a mathematical formalization of the theoretically best possible unpaired translation between a pair of domains w.r.t. the given similarity function. Inspired by the recent advances in neural optimal transport (OT), we propose a scalable algorithm to approximate ET maps as a limit of partial OT maps. We test our algorithm on toy examples and on the unpaired image-to-image translation task.
Real-world image super-resolution (SR) tasks often do not have paired datasets limiting the application of supervised techniques. As a result, the tasks are usually approached by unpaired techniques based on Generative Adversarial Networks (GANs) which yield complex training losses with several regularization terms such as content and identity losses. We theoretically investigate the optimization problems which arise in such models and find two surprising observations. First, the learned SR map is always an optimal transport (OT) map. Second, we empirically show that the learned map is biased, i.e., it may not actually transform the distribution of low-resolution images to high-resolution images. Inspired by these findings, we propose an algorithm for unpaired SR which learns an unbiased OT map for the perceptual transport cost. Unlike existing GAN-based alternatives, our algorithm has a simple optimization objective reducing the neccesity to perform complex hyperparameter selection and use additional regularizations. At the same time, it provides nearly state-of-the-art performance on the large-scale unpaired AIM-19 dataset.
Depth maps captured with commodity sensors often require super-resolution to be used in applications. In this work we study a super-resolution approach based on a variational problem statement with Tikhonov regularization where the regularizer is parametrized with a deep neural network. This approach was previously applied successfully in photoacoustic tomography. We experimentally show that its application to depth map super-resolution is difficult, and provide suggestions about the reasons for that.