Generative Adversarial Networks (GANs) are performant generative methods yielding high-quality samples. However, under certain circumstances, the training of GANs can lead to mode collapse or mode dropping. To address this problem, we use the last layer of the discriminator as a feature map to study the distribution of the real and the fake data. During training, we propose to match the real batch diversity to the fake batch diversity by using the Bures distance between covariance matrices in this feature space. The computation of the Bures distance can be conveniently done in either feature space or kernel space in terms of the covariance and kernel matrix respectively. We observe that diversity matching reduces mode collapse substantially and has a positive effect on sample quality. On the practical side, a very simple training procedure is proposed and assessed on several data sets.

In this paper, we propose a novel method for time-series modeling and forecasting. It is based on the temporal formulation of Restricted Kernel Machines leading to a dynamical equation in the latent-variables. Forecasting involves finding the next latent variable and then solving a pre-image problem to predict a new-point in the input space. Further, we benchmark our model on several standard data sets against other well-known time-series models. * European Research Council under the European Union's Horizon 2020 research and innovation programme: ERC Advanced Grants agreements E-DUALITY(No 787960) and Back to the Roots (No 885682). This paper reflects only the authors' views and the Union is not liable for any use that may be made of the contained information. Research Council KUL: Optimization frameworks for deep kernel machines C14/18/068; Research Fund (projects C16/15/059, C3/19/053, C24/18/022, C3/20/117, C3I-21-00316); Industrial Research Fund (Fellowships 13-0260, IOFm/16/004, IOFm/20/002) and several Leuven Research and Development bilateral industrial projects.

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