**Wasserstein Discriminant Analysis (WDA)**

Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have been usually considered, notably in LDA. Thanks to the the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples scale) interactions between classes. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; We show that the optimization of WDA can be tackled using automatic differentiation of Sinkhorn iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real life datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset. … **Distance Preservation to Local Mean (DPLM)**

In this paper, we propose a nonlinear dimensionality reduction algorithm for the manifold of Symmetric Positive Definite (SPD) matrices that considers the geometry of SPD matrices and provides a low dimensional representation of the manifold with high class discrimination. The proposed algorithm, tries to preserve the local structure of the data by preserving distance to local mean (DPLM) and also provides an implicit projection matrix. DPLM is linear in terms of the number of training samples and may use the label information when they are available in order to performance improvement in classification tasks. We performed several experiments on the multi-class dataset IIa from BCI competition IV. The results show that our approach as dimensionality reduction technique – leads to superior results in comparison with other competitor in the related literature because of its robustness against outliers. The experiments confirm that the combination of DPLM with FGMDM as the classifier leads to the state of the art performance on this dataset. … **Runge-Kutta Convolutional Neural Network (RKNet)**

A convolutional neural network for image classification can be constructed following some mathematical ways since it models the ventral stream in visual cortex which is regarded as a multi-period dynamical system. In this paper, a new point of view is proposed for constructing network models as well as providing a direction to get inspiration or explanation for neural network. If each period in ventral stream was deemed to be a dynamical system with time as the independent variable, there should be a set of ordinary differential equations (ODEs) for this system. Runge-Kutta methods are common means to solve ODE. Thus, network model ought to be built using these methods. Moreover, convolutional networks could be employed to emulate the increments within every time-step. The model constructed in the above way is named Runge-Kutta Convolutional Neural Network (RKNet). According to this idea, Dense Convolutional Networks (DenseNets) and Residual Networks (ResNets) were varied to RKNets. To prove the feasibility of RKNets, these variants were verified on benchmark datasets, CIFAR and ImageNet. The experimental results show that the RKNets transformed from DenseNets gained similar or even higher parameter efficiency. The success of the experiments denotes that Runge-Kutta methods can be utilized to construct convolutional neural networks for image classification efficiently. Furthermore, the network models might be structured more rationally in the future basing on RKNet and priori knowledge. …

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