Optimal transport theory has recently found many applications in machine learning thanks to its capacity to meaningfully compare various machine learning objects that are viewed as distributions. The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the Gromov–Wasserstein distance focuses on the relations between the elements, depicting the structure of the object, yet discarding its features. In this paper, we study the Fused Gromov-Wasserstein distance that extends the Wasserstein and Gromov–Wasserstein distances in order to encode simultaneously both the feature and structure information. We provide the mathematical framework for this distance in the continuous setting, prove its metric and interpolation properties, and provide a concentration result for the convergence of finite samples. We also illustrate and interpret its use in various applications, where structured objects are involved.

We propose an algorithm which performs a progressive approximation of a viability kernel, iteratively using a classificatio method. We establish the mathematical conditions that the classificatio method should fulfil to guarantee the convergence to the actual viability kernel. We study more particularly the use of support vector machines (SVMs) as classificatio techniques. We show that they make possible to use gradient optimisation techniques to fin a viable control at each time step, and over several time steps. This allows us to avoid the exponential growth of the computing time with the dimension of the control space. It also provides simple and efficien control procedures. We illustrate the method with some examples inspired from ecology.

Abstract. The SIFT framework has shown to be accurate in the image classification context. In [1], we designed a Bag-of-Words approach based on an adaptation of this framework to time series classification. It relies on two steps: SIFT-based features are first extracted and quantized into words; histograms of occurrences of each word are then fed into a classifier. In this paper, we investigate techniques to improve the performance of Bag-of-Temporal-SIFT-Words: dense extraction of keypoints and normalization of Bag-of-Words histograms. Extensive experiments show that our method significantly outperforms most state-of-the-art techniques for time series classification.