Exploratory visual analysis of multidimensional labeled data is challenging.Multidimensional Projections for labeled data attempt to separate classes while preserving neighborhoods.In this work, we consider the case where instances are assigned multiple labels with probabilities or weights: for example, the output of a probabilistic classifier, fuzzy membership functions in fuzzy logic, or the share of votes for each candidate in an election.We propose a new technique to better preserve neighborhoods of such data.Our experiments show improved qualitative results compared to unsupervised, and existing dimensionality reduction techniques.* The work of SM and SL has been realized with the participation of INES.2S.The work of DD has been

Nonlinear dimensionality reduction of high-dimensional data is challenging as the low-dimensional embedding will necessarily contain distortions, and it can be hard to determine which distortions are the most important to avoid. When annotation of data into known relevant classes is available, it can be used to guide the embedding to avoid distortions that worsen class separation. The supervised mapping method introduced in the present paper, called ClassNeRV, proposes an original stress function that takes class annotation into account and evaluates embedding quality both in terms of false neighbors and missed neighbors. ClassNeRV shares the theoretical framework of a family of methods descended from Stochastic Neighbor Embedding (SNE). Our approach has a key advantage over previous ones: in the literature supervised methods often emphasize class separation at the price of distorting the data neighbors' structure; conversely, unsupervised methods provide better preservation of structure at the price of often mixing classes. Experiments show that ClassNeRV can preserve both neighbor structure and class separation, outperforming nine state of the art alternatives.

We apply Computer algebra techniques, such as algebraic computations of resultants and discriminants, certified drawing (with a guaranteed topology) of plane curves, to a problem in Fluid dynamics: We investigate ``capillary-gravity'' solitary waves in shallow water, relying on the framework of the Serre-Green-Naghdi equations. So, we deal with 2 dimensional surface waves, propagating in a shallow water of constant depth. By a differential elimination process, the study reduces to describing the solutions of an ordinary non linear first order differential equation, depending on two parameters. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.