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.

Multidimensional scaling techniques are unsupervised Dimension Reduction (DR) techniques which use multidimensional data pairwise similarities to represent data into a plane enabling their visual exploratory analysis. Considering labeled data, the DR techniques face two objectives with potentially different priorities: one is to account for the data points' similarities, the other for the data classes' structures. Unsupervised DR techniques attempt to preserve original data similarities, but they do not consider their class label hence they can map originally separated classes as overlapping ones. Conversely, the state-of-the-art so-called supervised DR techniques naturally handle labeled data, but they do so in a predictive modeling framework where they attempt to separate the classes in order to improve a classification accuracy measure in the low-dimensional space, hence they can map as separated even originally overlapping classes. We propose ClassiMap, a DR technique which optimizes a new objective function enabling Exploratory Data Analysis (EDA) of labeled data. Mapping distortions known as tears and false neighborhoods cannot be avoided in general due to the reduction of the data dimension. ClassiMap intends primarily to preserve data similarities but tends to distribute preferentially unavoidable tears among the different-label data and unavoidable false neighbors among the same-label data. Standard quality measures to evaluate the quality of unsupervised mappings cannot tell about the preservation of within-class or between-class structures, while classification accuracy used to evaluate supervised mappings is only relevant to the framework of predictive modeling. We propose two measures better suited to the evaluation of DR of labeled data in an EDA framework. We use these two label-aware indices and four other standard unsupervised indices to compare ClassiMap to other state-of-the-art supervised and unsupervised DR techniques on synthetic and real datasets. ClassiMap appears to provide a better tradeoff between pairwise similarities and class structure preservation according to these new measures.

Abstract Multidimensional scaling is a must‐have tool for visual data miners, projecting multidimensional data onto a two‐dimensional plane. However, what we see is not necessarily what we think about. In many cases, end‐users do not take care of scaling the projection space with respect to the multidimensional space. Anyway, when using non‐linear mappings, scaling is not even possible. Yet, without scaling geometrical structures which might appear do not make more sense than considering a random map. Without scaling, we shall not make inference from the display back to the multidimensional space. No clusters, no trends, no outliers, there is nothing to infer without first quantifying the mapping quality. Several methods to qualify mappings have been devised. Here, we propose CheckViz, a new method belonging to the framework of Verity Visualization. We define a two‐dimensional perceptually uniform colour coding which allows visualizing tears and false neighbourhoods, the two elementary and complementary types of geometrical mapping distortions, straight onto the map at the location where they occur. As examples shall demonstrate, this visualization method is essential to help users make sense out of the mappings and to prevent them from over interpretations. It could be applied to check other mappings as well.

A novel network based on Linsker-type Hebbian learning is analyzed in its dynamical behavior. The network combines a coupled dynamics of fast and slow states and is prone to internal parametrical fluctuations as well as external noises. Robustness represents a crucial property of the network to attenuate the effects of internal fluctuation and external noise. In this study, we formulate this novel neural network as a coupled nonlinear differential systems operating at different time-scales under vanishing perturbations. We determine conditions for the existence of a global uniform attractor of the perturbed biological system. By using a Lyapunov function for the coupled system, we derive a maximal upper bound for the fast time scale associated with the fast state. Finally, two examples are given to confirm the applicability of the developed theoretical framework.

Visual exploratory data analysis represents a well-accepted imaging modality for high-dimensional DCE-MRI-derived breast cancer data. We employ this paradigm for discriminating between malignant and benign lesions based on different shape descriptors thanks to proven and novel dimension reduction algorithms. We demonstrate that shape structure changes such as weighted 3D Krawtchouck moments outperform global averaging moments such as geometric moment invariants in terms of discrimination of benign/malignant lesions. The best visualization of tumor shapes in a two-dimensional space is achieved based on nonlinear mapping methods, especially the ones that consider neighborhood ranks.

Mapping high-dimensional data in a low-dimensional space, for example, for visualization, is a problem of increasingly major concern in data analysis. This paper presents data-driven high-dimensional scaling (DD-HDS), a nonlinear mapping method that follows the line of multidimensional scaling (MDS) approach, based on the preservation of distances between pairs of data. It improves the performance of existing competitors with respect to the representation of high-dimensional data, in two ways. It introduces (1) a specific weighting of distances between data taking into account the concentration of measure phenomenon and (2) a symmetric handling of short distances in the original and output spaces, avoiding false neighbor representations while still allowing some necessary tears in the original distribution. More precisely, the weighting is set according to the effective distribution of distances in the data set, with the exception of a single user-defined parameter setting the tradeoff between local neighborhood preservation and global mapping. The optimization of the stress criterion designed for the mapping is realized by "force-directed placement" (FDP). The mappings of low- and high-dimensional data sets are presented as illustrations of the features and advantages of the proposed algorithm. The weighting function specific to high-dimensional data and the symmetric handling of short distances can be easily incorporated in most distance preservation-based nonlinear dimensionality reduction methods.