Advances in dissimilarity-based data visualisation

The amount of collected digital information grows with each day. This development is facilitated by advanced technology, allowing for more detailed and more complex measurements. As a result, huge data sets gathered in this way are no longer easily comprehensible for a human person. Dimensionality reduction constitutes an important tool to visualise highly complex data in two dimensions, allowing users to gain insight about the structure of the data. In this thesis, we give an overview on dimensionality reduction and discuss challenges arising in this context. Moreover, we elaborate in detail the capabilities of dimensionality reduction on two exemplary techniques: generative topographic mapping, as a parametric technique, generating a model of the data; and t-distributed stochastic neighbour embedding, as a nonparametric projection technique, based on a cost function. As a core contribution of this thesis, we will discuss four major limitations of dimensionality reduction techniques, and present solutions to overcome these problems. (i) One important requirement for a visualisation technique regards the extensibility to new data: after constructing a mapping for given data, new data should be visualised in a consistent way. While this is readily available for parametric techniques, nonparametric methods are lacking this property, and we present a general approach to provide an explicit mapping. (ii) Another limitation regards the treatment of complex data: in many scenarios classical feature vectors are no longer sufficient, instead, the use of pairwise data relations in the form of similarity or dissimilarity measures becomes mandatory. Some algorithms are not capable of dealing with dissimilarity data, therefore we develop an approach to solve this problem via an implicit vectorial embedding. (iii) The techniques working on (dis-)similarity data suffer from high computational and memory complexity, since the matrix with pairwise relations grows quadratically with the number of data points. To drastically reduce the complexity, the Nyström approximation technique can be applied. In this context, we generalize the Nyström method for arbitrary symmetrical (dis-)similarity matrices, and define a universal framework resulting in linear time algorithms. (iv) Finally, we address the problem of data visualisation being ill-posed, since there are multiple ways to reduce the dimensionality and it is in general not clear which one is the best. Therefore, we follow the metric learning paradigm to focus on dimensions which are important for class separation. We investigate this approach for a parametric technique on the one hand and introduce a general framework for nonparametric techniques on the other hand.