We study the extraction of nonlinear data models in high dimensional spaces with modified self-organizing maps. Our algorithm maps lower dimensional lattice into a high dimensional space without topology violations by tuning the neighborhood widths locally. The approach is based on a new principle exploiting the specific dynamical properties of the first order phase transition induced by the noise of the data. The performance of the algorithm is demonstrated for one- and two-dimensional principal manifolds and for sparse data sets.

Several ready-to-use supervised classifiers perform predictively well in large-sample cases, but generally, the same cannot be expected when transitioning to high-dimensional settings. This can be explained by the classical supervised theory that has not been developed within high-dimensional spaces, giving several classifiers a hard combat against the curse of dimensionality. A rise in parsimonious classification procedures, particularly techniques incorporating feature selectors, can be observed. It can be interpreted as a two-step procedure: allowing an arbitrary selector to obtain a feature subset independent of a ready-to-use model and subsequently classify unlabelled instances within the selected subset. Modeling the two-step procedure is often heavy in motivation, and theoretical and algorithmic descriptions are frequently overlooked. In this thesis, we aim to describe the theoretical and algorithmic framework when employing a feature selector as a pre-processing step for Support Vector Machine and assess its validity in high-dimensional settings. The validity of the proposed classifier is evaluated based on predictive performance through a comparative study with a state-of-the-art algorithm designed for advanced learning tasks. The chosen algorithm effectively employs feature relevance during training, making it suitable for high-dimensional settings. The results suggest that the proposed classifier performs predicatively superior to the Support Vector Machine in lower input dimensions; however, a high rate of convergence towards a performance comparable to the Support Vector Machine tends to emerge for input dimensions beyond a certain threshold. Additionally, the thesis could not conclude any strict superior performance between the chosen state-of-the-art algorithm and the proposed classifier. Nonetheless, the state-of-the-art algorithm imposes a more balanced performance across both labels.