High-dimensional data analysis : optimal metrics and feature selection/

High-dimensional data are everywhere: texts, sounds, spectra, images, etc. are described by thousands of attributes. However, many data analysis tools at disposal (coming from statistics, artificial intelligence, etc.) were designed for low-dimensional data. Many of the explicit or implicit assumptions made while developing the classical data analysis tools are not transposable to high-dimensional data. For instance, many tools rely on the Euclidean distance, to compare data elements. But the Euclidean distance concentrates in high-dimensional spaces: all distances between data elements seem identical. The Euclidean distance is furthermore incapable of identifying important attributes from irrelevant ones. This thesis therefore focuses the choice of a relevant distance function to compare high-dimensional data and the selection of the relevant attributes. In Part One of the thesis, the phenomenon of the concentration of the distances is considered, and its consequences on data analysis tools are studied. It is shown that for nearest neighbours search, the Euclidean distance and the Gaussian kernel, both heavily used, may not be appropriate; it is thus proposed to use Fractional metrics and Generalised Gaussian kernels. Part Two of this thesis focuses on the problem of feature selection in the case of a large number of initial features. Two methods are proposed to (1) reduce the computational burden of feature selection process and (2) cope with the instability induced by high correlation between features that often appear with high-dimensional data. Most of the concepts studied and presented in this thesis are illustrated on chemometric data, and more particularly on spectral data, with the objective of inferring a physical or chemical property of a material by analysis the spectrum of the light it reflects.