Kernel methods for high dimensional data analysis

TL;DR: This thesis aims at introducing a new point of view in the use of distances and probability measures defined on the data set, and shows that kernel methods, already used in the intrinsic low dimensional scenario in order to reduce dimensionality, can be investigated under purely high dimensional hypotheses.

Abstract: Since data are being collected using an increasing number of features, datasets are of increasingly high dimension. Computational problems, related to the apparent dimension, i.e. the dimension of the vectors used to collect data, and theoretical problems, which depends notably on the effective dimension of the dataset, the so called intrinsic dimension, have affected high dimensional data analysis. In order to provide a suitable approach to data analysis in high dimensions, we introduce a more comprehensive scenario in the framework of metric measure spaces. The aim of this thesis, is to show how to take advantage of high dimensionality phenomena in the pure high dimensional regime. In particular, we aim at introducing a new point of view in the use of distances and probability measures defined on the data set. More specifically, we want to show that kernel methods, already used in the intrinsic low dimensional scenario in order to reduce dimensionality, can be investigated under purely high dimensional hypotheses, and further applied to cases not covered by the literature.