Performances of some learning algorithms: Kernel Projection Machine and kernel principal component analysis
PhD thesis, University Paris XI (Orsay).
This thesis takes place within the framework of statistical
learning. It brings contributions to the machine learning community using
modern statistical techniques based on progress in the study of empirical
processes. The first part investigates the statistical properties of Kernel
Principal Component Analysis (KPCA). The behavior of the reconstruction error is
studied with a non-asymptotic point of view and concentration inequalities of
the eigenvalues of the kernel matrix
are provided. All these results
correspond to fast convergence rates. Non-asymptotic results concerning the
eigenspaces of KPCA themselves are also provided. A new algorithm of
classification has been designed in the second part: the Kernel Projection
Machine (KPM). It is inspired by the Support Vector Machines (SVM). Besides,
highlights that the selection of a vector space by a dimensionality
reduction method such as KPCA regularizes suitably. The choice of the vector
space involved in the KPM is guided by statistical studies of model selection
using the penalized minimization of the empirical loss.
This regularization procedure is intimately connected with the finite
dimensional projections studied in the statistical work of Birg\'e and
Massart. The performances of KPM and SVM are then compared on some data
sets. Each topic tackled in this thesis raises new questions.