Kernel Methods for Unsupervised Learning Kernel Methods for Unsupervised Learning Title: Kernel Methods for Unsupervised Learning

Kernel Methods are algorithms that projects input data by a nonlinear mapping in a new space (Feature Space). In this thesis we have investigated Kernel Methods for Unsupervised learning, namely Kernel Methods that do not require targeted data. Two classical unsupervised learning problems using Kernel Methods have been tackled. The former is the Data Dimensionality Estimation, the latter is the Clustering. The dimensionality of a data set, called Intrinsic Dimension (ID), is the minimum number of free variables needed to represent the data without information loss. The Kernel PCA, has been investigated and compared, as ID estimator, with a classical dimensionality estimation method, namely the Principal Component Analysis (PCA). The study has been carried out both on a synthethic data set of known dimensionality and on real data benchmarks, i.e. MIT-CBCL Face database and Indoor-Outdoor Image database. The investigations have enlighted that Kernel PCA performs better than PCA, as dimensionality estimator, only when the kernel adopted is very close to the function that has generated the data set. Otherwise Kernel PCA can perform even worst than PCA. With regard to the Clustering problem, we have proposed a novel Kernel Method, the Kernel Grower [Cam04]. The Kernel Grower convergence is guaranteed, unlike several classical clustering algorithm, since it is an Expectation-Maximization algorithm. The main Kernel Grower quality consists, unlike all clustering algorithms published in the literature, in producing nonlinear separation surfaces among data. Kernel Grower compares better with popular clustering algorithms, namely K-MEANS, Neural Gas and Self Organizing Maps, on a synthethic dataset and two UCI real data benchmarks, i.e. IRIS data and Wisconsin breast cancer database. Kernel Grower is the main original result of the thesis.