Advances in independent component analysis and nonnegative matrix factorization

Abstract

A fundamental problem in machine learning research, as well as in many other disciplines, is finding a suitable representation of multivariate data, i.e. random vectors. For reasons of computational and conceptual simplicity, the representation is often sought as a linear transformation of the original data. In other words, each component of the representation is a linear combination of the original variables. Well-known linear transformation methods include principal component analysis (PCA), factor analysis, and projection pursuit. In this thesis, we consider two popular and widely used techniques: independent component analysis (ICA) and nonnegative matrix factorization (NMF). ICA is a statistical method in which the goal is to find a linear representation of nongaussian data so that the components are statistically independent, or as independent as possible. Such a representation seems to capture the essential structure of the data in many applications, including feature extraction and signal separation. Starting from ICA, several methods of estimating the latent structure in different problem settings are derived and presented in this thesis. FastICA as one of most efficient and popular ICA algorithms has been reviewed and discussed. Its local and global convergence and statistical behavior have been further studied. A nonnegative FastICA algorithm is also given in this thesis. Nonnegative matrix factorization is a recently developed technique for finding parts-based, linear representations of non-negative data. It is a method for dimensionality reduction that respects the nonnegativity of the input data while constructing a low-dimensional approximation. The non-negativity constraints make the representation purely additive (allowing no subtractions), in contrast to many other linear representations such as principal component analysis and independent component analysis. A literature survey of Nonnegative matrix factorization is given in this thesis, and a novel method called Projective Nonnegative matrix factorization (P-NMF) and its applications are provided.