Polytopic matrix factorization (PMF): A new data decomposition tool

Matrix factorization methods are widely used in signal processing and machine learning applications. These methods lay the foundation of a large number of algorithms utilized in problems of those areas. As one of the main problems, we try to discover the information hidden inside input data. As a general solution strategy for this problem, input data is modeled as a product of two factors/matrices. This is known as Blind Source Separation (Blind Data Decomposition) in the signal processing literature. In this thesis, we introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. We model input data as unknown linear transformations of some latent vectors drawn from a polytope. The choice of polytope determines the presumed structure of latent vectors and their relationships. We first propose the identifiability criterion and identifiability conditions of PMF regarding the latent vectors. We introduce a sufficient condition for identifiability, which requires that the maximum volume inscribed ellipsoid of the polytope is contained in the convex hull of the latent vectors with a particular tightness constraint. We propose PMF identifiability results of special polytope cases corresponding to widely utilized feature attributes in applications. Then, a generalized version of PMF is presented with the characterization of eligible polytope choices and we propose to use a decision algorithm to determine these eligible polytopes. The proposed PMF tool is extended by considering a special class of linear mappings on eligible polytopes. We further extend PMF as Bounded Matrix Factorization and provide identifiability results of some bounded sets referring to polytopes. We furthermore present a PMF algorithm and interesting examples that utilize PMF as a data decomposition tool. PMF enables us to use infinitely many polytope choices and bounded sets in characterizing latent vectors. Therefore, it is possible to define different presumed structures and relations for latent vectors such as nonnegativity, sparsity and such attributes in subvector level. In brief, we present a novel data decomposition tool that provides a high degree of flexibility in terms of presumed structures of latent vectors and offer examples illustrating this flexibility with different eligible sets and corresponding feature attributes.