Rapid technological advances have led to the production of different types of biological data and enabled construction of complex networks with various types of interactions between diverse biological entities. Standard network data analysis methods were shown to be limited in dealing with such heterogeneous networked data and consequently, new methods for integrative data analyses have been pr...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n × d matrix W and a nonnegative d ×m matrix H. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negat...

Nonnegative matrix factorisation and tri-factorisation Nonnegative matrix factorisation (NMF) and tri-factorisation (NMTF) methods decompose a given matrix R into two or three smaller matrices so that R ≈ UV T or R ≈ FSG , respectively. Schmidt, Winther and Hansen (2009) introduced a Bayesian version of nonnegative matrix factorisation (left), which we extend to matrix tri-factorisation (right)...

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Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can impose on it to necessitate analogs of unique factorization. The trivial ideal structure of a field, the extent to which primary decomposition is unique, that a Noetherian ring necessarily has one, that a principal ideal domain is a unique factorization domain, and that a Dedekind domain h...

High tempo detection accuracies have been reported for the analysis of percussive, constant-tempo, Western music audio signals. As a consequence, active research in the tempo detection domain has been shifted to yet open tasks like tempo analysis of non-percussive, expressive, or non-western music. Also, tempo detection is included in a large range of music-related software. In DJ software, fea...

Non-Negative Matrix Factorization (NMF) is a dimensionality reduction method that has been shown to be very useful for a variety of tasks in machine learning and data mining. One of the fastest algorithms for NMF is the Block Principal Pivoting method (BPP) of [6], which follows a block coordinate descent approach. The optimization in each iteration involves solving a large number of expensive ...