On factorisation forests

Factorisation forests for infinite words Application to countable scattered linear orderings

The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. We provide two improvements to the standard result. First we improve on all previously known bounds for the standard theorem. Second, we extend it to every ‘complete linear ordering’. We use this variant in a...

On factorisation forests And some applications

The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show th...

Factorisation Forests for Infinite Words

Probabilistic non-negative matrix factorisation and extensions

Matrix factorisation models have had an explosive growth in popularity in the last decade. It has become popular due to its usefulness in clustering and missing values prediction. We review the main literature for matrix factorisation, focusing on nonnegative matrix factorisation and probabilistic approaches. We also consider several extensions: matrix tri-factorisation, Tensor factorisation, T...

Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation

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)...