Multiplicative update algorithms have encountered a great success to solve optimization problems with nonnegativity constraints, such as the famous non-negative matrix factorization and its many variants. However, despite several years of research on the topic, the understanding of their convergence properties is still to be improved. In this paper, we show that Lyapunov’s stability theory prov...

We introduce a new design for visualizing high dimensional data. Points in high dimensional space are often illustrated through 2D or 3D embeddings, which are often cluttered and may-be unaccessible to non-scientific audience. In our design, instead of representing data points as 2D/3D vectors, each data point is represented by a smoothly varying function of time. These smooth functions are use...

In this short note, we focus on the use of the generalized Kullback–Leibler (KL) divergence in the problem of non-negative matrix factorization (NMF). We will show that when using the generalized KL divergence as cost function for NMF, the row sums and the column sums of the original matrix are preserved in the approximation. We will use this special characteristic in several approximation prob...

In this paper, the problem of nonnegative matrix factorization (NMF) is considered. It is formulated as the optimization of a criterion with bound constraints. We propose an approach based on Givens parameterization of some positive vector, and criterion minimization is achieved using Levenberg-Marquardt algorithm. The performance of the developed NMF method is illustrated for the separation of...

Matrix factorization (MF) is one of the most powerful approaches used in the frame of recommender systems. It aims to model the preferences of users about items through a reduced set of latent features. One main drawback of MF is the difficulty to interpret the automatically formed features. Following the intuition that the relation between users and items can be expressed through a reduced set...

In this paper, I will give a brief introduction to a data analysis technique called non-negative matrix factorization (NMF), which has attracted a lot of attention in the field of audio signal processing in recent years. I will mention some basic properties of NMF, effects induced by the non-negative constraints, how to derive an iterative algorithm for NMF, and some attempts that have been mad...

Despite its relative novelty, non-negative matrix factorization (NMF) method knew a huge interest from the scientific community, due to its simplicity and intuitive decomposition. Plenty of applications benefited from it, including image processing (face, medical, etc.), audio data processing or text mining and decomposition. This paper briefly describes the underlaying mathematical NMF theory ...

In this paper, we investigate the problem of real-time polyphonic music transcription by employing non-negative matrix factorization techniques and the β-divergence as a cost function. We consider real-world setups where the music signal arrives incrementally to the system and is transcribed as it unfolds in time. The proposed transcription system is addressed with a modified non-negative matri...