A Dynamical System Approach for Continuous Nonnegative Matrix Factorization

Abstract

Nonnegative matrix factorization is a linear dimensionality reduction technique used for decomposing high-dimensional nonnegative data matrices for extracting basic and latent features. This technique plays fundamental roles in music analysis, signal processing, sound separation, and spectral data analysis. Given a time-varying objective function or a nonnegative time-dependent data matrix Y(t), the nonnegative factors of Y(t) can be obtained by taking the limit points of the trajectories of the corresponding ordinary differential equations. When the data are time dependent, it is natural to devise factorization techniques that capture the time dependency. To achieve this, one needs to solve continuous-time dynamical systems derived from iterative optimization schemes and construct nonnegative matrix factorization algorithms based on the solution curves. This article presents continuous nonnegative matrix factorization methods based on the solution of systems of ordinary differential equations associated with time-dependent data. In particular, we propose two new continuous-time algorithms based on the Kullback–Leibler divergence and the Amari α -divergence

Adaptive Graph Regularization Discriminant Nonnegative Matrix Factorization for Data Representation

“…This attribute is consistent with the physiological and psychological elements based on the representation of the parts of the human brain. As an ideal algorithm, NMF has shown advantages in image clustering [11]- [13], face recognition [14]- [16], pattern recognition [17]- [19], and text clustering [20]- [22] and so on [23]- [25]. In recent years, some researchers put forward some new algorithms by adding additional constraints to NMF.…”