A Dynamical System Approach for Continuous Nonnegative Matrix Factorization

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• 作者：Melisew Tefera Belachew ; Nicoletta Del Buono
• 关键词：Nonnegative matrix factorization ; time ; dependent data ; gradient flow ; continuous ; time dynamical systems
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：14
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1660-5454
• 卷排序：14

文摘

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 \(\alpha \)-divergence.