Nonnegative Matrix Factorization (NMF) is an unsupervised learning method for extracting latent features in high-dimensional nonnegative data. This report presents NMF strategies based on the solution of the systems of ordinary differen- tial equations associated with both static and time dependent data. Here the NMF minimization problem is treated in two ways: we start off by deriving dynami- cal systems using the gradient descent approach and finish off with quasi-Newton optimization. The steepest descent approach by itself is divided into two cases: con- strained and unconstrained minimization. Given a time-varying objective function or a nonnegative time dependent (static) data matrix Y , the approximate nonneg- ative factors of Y can be obtained by taking the limit points of the trajectories of the corresponding ODEs. In the case when the data to deal with is time dependent, it is a good practice to devise an NMF strategy that captures the time dependency. In either of the above cases the natural thing to do is solve the continuous-time dy- namical systems derived from iterative optimization schemes and construct NMF algorithms based on the solution curves.

Nonnegative Matrix Factorization based on the Solution of Systems of ODEs

DEL BUONO, Nicoletta
2014

Abstract

Nonnegative Matrix Factorization (NMF) is an unsupervised learning method for extracting latent features in high-dimensional nonnegative data. This report presents NMF strategies based on the solution of the systems of ordinary differen- tial equations associated with both static and time dependent data. Here the NMF minimization problem is treated in two ways: we start off by deriving dynami- cal systems using the gradient descent approach and finish off with quasi-Newton optimization. The steepest descent approach by itself is divided into two cases: con- strained and unconstrained minimization. Given a time-varying objective function or a nonnegative time dependent (static) data matrix Y , the approximate nonneg- ative factors of Y can be obtained by taking the limit points of the trajectories of the corresponding ODEs. In the case when the data to deal with is time dependent, it is a good practice to devise an NMF strategy that captures the time dependency. In either of the above cases the natural thing to do is solve the continuous-time dy- namical systems derived from iterative optimization schemes and construct NMF algorithms based on the solution curves.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11586/93335
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