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Non-negative matrix factorization in polynomial feature space

Domenii publicaţii > Ştiinţe informatice + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: Ioan Buciu, Nikos Nikolaidis, Ioannis Pitas

Editorial: IEEE Transactions on Neural Networks, 19 (6), p.1090-695, 2008.


Plenty of methods have been proposed in order to discover latent variables (features) in data sets. Such approaches include the Principal Component Analysis (PCA), Independent Component Analysis (ICA), Factor Analysis (FA), etc., to mention only a few. A recently investigated approach to decompose a data set with a given dimensionality into a lower dimensional space is the so-called Non-negative Matrix Factorization (NMF). Its only requirement is that both decomposition factors are non-negative. To approximate the original data, the minimization of the NMF objective function is performed in the Euclidean space, where the difference between the original data and the factors can be minimized by employing $L_2$ norm. We propose a generalization of the NMF algorithm by translating the objective function into a Hilbert space (also called feature space) under nonnegativity constraints. With the help of kernel functions we developed an approach that allows high-order dependencies between the basis images while keeping the non-negativity constraints on both basis images and coefficients. Two practical applications, namely facial expression and face recognition, show the potential of the proposed approach.

Cuvinte cheie: Non-negative Matrix Factorization, Hilbert space, polynomial kernel