Weighted Nonnegative Matrix Factorization and Face Feature Extraction

In this paper we consider weighted nonnegative matrix factorizations and we show that the popular algorithms of Lee and Seung can incorporate such a weighting. We then prove that for appropriately chosen weighting matrices, the weighted Eu-clidean distance function and the weighted generalized Kullback-Leibler divergence function are essentially identical. We finally show that the weighting can be chosen to emphasize parts of the data matrix to be approximated and we can apply this to the low rank fitting of a face image database. Key words: Non-negative matrix factorization, weighting, Euclidean distance, generalized Kullback-Leibler divergence