Non-Negative Matrix Factorization (NMF)

Abstract

In this chapter we introduce the Non-Negative Matrix Factorization (NMF), which is an unsupervised algorithm that projects data into lower dimensional spaces, effectively reducing the number of features while retaining the basis information necessary to reconstruct the original data. Basically, it decomposes a matrix, containing only non-negative coefficients, into the product of two other non-negative matrices with reduced ranks. Since negative coefficients are not allowed, the original data is reconstructed through additive combinations of the parts-based factorized matrix representation. Following, we present the multiplicative and the additive GPU implementations of the NMF algorithm for the Euclidean distance as well as for the divergence cost function. In addition, a new semi-supervised approach that reduces the computational cost while improving the accuracy of NMF-based models is also presented. Finally, we present results for well-known face recognition benchmarks that demonstrate the advantages of both the proposed method and the GPU implementations.