Local Coordinate Projective Non-negative Matrix Factorization

Non-negative matrix factorization (NMF) decomposes a group of non-negative examples into both lower-rank factors including the basis and coefficients. It still suffers from the following deficiencies: 1) it does not always ensure the decomposed factors to be sparse theoretically, and 2) the learned basis often stays away from original examples, and thus lacks enough representative capacity. This paper proposes a local coordinate projective NMF (LCPNMF) to overcome the above deficiencies. Particularly, LCPNMF induces sparse coefficients by relaxing the original PNMF model meanwhile encouraging the basis to be close to original examples with the local coordinate constraint. Benefitting from both strategies, LCPNMF can significantly boost the representation ability of the PNMF. Then, we developed the multiplicative update rule to optimize LCPNMF and theoretically proved its convergence. Experimental results on three popular frontal face image datasets verify the effectiveness of LCPNMF comparing to the representative methods.