Projective Nonnegative Matrix Factorization with α-Divergence

A new matrix factorization algorithm which combines two recently proposed nonnegative learning techniques is presented. Our new algorithm, alpha-PNMF, inherits the advantages of Projective Nonnegative Matrix Factorization (PNMF) for learning a highly orthogonal factor matrix. When the Kullback-Leibler (KL) divergence is generalized to alpha-divergence, it gives our method more flexibility in approximation. We provide multiplicative update rules for alpha-PNMF and present their convergence proof. The resulting algorithm is empirically verified to give a good solution by using a variety of real-world datasets. For feature extraction, alpha-PNMF is able to learn highly sparse and localized part-based representations of facial images. For clustering, the new method is also advantageous over Nonnegative Matrix Factorization with alpha-divergence and ordinary PNMF in terms of higher purity and smaller entropy.