Box-constrained projective nonnegative matrix factorization via augmented Lagrangian method

Projective non-negative matrix factorization (P-NMF) projects a set of examples onto a subspace spanned by a non-negative basis whose transpose is regarded as the projection matrix. Since PNMF learns a natural parts-based representation, it has been successfully used in text mining and pattern recognition. However, it is non-trivial to analyze the convergence of the optimization algorithms for PNMF because its objective function is non-convex. In this paper, we propose a Box-constrained PNMF (BPNMF) method to overcome this deficiency of PNMF. In particular, BPNMF introduces an auxiliary variable, i.e., the coefficients of examples, and incorporates the following two types of constraints: 1) each entry of the basis is non-negative and upper-bounded, i.e., box-constrained, and 2) the coefficients equal to the projected points of the examples. The first box constraint makes the basis to be bound and the second equality constraint keeps its equivalence to PNMF. Similar to PNMF, BPNMF is difficult because the objective function is non-convex. To solve BPNMF, we developed an efficient algorithm in the frame of augmented Lagrangian multiplier (ALM) method and proved that the ALM-based algorithm converges to local minima. Experimental results on two face image datasets demonstrate the effectiveness of BPNMF compared with the representative methods.