Nonnegative matrix factorization with quadratic programming

Nonnegative matrix factorization (NMF) solves the following problem: find such nonnegative matrices A∈R+I×J and X∈R+J×K that Y≅AXY≅AX, given only Y∈RI×KY∈RI×K and the assigned index J   (K⪢I⩾JK⪢I⩾J). Basically, the factorization is achieved by alternating minimization of a given cost function subject to nonnegativity constraints. In the paper, we propose to use quadratic programming (QP) to solve the minimization problems. The Tikhonov regularized squared Euclidean cost function is extended with a logarithmic barrier function (which satisfies nonnegativity constraints), and then using second-order Taylor expansion, a QP problem is formulated. This problem is solved with some trust-region subproblem algorithm. The numerical tests are performed on the blind source separation problems.