A sparse optimization approach to supervised NMF based on convex analytic method

In this paper, we propose a novel scheme to supervised nonnegative matrix factorization (NMF). We formulate the supervised NMF as a sparse optimization problem assuming the availability of a set of basis vectors, some of which are irrelevant to a given matrix to be decomposed. The proposed scheme is presented in the context of music transcription and musical instrument recognition. In addition to the nonnegativity constraint, we introduce three regularization terms: (i) a block ℓ₁ norm to select relevant basis vectors, and (ii) a temporal-continuity term plus the popular ℓ₁ norm to estimate correct activation vectors. We present a state-of-the-art convex-analytic iterative solver which ensures global convergence. The number of basis vectors to be actively used is obtained as a consequence of optimization. Simulation results show the efficacy of the proposed scheme both in the case of perfect/imperfect basis matrices.