Supervised nonnegative matrix factorization via minimization of regularized Moreau-envelope of divergence function with application to music transcription

We propose a convex-analytic approach to supervised nonnegative matrix factorization (NMF), using the Moreau envelope, a smooth approximation, of the β-divergence as a loss function. The supervised NMF problem is cast as minimization of the loss function penalized by four terms: (i) a time-continuity enhancing regularizer, (ii) the indicator function enforcing the nonnegativity, (iii) a basis-vector selector (a block ℓ1 norm), and (iv) a sparsity-promoting regularizer. We derive a closed-form expression of the proximity operator of the sum of the three non-differentiable penalty terms (ii)-(iv). The optimization problem can thus be solved numerically by the proximal forward-backward splitting method, which requires no auxiliary variable and is therefore free from extra errors. The source number is automatically attained as an outcome of optimization. The simulation results show the efficacy of the proposed method in an application to polyphonic music transcription.