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In this letter, we address an audio signal separation problem and propose a new effective algorithm for solving a bilevel optimization in discriminative nonnegative matrix factorization (NMF). Recently, discriminative training of NMF bases has been developed for better signal separation in supervised NMF (SNMF), which exploits a priori training of given sample signals. The optimization in this method consists of a simultaneous minimization of two objective functions, resulting in a bilevel optimization problem with SNMF (BiSNMF), where conventional methods approximately solve this optimization. To strictly solve BiSNMF, we introduce a new algorithm with the following two features: (a) conversion of the optimization constraint into a penalty term and (b) optimization of the reformulated problem on the basis of a multiplicative steepest descent, ensuring the nonnegativity of variables. Experiments on music signal separation show the efficacy of the proposed algorithm.
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