Lp-norm non-negative matrix factorization and its application to singing voice enhancement

Abstract

Measures of sparsity are useful in many aspects of audio signal processing including speech enhancement, audio coding and singing voice enhancement, and the well-known method for these applications is non-negative matrix factorization (NMF), which decomposes a non-negative data matrix into two non-negative matrices. Although previous studies on NMF have focused on the sparsity of the two matrices, the sparsity of reconstruction errors between a data matrix and the two matrices is also important, since designing the sparsity is equivalent to assuming the nature of the errors. We propose a new NMF technique, which we called L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> -norm NMF, that minimizes the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> norm of the reconstruction errors, and derive a computationally efficient algorithm for L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> -norm NMF according to an auxiliary function principle. This algorithm can be generalized for the factorization of a real-valued matrix into the product of two real-valued matrices. We apply the algorithm to singing voice enhancement and show that adequately selecting p improves the enhancement.