An application of multiple-stage non-negative matrix factorization to music analysis

Non-negative matrix factorization (NMF) is an analysis technique for matrix with non-negative elements. NMF decomposes a sound spectrogram X obtained by a short-time Fourier transform into the product of two non-negative matrices H and U. The matrix H represents the bases of X and the matrix U represents the activation gains of H. As an extension of NMF, multiple-stage NMF has been proposed in order to represent the hierarchy of data. This research focuses on the hierarchy of music data, and attempts to apply multiple-stage NMF to music analysis. For simplicity, sounds treated in this research are assumed as follows: single tone consists of fundamental tone and harmonic overtones and a chord consists of some single tones. According to this assumption, observed sound spectrogram can be decomposed into three stages. In the first stage, the basis matrix is approximated by a Gaussian distribution in order to represent a spectrum of fundamental tone and harmonic overtones. In the second stage, the basis matrix...