Estimating the Rank of a Nonnegative Matrix Factorization Model for Automatic Music Transcription Based on Stein’s Unbiased Risk Estimator

Abstract

In this paper, methods to estimate the number of basis vectors of the nonnegative matrix factorization (NMF) of automatic music transcription (AMT) systems are proposed. Previously, studies on NMF-based AMT have demonstrated that the number of basis vectors affects the performance and that the number of note events can be a good selection as the rank of NMF. However, many NMF-based AMT methods do not provide a method to estimate the appropriate number of basis vectors; instead, the number is assumed to be given in advance, even though the number of basis vectors significantly affects the algorithm’s performance. Recently, based on Bayesian methods, certain estimation algorithms for the number of basis vectors have been proposed; however, they are not designed to be used as music transcription algorithms but are components of specific NMF methods and thus cannot be used generally as NMF-based transcription algorithms. Our proposed estimation algorithms are based on eigenvalue decomposition and Stein’s unbiased risk estimator (SURE). Because the SURE method requires variance in undesired components as a priori knowledge, the proposed algorithms estimate the value using random matrix theory and first and second onset information in the input music signal. Experiments were then conducted for the AMT task using the MIDI-aligned piano sounds (MAPS) database, and these algorithms were compared with variational NMF, gamma process NMF, and NMF with automatic relevance determination algorithms. Based on experimental results, the conventional NMF-based transcription algorithm with the proposed rank estimation algorithms demonstrated enhanced F1 score performances of 2–3% compared to the algorithms. While the performance advantages are not significantly large, the results are meaningful because the proposed algorithms are lightweight, are easy to combine with any other NMF methods that require an a priori rank parameter, and do not have setting parameters that considerably affect the performance.