## 3 Method for Estimating the NMF Rank
### Noise Variance Estimation  
While the detailed derivation process differs from the method in [16], the developed results (22) are similar to those of SURE-based algorithm in [16; 18]. However, (22) requires noise variance information, which has a significant impact on performance. To estimate the noise variance, the method used in [16] crudely estimates the rank based on the probability model, and that used in [18] performs noise-component extraction based on the median filtering. While neither of these methods use any characteristics of NMF-based AMT techniques, this study proposes a new noise variance method by considering such characteristics. As per random matrix theory, noise eigenvalues follow a Marchenko-Pastur (MP) density [16; 27]. The MP density for \(a\leq x\leq b\) is  
\[f_{\gamma,\sigma_{e}^{2}}(x)=\frac{\gamma}{2\pi\sigma_{e}^{2}x}\sqrt{(b-x)(x- a)},\quad(a\leq x\leq b) \tag{23}\]  
and \(f_{\gamma,\sigma_{e}^{2}}(x)=0\) for \(x<a\) or \(x>b\), where  
\[a=\sigma_{e}^{2}(1-\gamma^{-1/2})^{2}, \tag{24}\] \[b=\sigma_{e}^{2}(1+\gamma^{-1/2})^{2}, \tag{25}\]  
and \(\gamma=N/K\). According to [16; 27], the largest noise eigenvalue converges as  
\[l_{R+1}\to\sigma_{e}^{2}(1+\gamma^{-1/2})^{2},\quad\text{{almost surely}}. \tag{26}\]  
Furthermore, the median value satisfies the relationship [16]  
\[med(l_{R+1},\cdots,l_{\rho})\to F_{\gamma,\sigma_{e}^{2}}^{-1}\left(\frac{1}{ 2}\right),\ \text{{almost surely}}, \tag{27}\]  
where \(med(C)\) is a median value of a set \(C\), \(F_{\gamma,\sigma_{e}^{2}}^{-1}\) is the inverse of the cumulative distribution function for the MP distribution, and \(\rho\) is a smaller value of \(K\) and \(N\). Consequently, if we know the maximum noise eigenvalue, we can estimate \(\sigma_{e}^{2}\) from that eigenvalue. Generally, it is necessary to know the value of \(R\) to determine the maximum noise eigenvalue \(l_{R+1}\). Thus, model order estimation algorithms based on SURE and RMT use the following  
Figure 6a shows that a large eigenvalue \((j=1)\) corresponds to the signal \(\mathbf{v}_{0}\), and 20 small eigenvalues \((2\leq k\leq 21)\) correspond to the remaining components \(\mathbf{e}(n)\). The remaining eigenvalues \((k\geq 22)\) are not shown in the graph because their values are zero, agreeing with the theory in [27]. Figure 7a shows that four large eigenvalues \((1\leq k\leq 4)\) correspond to the signal \(\mathbf{v}_{0}\), and 83 small eigenvalues \((5\leq k\leq 87)\) correspond to the remaining noisy components. Figures 6b and 7b show the eigenvalues within the range of 0-100 for comparison with the noisy signal properties. Comparing the \"o\" marks and the \"+\" marks in the Figure 7b, the graph shows that the noise eigenvalues of the data for \(1\leq n\leq 21\) and \(1\leq n\leq 87\) have similar properties, i.e., maximum value, median value, and shape. Because \(\gamma\) is different when \(N_{p}=21\) and \(N_{p}=87\), the maximum noise eigenvalue is different even if \(\sigma_{e}^{2}\) is the same. If we compensate the difference of \(\gamma\) between two points, the magnitude of \(l_{2}(N_{p}=21)\), 78.04, is equivalent to the value, 26.02, when \(N=87\). While the maxima of the noise eigenvalues are not exactly the same, the difference between the noise eigenvalues of the data for \((N_{p}=21)\) and \((N_{p}=87)\) is very small ( 9.4 if we compensate for the change of \(\gamma\)) compared to the differences between the magnitudes of the signal eigenvalues and the noise eigenvalues for each data case. The difference between the median values is small. Thus, this data can be used to make a crude estimation of the MP distribution parameters and \(\sigma_{e}^{2}\). We defined \(N_{rank1}\) as the frame number for an arbitrary frame between the first and the second onset frames. If there is only one note in \(1\geq n\geq N_{rank1}\), as shown in the example, \(\mathbf{R_{v}}(N_{rank1})\) is a rank-1 matrix.