## 3 Proposed System
### Iterative and Non-Iterative Feature Extraction Methods  
Once the frequency basis matrix \(\mathbf{W}\) is trained, the feature matrix \(\mathbf{H}_{clip}\) can be extracted from the data matrix \(\mathbf{V}_{clip}\) of an audio clip via the iteration of [31]  
\[\mathbf{H}_{clip}\leftarrow\mathbf{H}_{clip}\otimes\frac{\mathbf{W}^{T}\Big{[} \mathbf{V}_{clip}/(\mathbf{W}\mathbf{H}_{clip})\Big{]}}{\mathbf{W}^{T}\mathbf{ 1}_{K\times N_{clip}}}. \tag{11}\]  
with minimization of the KLD (as Equation (5)). In this paper, we also develop a non-iterative feature extraction method. To achieve this goal, we develop a closed-form solution for \(\mathbf{H}_{clip}\) which makes the gradient of the divergence function with regard to \(\mathbf{H}_{clip}\) zero (that is, \(\
abla_{\mathbf{H}_{clip}}D\Big{(}\mathbf{V}_{clip}|\mathbf{W}\mathbf{H}_{ clip}\Big{)}=0\)). In addition, a solution with simple operations, e.g. matrix multiplication, is a preferred one because it is easy to implement. The NMF-based methods for acoustic signal processing mainly use three types of distance functions: KLD, Euclidean distance, and Itakura-Saito divergence (ISD). The three distance functions are in the \(\beta\)-divergence family, which is expressed by [38]:  
\[D_{\beta}(x|y)=\left\{ \begin{array}{ll}\frac{x}{y}-\log\frac{x}{y}-1&\beta= 0\\ x(\log x-\log y)+(y-x)&\beta=1\\ \frac{1}{\beta(\beta-1)}\big{(}x^{\beta}+(\beta-1)y^{\beta}-\beta xy^{\beta-1} \big{)}&\text{otherwise}\end{array} \right. \tag{12}\]  
where \(x\) and \(y\) are arbitrary values and \(\beta\) is a constant to adjust the cost function among the Euclidean (\(\beta=2\)), KLD (\(\beta=1\)), and ISD (\(\beta=0\)). In our problem, \(x\) and \(y\) are elements of the matrices \(\mathbf{V}_{clip}\) and \(\mathbf{W}\mathbf{H}_{clip}\), respectively, and the gradient of the beta-divergence with regard to \(\mathbf{H}_{clip}\) is [38]  
\[\
abla_{\mathbf{H}_{clip}}D_{\beta}\Big{(}\mathbf{V}_{clip}\big{|}\mathbf{W} \mathbf{H}_{clip}\Big{)}=\mathbf{W}^{T}\bigg{\{}\Big{(}\mathbf{W}\mathbf{H}_ {clip}\Big{)}^{\cdot[\beta-2]}\otimes\Big{(}\mathbf{W}\mathbf{H}_{clip}- \mathbf{V}_{clip}\Big{)}\bigg{\}} \tag{13}\]  
where \(\mathbf{A}^{[n]}\) denotes element-wise \(n\)th power of matrix \(\mathbf{A}\). Because Equation (13) includes matrix multiplications, element-wise products and power, and the relationship between these operations is also complex. Therefore, it is not easy to find a solution that makes Equation (13) zero, and it is even more difficult to find a solution consisting of simple matrix multiplications. One easy way is to make the term of the left of the Hadamard product, \((\mathbf{W}\mathbf{H})^{\cdot[\beta-2]}\), equal to one. In this case, \(\beta\) becomes 2.0, and the divergence function becomes the Euclidean distance. The Euclidean distance function can be defined using the Frobenius norm as  
\[D_{EUC}\Big{(}\mathbf{V}_{clip}|\mathbf{W}\mathbf{H}_{clip}\Big{)}=\frac{1}{2 }\big{\|}\mathbf{V}_{clip}-\mathbf{W}\mathbf{H}_{clip}\big{\|}_{F}^{2}, \tag{14}\]  
and the gradient of the cost function with respect to \(\mathbf{H}_{clip}\) is  
\[\
abla_{\mathbf{H}_{clip}}D_{EUC}\Big{(}\mathbf{V}_{clip}\big{|}\mathbf{W} \mathbf{H}_{clip}\Big{)}=-\mathbf{W}^{T}\Big{[}\mathbf{V}_{clip}-\mathbf{W} \mathbf{H}_{clip}\Big{]}. \tag{15}\]  
Since the cost function in Equation (14) is convex, the optimal solution of the cost function makes the gradient zero. Therefore, the optimal solution of \(\mathbf{H}_{clip}\) needs to satisfy the relationship:  
\[\mathbf{W}^{T}\mathbf{V}_{clip}-\mathbf{W}^{T}\mathbf{W}\mathbf{H}_{clip}= \mathbf{0} \tag{16}\]  
and therefore  
\[\mathbf{H}_{clip} =\Big{[}\mathbf{W}^{T}\mathbf{W}\Big{]}^{-1}\mathbf{W}^{T} \mathbf{V}_{clip}\] \[=\mathbf{W}^{T}\mathbf{V}_{clip}, \tag{17}\]where \(\mathbf{W}^{\dagger}\) denotes the Moore-Penrose pseudo-inverse of the matrix \(\mathbf{W}\). Since the frequency basis matrix \(\mathbf{W}\) is pre-trained and fixed, \(\mathbf{W}^{\dagger}\) is also defined a priori.