## 3 Proposed System  
### Strategy for the Frequency Basis Learning  
Similar to the previous studies [5; 31], the temporal basis matrix **H** is used as a feature. To extract the feature matrix, the frequency basis matrix needs to be estimated in advance. Inspired by speech denoizing [27] and reverberation suppression [37] techniques based on NMF [27], the frequency basis matrix of each class is independently estimated from the spectrogram set of the class, as presented in Figure 3a. In this method, the frequency basis matrix is divided into \(C\) groups as [31]  
\[\textbf{W}=[\textbf{W}_{1}\quad\textbf{W}_{2}\quad\cdots\quad\textbf{W}_{C}]. \tag{8}\]  
Each group denoted by \(\textbf{W}_{c}\) is a \(K\times R_{C}\) matrix, where \(R_{C}\) denotes the basis number of each class. The frequency and temporal basis matrices can be estimated by optimizing various cost functions. KLD is one of the common choices in the acoustic signal processing field. Therefore, the frequency and temporal basis matrices are iteratively updated by  
\[\textbf{W}_{c}\leftarrow\textbf{W}_{C}\otimes\frac{[\textbf{V}_{c}/( \textbf{W}_{c}\textbf{H}_{c})]\textbf{H}_{c}^{T}}{\textbf{1}_{K\times N_{c}} \textbf{H}_{c}^{T}} \tag{9}\]  
\[\textbf{H}_{c}\leftarrow\textbf{H}_{c}\otimes\frac{\textbf{W}_{c}^{T}[\textbf {V}_{c}/(\textbf{W}_{c}\textbf{H}_{c})]}{\textbf{W}_{c}^{T}\textbf{1}_{K\times N _{c}}}, \tag{10}\]  
which is similar to Equations (4) and (5), where \(\textbf{V}_{c}\) and \(\textbf{H}_{c}\) denote \(K\times N_{c}\) data matrix and \(R_{C}\times N_{c}\) temporal basis matrix, respectively, and \(N_{c}\) stands for the total number of frames in the data matrix of the \(c\)th class.  
Since weakly-supervised SED has a heterogeneous database, different learning strategies for each type of data need to be developed. In the case of strongly-labeled data, which contains the class and temporal information, the data matrix is a set of audio clips with a cut-off part, as presented in Figure 3b. As can be seen in the figure, there is a risk of data mixing from different classes. For example, if we assume that we want to generate the data matrix **V** for Class A, there may be various combinations, e.g., A-B, A-C, and A-D. However, even in this case, the data from Class A is expected to be dominant. Thus, the  
Figure 2: Schematic diagram of the applications of the non-negative matrix factorization (NMF) methods to the acoustic signal processing systems.  
frequency matrix **W** can be expected to represent the characteristics of Class A if we choose a sufficiently small rank for matrix **W**.  
For the weakly-labeled data that do not contain an onset and offset information, the temporal information cannot be utilized. Therefore, we compose the data matrix **V** by just cascading the clips, as shown in Figure 3c. Unlike the case of the strongly-labeled data, where only parts of the event overlap, in the weakly-labeled data, waveforms from different classes are completely mixed when two or more classes are in one clip. Therefore, it is expected that the training of matrix **W** is more affected by the unwanted class interference problem compared to strongly-labeled data. Therefore, we evaluate the effect of the unwanted class interference problem on the weakly-labeled data.  
Figure 3: Block diagrams for: (**a**) the learning of the frequency basis; (**b**) the learning of the composition of the data matrix from strongly-labeled data; and (**c**) the learning of the composition of the data matrix from weakly-labeled data.