## 6 Results and Discussion
### Disentanglement of Bass Patterns  
Next, we look at the capability of the models to disentangle the bass patterns that were defined in the artificial datasets. To that end, we plot the latent variable for a subset of the data, coloring each point according to the corresponding bass pattern in the ground truth annotations (Figure 5). More specifically, each point in the plots is the mean \(\mu^{(x,y)}\) of the posterior distribution \(\mathcal{N}^{(x,y)}\) for a random pair \((x,y)\) from the validation data, and it is colored according to the bass pattern that was used to produce the bass from the mix when creating the artificial data.  
Even if the plots already give a good visual impression of the disentanglement, we can quantify the separation of patterns in the latent space using quadratic discriminant analysis. This analysis consists of estimating a quadratic classifier to predict the pattern \(c\) from the latent variable \(z\). It works by defining the pattern-conditional distribution of the latent variable \(p(z|c)\) as a multivariate gaussian, and using Bayes rule to infer \(p(c|z)\). For a given instance \((x,y)\), the predicted class is defined as the class with the highest conditional probability \(argmax_{c}P(c|\mu^{(x,y)})\). The background colors in Figure 5 show how the classifiers partition the latent space into classes. The plots also include the isolines that correspond to \(p(c|z)=0.75\) for each pattern \(c\). The prediction accuracies of the classifiers serve as a quantitative measure of the disentanglement of the patterns in the learned latent spaces. Table 2 lists the different models and datasets.  
The results show that, in most of the cases, but not all, the latent space is organized in order to clearly separate the bass patterns. Models with window size 20/20 and those with expanded window sizes of 40 (i.e., 10/40, 20/40, and 40/40) produce virtually complete disentanglement of bass patterns. Models with longer windows (10/640, 20/640, and 40/640) have difficulty separating the patterns in datasets A2 and A3. In A2, the separation of the bass patterns requires models to successfully recognize the chord type (major vs minor), whereas, in A3, the separation requires sensitivity to instrument timbre in the mix. It is not surprising that, with longer windows, it is more difficult to recognize these aspects of the mix, because the longer windows will almost always span multiple chords, likely with different chord types, and in the case of A3 different instrument timbres. For A1 and A4, the separation of the models with longer windows is clearly visible, but not as clean as the shorter window models. A more surprising result is confusion of patterns in A3 for the 10/10 and 10/20 models, especially since the 20/20 model has no problem finding a clear separation. When comparing the entries of column A3/f0 for models 10/20 and 10/20 in Table 1, the lack of separation of the patterns appears to have a negative impact on the cross-reconstruction loss. A possible explanation for the fact that the short window models fail to capture  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**A1**} & \multicolumn{3}{c}{**A2**} & \multicolumn{3}{c}{**A3**} & \multicolumn{3}{c}{**A4**} & \multicolumn{3}{c}{**PR**} \\ \cline{2-13}
**Size** & \(f_{0}\) & \(l\) & \(o\) & \(f_{0}\) & \(l\) & \(o\) & \(f_{0}\) & \(l\) & \(o\) & \(f_{0}\) & \(l\) & \(o\) & \(f_{0}\) & \(l\) & \(o\) \\ \hline
10/10 & **0.263** & **0.479** & **0.050** & **0.255** & **0.480** & **0.064** & 0.438  
mass patterns that are conditioned on instrument in A3, is that the windows are too short to capture the temporal dynamics that help to distinguish between guitar and organ.  
Figure 5: _Cont._