## 6 Results and Discussion
### Effect of Cross-Reconstruction Loss  
The previous section shows that training models with intermediate size windows leads to a disentanglement of bass patterns in the latent space. In this section, we examine to what degree this organization of the latent space is induced by the constraint of temporal stability of latent space trajectories. We do so by way of an ablation study in which we train a model while using the regular reconstruction loss instead of the cross-reconstruction loss. Recall that the original training objective (Equation (9)) involves an instance \((x,y)\) and a randomly selected nearby neighbor \((u,v)\). In that objective, the reconstruction terms \(\hat{y}\), \(\hat{f_{0}}\), \(\hat{\delta}\), and \(\hat{l}\) for the instance \((x,y)\) are computed based on a sample from latent space distribution \(p(z|u,v)\) inferred from its neighbor instance \((u,v)\), and vice versa. In this ablation study, we compute the reconstruction terms for \((x,y)\) based on the latent position inferred from \((x,y)\) itself. We also omit the Kullback-Leibler divergence between the latent distributions \(p(z|x,y)\) and \(p(z|u,v)\)--the first right-hand term in Equation (9). This yields a training objective that is identical to the test objective given in Equation (10). With this objective, we train the 40/40 model architecture from the previous experiment, which generally produced clearly disentangled latent representations of the bass patterns. Apart from the training objective, all other training specifics are as in the previous experiment. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Size** & **A1** & **A2** & **A3** & **A4** \\ \hline
10/10 & 0.999 & 0.996 & 0.515 & 0.982 \\
20/20 & 1.000 & 0.998 & 0.999 & 0.985 \\
40/40 & 1.000 & 0.996 & 1.000 & 0.999 \\
10/20 & 0.999 & 1.000 & 0.540 & 0.988 \\
10/40 & 0.991 & 0.991 & 0.995 & 0.993 \\
20/40 & 1.000 & 0.994 & 0.996 & 0.999 \\
10/640 & 0.965 & 0.551 & 0.536 & 0.729 \\
20/640 & 0.975 & 0.560 & 0.526 & 0.912 \\
40/640 & 0.965 & 0.599 & 0.554 & 0.919 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Predictive accuracy of the quadratic classifiers. Figure 5: Distribution of a subset of the validation data in the latent space. The points are colored according to their corresponding groundtruth bass pattern. The background colors and isolines show the predictions of quadratic classifiers that predict the bass pattern from the latent variable. Figure 6 and Table 3 show the latent space projection for a subset of the validation set, and the corresponding quadratic classifier prediction accuracies, respectively. Although some clusters form in the latent space, these clusters only occasionally coincide with the bass patterns that underlie the artificial data. Further investigation revealed that the organization sometimes coincides with other factors, such as the absolute pitch of the mix/bass track contents. This shows that the disentanglement of bass patterns can be largely accounted for by the minimization of the cross-reconstruction loss. The explanation for this is that the cross-reconstruction loss drives the latent space to encode features of the data that remain largely stable over time. Note that the bass patterns in the artificial data only change infrequently, which means that pairs of instances used to compute the cross-reconstruction loss are likely to be generated from the same pattern. The effect of the cross-reconstruction loss can also be observed when plotting the latent space trajectories over time. Figure 7 shows that the latent space trajectory of the model trained with the regular reconstruction loss (right column) fluctuates more strongly at short time scales, corresponding to changes in absolute pitch. However, because the bass and the guitar play an almost identical riff in this song, encoding the relationship between bass and guitar in terms of relative pitch yields a more stable representation.