## 3 Learned Separation
### Training
At the same time, the value of the regularization loss (which, with \(\mu_{3}=1\), has a much lower weight than the others) increases. Therefore, this particular sample could potentially benefit from more regularization. In Figure 8, the learning curves for both samples over all the respective runs are displayed. It is obvious that the results for the sample with recorder and violin (Figure 7(a)) are more consistent than those for the sample with clarinet and piano (Figure 7(b)). Moreover, we can see that in the former sample, separation quality generally deteriorates after too many iterations while in the latter, some runs only achieve peak performance near the end of the training. Thus, different samples can benefit from taking the results at different points in the training process. Some curves in Figure 7(b) terminate early, which was due to numerical failures in the training process. One difficulty with the piano as an instrument is that it exhibits significant inharmonicity, cf. [34]. While the algorithm from [20] has been shown to correctly identify the inharmonicity parameter on the isolated piano track, it relies on cross-correlation _without_ inharmonicity for tone detection. By contrast, the algorithm presented here is based on a neural network, so correctly dealing with inharmonicity at the input stage is merely a matter of training. Since inharmonicity mostly affects higher harmonics which have low volume, the ability to represent it in the output stage does not show up as much in the \(\ell_{2}\)-based \(\mathrm{SDR/SAR/SIR}\) figures. However, due to the lifting property of \(d_{2,\delta}^{\mathrm{q,rad}}\) and \(d_{2,\delta}^{\mathrm{q,abs}}\) (with \(q=0.5<1\)), it does influence the losses. Figure 7: Separation performance and loss values while training on the sample with clarinet and piano in the best-case run. The vertical gray lines indicate the point at which the result was taken (70,000 iterations). To illustrate the effect, we also ran our algorithm on the isolated piano track, once with and once without the inharmonicity parameter in the model, with 4 distinct random seeds each. The results for the random seeds of 11 are displayed in Figure 9. While the difference appears small on a global scale, it is consistent. ### Urmp

The URMP dataset [42] consists of samples with two or more acoustic instruments. It was not created for blind separation, so the samples are generally too \"hard\" to be used in that context. Nevertheless, in [20], a subset of potentially appropriate samples was determined, and we compare the performance of our new algorithm on these samples. As can be seen in Table 2, the results vary widely. For the first and the fourth sample, we have to declare a failure compared to the other two algorithms. On the second and the third sample, however, our algorithm is universally dominant. The good performance on the sample with trumpet and saxophone is especially surprising, since we deemed this a very challenging sample due to the similarity of the sounds of the instruments. Figure 8: Mean separation performance over the instruments in the samples based on the piece by [PERSON]. Each line represents a different run with specific random seeds. The vertical gray lines indicate the point at which the result was taken (70,000 iterations). Figure 9: Observed loss convergence on the isolated piano track. In order to investigate the failure of the separation method on the first and the fourth sample, we first train _oracle_ dictionaries by providing the algorithm with the ground-truth individual tracks for the respective instruments. We then supply the separation procedures with these dictionaries as initial values. In one instance, we keep the dictionaries fixed throughout the training, and in another one, we train then at the normal rate, starting from oracle dictionaries. For each separation, we use 4 different random seeds.