## 3 Learned Separation
### Training
### [PERSON]'s Duo for Two Instruments

Like in [20], we use the 8 th piece from the 12 Basset Horn Duos by [PERSON] (K. 487) in an arrangement by [PERSON] for two recorders ([[https://imslp.org/wiki/12_Hom_Duos_K.487/496a_](https://imslp.org/wiki/12_Hom_Duos_K.487/496a_)]([https://imslp.org/wiki/12_Hom_Duos_K.487/496a_](https://imslp.org/wiki/12_Hom_Duos_K.487/496a_))([PERSON]_[PERSON]_[PERSON]), accessed on 6 September 2021) as an example piece. In one sample, it is played with recorder and violin, and in the second sample, it is played with clarinet and piano. All the instruments are acoustic. In Table 1, we compare the performance of our algorithm to that of the other two on both samples. The sample with recorder and violin is comparatively \"easy.\" While our proposed algorithm universally gives the best SIR values (indicating low interference between the instrument tracks), the algorithm from [20] outperforms it for the recorder track in terms of SDR and SAR. Possible explanations include a more effective optimization process of the respective objective function, but it could also be a result of the different data representation used in that method (namely the special log-frequency spectrogram) or a regularizing effect of the sparse pursuit algorithm. In preliminary experiments, we found that in this particular sample, a lower value for \(\mu_{1}\) can increase separation quality (putting more emphasis on the correctness of the direct prediction rather than the dictionary-based prediction), but requiring the user to guess the difficulty beforehand violates our notion of blind separation. The first few seconds of the separation result are displayed in spectrogram form in Figure 6. Overall, the direct predictions are very accurate, but like it was observed in [20], the last tone visible in Figure (b)b jumps up a fifth, which it does not do in the ground truth. This is because the recorder tone is actually one octave above the violin tone, and the overlap is virtually perfect. In such cases, since the dictionary model is never fully accurate, it may happen that an incorrect solution actually yields a lower loss. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Instrument** & **SDR** & **SIR** & **SAR** \\ \hline \multirow{4}{*}{Ours} & Recorder & 13.1 & 34.8 & 13.2 \\ \cline{2-5}  & Violin & 13.4 & 34.2 & 13.5 \\ \cline{2-5}  & Clarinet & 12.4 & 28.0 & 12.6 \\  & Piano & 8.1 & 42.2 & 8.1 \\ \hline \multirow{4}{*}{[20]} & Recorder & 15.1 & 32.4 & 15.2 \\  & Violin & 11.9 & 23.8 & 12.2 \\ \cline{2-5}  & Clarinet & 4.1 & 24.3 & 4.1 \\  & Piano & 2.1 & 9.3 & 3.5 \\ \hline \multirow{4}{*}{[18]} & Recorder & 10.6 & 21.4 & 11.0 \\  & Violin & 5.8 & 18.4 & 6.1 \\ \cline{1-1} \cline{2-5}  & Clarinet & 6.7 & 21.3 & 6.9 \\ \cline{1-1}  & Piano & 5.5 & 16.4 & 5.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the separation algorithms on the samples based on the piece by [PERSON]. Best numbers are highlighted. Figure 6: Excerpt of the separation result for the piece by [PERSON], played on recorder and violin. Displayed are the original STFT magnitude spectrogram as well as the direct predictions for each instrument. In the highlighted section, the last tone is supposed to be a constant octave interval between the violin and the recorder, but the prediction for the recorder contains an erroneous jump. The color axes of the plots are normalized individually to a dynamic range of 100 dB. By comparison, the sample with clarinet and piano is rather \"hard,\" and our algorithm clearly shows superior performance, especially in the separation quality of the piano track. An interesting observation in Figure 6(a) is that while the separation performance first reaches a plateau around \(10\,\mathrm{dB}\) (in the mean), it then declines to around \(8\,\mathrm{dB}\). However, as shown in Figure 6(b), the values of both \(d_{2,\delta}^{\mathrm{q,abs}}(\gamma,y^{\mathrm{spr}})\) and \(d_{2,\delta}^{\mathrm{q,rad}}(\gamma,y^{\mathrm{dir}})\) decrease, indicating this is not an optimization failure.