## 3 Learned Separation
### Training
The results are displayed in Table 3. While we usually use the direct prediction \(z^{\text{dir}}[k,l]\) for resynthesis, we here also include the resynthesis based on the dictionary prediction \(z[k,l]\) for analysis. With the fixed oracle dictionary, the results are generally much better than with the normal training in Table 2. However, when allowing training from the oracle dictionary, the separated flute and the cello tracks become unacceptably bad again. When using the dictionary prediction for resynthesis, all the results are of very poor quality, indicating that the dictionary model (10) is not an accurate representation of the spectral characteristics of the tones. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Instrument** & **SDR** & **SIR** & **SAR** \\ \hline \multirow{6}{*}{Ours} & Flute & \(-4.7\) & \(17.5\) & \(-4.6\) \\  & Clarinet & \(5.0\) & \(10.1\) & \(7.0\) \\ \cline{2-5}  & Trumpet & \(7.7\) & \(19.9\) & \(8.0\) \\  & Violin & \(9.7\) & \(30.7\) & \(9.7\) \\ \cline{2-5}  & Trumpet & \(8.4\) & \(30.3\) & \(8.4\) \\  & Saxophone & \(13.0\) & \(24.9\) & \(13.3\) \\ \cline{2-5}  & Oboe & \(2.9\) & \(6.9\) & \(5.9\) \\  & Cello & \(-0.6\) & \(19.2\) & \(-0.5\) \\ \hline \multirow{6}{*}{[20]} & Flute & \(2.4\) & \(9.5\) & \(3.9\) \\  & Clarinet & \(6.2\) & \(25.3\) & \(6.3\) \\ \cline{2-5}  & Trumpet & \(5.3\) & \(16.6\) & \(5.7\) \\  & Violin & \(7.7\) & \(25.1\) & \(7.8\) \\ \cline{2-5}  & Trumpet & \(-2.4\) & \(1.1\) & \(2.7\) \\  & Saxophone & \(0.1\) & \(22.5\) & \(0.2\) \\ \cline{2-5}  & Oboe & \(6.3\) & \(17.0\) & \(6.8\) \\  & Cello & \(4.2\) & \(17.1\) & \(4.5\) \\ \hline \multirow{6}{*}{[18]} & Flute & \(3.4\) & \(19.6\) & \(3.6\) \\  & Clarinet & \(2.1\) & \(5.9\) & \(5.4\) \\ \cline{2-5}  & Trumpet & — & — & — \\  & Violin & — & — & — \\ \cline{2-5}  & Trumpet & \(1.2\) & \(9.4\) & \(2.3\) \\  & Saxophone & \(6.9\) & \(17.2\) & \(7.4\) \\ \cline{2-5}  & Oboe & \(-0.8\) & \(13.1\) & \(-0.4\) \\ \cline{2-5}  & Cello & \(3.4\) & \(6.4\) & \(7.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the separation algorithms on a selection of samples from the URMP [42] dataset. Best numbers are highlighted. Upon manual inspection of the flute track, we noticed that it contains a number of tones which are \"half-overblown,\" such that the spectra of both the higher octave and of the lower octave are present. This does not represent the normal spectral characteristics of the flute sound, so the oracle dictionary contains a \"compromise,\" while the trained dictionary fails to represent these half-overblown tones. In the cello track, there is no obvious technical peculiarity, but the tones are simply very diverse, involving different open strings and also different articulation between the tones, so even the training of the oracle dictionary is problematic. Generally, for instruments with inconsistent spectral characteristics, the method from [20] may be at an advantage since it prunes and randomly reinitializes parts of the dictionary in regular intervals; so, given enough tries, it can reach an appropriate dictionary by chance, even if it is potentially suboptimal with respect to the loss function. ### [PERSON] et al. In [18], a number of original samples are used. As we mentioned, their algorithm has the unique ability of separating a representable instrument track out of a mixture with a non-representable residual, which can, for instance, be a singing voice. Since our algorithm is not designed for such signals, we selected the samples for which all the instruments can be represented. In total, these are one sample with acoustic oboe and euphonium, one sample with _synthetic_ piccolo and organ, and a third sample with a synthetic oboe track added to the previous sample. The problems with these samples are that they have a different sampling frequency (\(f_{\mathrm{s}}=22.05\,\mathrm{kHz}\)) and they are also very short.