## 3 Learned Separation
### Training
Whereas in [20], the signals were converted to a different sampling frequency as a preprocessing step in order to reduce the loss of resolution due to smoothing, we do not have this problem here. While we keep the value of \(\zeta\) from (9) constant in terms of absolute units, the relation to the sampling frequency consequently changes to \(\zeta f_{\mathrm{s}}=470.4\). The frequency constant changes proportionately with the sampling frequency to \(\beta=1.794\,433\,593\,75\,\mathrm{Hz}\). Since the samples are short, we choose an even smaller time constant compared to (9) by setting \(\alpha=16/f_{\mathrm{s}}\approx 0.73\,\mathrm{ms}\) in the acoustic sample and \(\alpha=128/f_{\mathrm{s}}\approx 5.80\,\mathrm{ms}\) in the synthetic ones, each with \(\bar{\alpha}=\alpha/4\) for training. The results are displayed in Table 4.

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline

While the results for the acoustic sample are better than in the original publication, they are still not nearly as good as those in [20]. Our explanation is that while the time resolution of the spectrogram is almost as high as that of the time-domain signal (\(\bar{\alpha}=4/f_{\text{s}}\)), there is still just not enough data in the sample to train the neural network, and thus hand-crafted methods are at an advantage. By contrast, our method delivers very good results with synthetic instruments, clearly and universally outperforming the other methods on the sample with two instruments and providing the best average performance on the sample with three instruments. ## 5 Conclusions

We have developed a blind source separation method that unmixes the contributions of different instruments in a polyphonic music recording via a parametric model and a dictionary. The model parameters are predicted by a deep convolutional neural network, and with respect to those that do not possess a useful backpropagation gradient, we use the policy gradient instead. Unlike other algorithms, ours operates directly on the complex output of the STFT, which is linear and preserves the phase. Rather than using spectral masking, we let the network give a direct prediction for the complex amplitudes of the harmonics. In general, the algorithm exhibits very good performance on a variety of samples. It is especially dominant in terms of SIR, which is relevant since eliminating cross-talk is the main objective in separation. We attribute this to the use of a complex-valued direct prediction for the individual instruments that can properly handle interference between the instrument tones in the spectral domain. Such interference is particularly prevalent in synthetic samples, on which the performance of our algorithm surpasses that of the competing methods. It also clearly outperforms the other methods on the sample with the acoustic piano, which poses the challenge of detecting tones with inharmonicity. As is usual with blind separation, however, problems arise when the structural assumptions are not satisfied. In two samples from the URMP database, there was one instrument each whose sound in the recording varied too much to be accurately represented by the dictionary. While we found that using oracle dictionaries, satisfactory