  We propose a method for the blind separation of sounds of musical instruments
in audio signals. We describe the individual tones via a parametric model,
training a dictionary to capture the relative amplitudes of the harmonics. The
model parameters are predicted via a U-Net, which is a type of deep neural
network. The network is trained without ground truth information, based on the
difference between the model prediction and the individual time frames of the
short-time Fourier transform. Since some of the model parameters do not yield a
useful backpropagation gradient, we model them stochastically and employ the
policy gradient instead. To provide phase information and account for
inaccuracies in the dictionary-based representation, we also let the network
output a direct prediction, which we then use to resynthesize the audio signals
for the individual instruments. Due to the flexibility of the neural network,
inharmonicity can be incorporated seamlessly and no preprocessing of the input
spectra is required. Our algorithm yields high-quality separation results with
particularly low interference on a variety of different audio samples, both
acoustic and synthetic, provided that the sample contains enough data for the
training and that the spectral characteristics of the musical instruments are
sufficiently stable to be approximated by the dictionary.
