Convolutional neural networks (CNNs) excel in image recognition and generation. Among many efforts to explain their effectiveness, experiments show that CNNs carry strong inductive biases that capture natural image priors. Do deep networks also have inductive biases for audio signals? In this paper, we empirically show that current network architectures for audio processing do not show strong evidence in capturing such priors. We propose Harmonic Convolution, an operation that helps deep networks distill priors in audio signals by explicitly utilizing the harmonic structure within. This is done by engineering the kernels to be supported by sets of harmonic series, instead of local neighborhoods for convolutional kernels. We show that networks using Harmonic Convolution can reliably model audio priors and achieve high performance in unsupervised audio restoration tasks. With Harmonic Convolution, they also achieve better generalization performance for sound source separation.
The paper considers the effectiveness of standard convolutional blocks for modelling learning tasks with audio signals. The effectiveness of a neural network architecture is assessed by evaluating its ability to map a random vector to a signal corrupted with an additive noise. Figure 1 illustrates this process with a network taking a single standard normal vector as input and having a single target output consisting of some signal corrupted with additive noise.
The paper is not well written and it is rather difficult to follow. It is also not well structured with a number of relevant concepts properly described only sections after they appear for the first time.
The first issue I had with the paper was the notion of audio prior. It was only after reading the whole paper that I have realized what this means. Having said this, it is unclear why the employed notion would work in general. I see why it could work when the distribution of the input vector and additive noise are correlated. This has not been clarified nor discussed and I believe it merits a couple of sentences.
In the introduction, the paper states "... unlike CNNs for image modelling, the design of deep neural networks for auditory signals has not yer converged". First, it is not clear what it means for the architecture to converge. If we assume that it refers to standard convolutions with a couple of widely accepted filter size and max pooling, the I would say that in speech recognition the structures that work are quite similar for mel-frequency coefficients or fbank features as inputs (which are again convolutional feature extraction layers).
Shortly after this, there is a question on justification of network designs. I disagree with a potential implication that this is well understood for image processing. For some insights relevant to speech, the work by Mallat ("Group invariant scattering", 2012) might be useful.
Figure 1 and the paragraph just below its caption are not clear. It is not explained what is the input/output of the network and this is of great importance for the understanding of the illustration in Figure 1.
The introduction does not explicitly define the notion of audio prior and the whole paper is about this. In my opinion, it is wrong to assume that a reader has seen the paper by Lempitsky et al (2018).
Section 2.1, the optimization objective as formulated implies that z and x_0 are completely independent. I do not see how any meaningful conclusion can be derived by fitting a map between independent input and output vectors. Some assumption is required for the proper notion of "audio prior" (if not, then a discussion arguing for the opposite).
Section 3, opening paragraph concludes that standard CNNs are not the best blocks to model learning tasks with audio signals. For this implication, one needs the exact structure of CNN network and more details with regard to the experiment itself. In particular, there are deep CNNs (with mel-frequency coefficients as inputs) that work very well in speech recognition (e.g., on noisy datasets such as aurora4). This illustration does not say anything about the influence of the depth and number of convolutional blocks on a learning task. The language should be more moderate here and, in general, some additional work is required on the motivation of harmonic convolutions.
In my understanding, harmonic convolutions are a special case of deformable convolutions (Dai et al., 2017). In essence, standard convolution is applied over time and deformable over the frequency axis of a spectrogram. The main contribution seems to be in that the work provides a structure to the offsets in Dai et al. (Section 2.1, 2017). If I am correct, then this should be discussed in details and the harmonic convolution needs to be placed in the context of prior work. It might help by starting with a review of that work and then introducing the imposed structure on the offset vectors. I am having problems understanding the illustration in Figure 3.
In the experiments, the work is evaluated on signal de-noising (audio restoration) and sound separation. 
The first task is carried out under the assumption that the signal has been corrupted with Gaussian noise and shows advantages of the approach over baselines which include standard convolutional networks. It would be interesting here to see how the depth of a convolutional network affects the performance. Also, as the approach is (in my understanding) a special case of deformable convolutions it would be insightful to show an experiment with that baseline. While additive noise is difficult on its own, many signals are corrupted by channel noise. It would be interesting to add an experiment with different types of channel noise and which network design is more likely to de-convolve the noise from the signal.
The second experiment deals with separation of sounds of different musical instruments and the results again show advantages of harmonic convolutions over the baselines.