Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper proposes distilling a DNN into a learnable wavelet transform. Producing an interpretable architecture, that was used in two scientific applications cosmology and molecular-partner prediction.
The paper proposed 3 losses: In AWD the wavelet transform should be invertible, allowing for reconstruction of the original data. This ensures that the transform does not lose any information in the input data "reconstruction loss".  The distillation part of AWD is added by calculating the attribution scores of a  given model for each coefficient in the wavelet representation and try to find a wavelet function that makes these attributions sparse "interpretation loss". Finally, the wavelet loss which ensures that the learned filters yield a valid wavelet transform.

The paper provides two scientific data problems where the resulting wavelet model is sufficiently interpretable for use and provides similar prediction performance as the DNN.  The paper also evaluated AWD on Synthetic data showing that AWD can recover groundtruth wavelets from noisy data.


main_review: Strength:
- The work is novel, and original.
- The paper is well written and clear.
- Significance of work is high since learned wavelets can be used as interpretations for scientific domains.
- The submission is technically sound.

Weakness:
- No comparison was done with other distillation methods.
- Transformation Importance relies on gradients for importance identifications which are noisy.

Questions:
- Do you have any insights on why distillation improves accuracy above DNN? Typically the upper bound is DNN accuracy.

Typos:
- line 106 bijective-> objective
- line 176 think you mean Fig 2?
limitations_and_societal_impact: N/A
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper proposes learning an informative wavelet decomposition of the input space of a neural network that is able to come up with sparse attributions of the features used in the output. The idea is that in some scientific domains which collect time or space oriented data (in the form of images or signals), the features that lead to a particular prediction are hard to interpret because of complex patterns, and that wavelets can instead provide a concise description since they are localized in both frequency and space. 
main_review: Overall, I find this to be a valuable contribution in overcoming the black boxness of neural networks used in scientific domains. The example case studies in biology and cosmology are convincing and clearly written even to those with a limited background in this field.  The summary of the wavelet transform (S2.1) is clear enough to make this paper accessible to a reader who is unfamiliar with wavelets.  However, the overall way the paper is written could be improved.  First, the authors should make it clear what sort of domains benefit from these---clearly structured domains. For example frequency domain features would likely be of little use in interpreting DNA sequences or text data. It also probably more useful in multi-granular settings where despite the high dimensionality of the underlying space, the phenomenon is a signal over the space. From this standpoint, this work could probably also be extended to graph signals. 

Second, the authors don't use much bandwidth to discuss selection of wavelets, scales, or levels (in cases of multilevel transforms), but these are precisely what would be domain-specific, as the authors note that interpretations often are. Indeed it seems as if first learning a useful dictionary of wavelets may be helpful. See Aharon et al. 2006, or Ophir et al. 2011 for wavelet dictionary learning algorithms. 

Figure 1 does not seem very helpful. The authors instead should showcase that the transform is trained with several losses including an autoencoder-like loss reconstruction loss, and a sparsity loss which the authors call 'interpretability loss.' This could be visualized along side a neural network. 

In Line 43, the statement that wavelets have not been used for interpretability is not quite true. I believe there are quite a few papers applying wavelet and using it in terms of model interpretability. Just a few sample papers:  
Wang et al., Multilevel Wavelet Decomposition Network for Interpretable Time Series Analysis
Li et al, WaveletKernelNet: An Interpretable Deep Neural Network for Industrial Intelligent Diagnosis
Khan et al, Learning filter widths of spectral decompositions with wavelets

Additional points of clarification: 

1) Do the wavelets have to be initialized to a known filter or it can be random? Is the distance loss added during training to recover the ground truth wavelet? 

2) In the experimental section, the authors showed the AWD method helped enhance the prediction accuracy, but how is the interpretability compared to other methods? Can the authors provide some comparisons of AWD and other methods so the idea of enhancing interpretability is further illustrated?

I would also suggest the authors to conduct an ablation study to see the effects of each loss term on learning the wavelet transforms? Since the key of this method is the combination of three losses, it’s good to show how these loss terms affect the model.

I am satisfied by the author responses and will maintain my high score. 

limitations_and_societal_impact: The authors should clearly set out the use case for this contribution. Clearly this is only useful for signals in time or space domains.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The authors propose an extended approach for learnable wavelet transforms, and add a distillation-based regularization term to the loss, which penalizes parts of the wavelet with low importance scores in the wavelet-transformed space given a pre-trained neural network.
In experiments on synthetic data, as well as two scientific datasets, their predictive performance is quantitatively analyzed.
Additionally, the interpretability of the learned wavelets is analyzed qualitatively.

main_review: ### Originality:
Previous approaches to learned sparse wavelet transforms are extended by additional constraints from previous literature.
The main originality is the proposed neural network-attribution-based distillation regularization term introduced into the learned wavelet loss.
The paper is fairly original, and differentiates itself well from previous work.

### Quality:
It is not clear, how much the teacher model actually contributes to the increased performance with the intepretation loss, especially in Section 4.3, where the AWD outperforms its teacher.
Reporting the RSME for the case where the interpretation penalty is set to zero in Tables 1 and 2 may give insight to whether and how much the interpretation contributes to increased performance.

The authors claim high interpretability of the learned wavelets.
This may be highly dependent on the problem domain.
Introducing an experiment for which wavelets may not be the best choice, may better show the limitations of the interpretability of method, and make it more convincing.
While it is intuitive, that these learned wavelets give a much clearer idea of their produced predictions, the analysis to support the claim is purely qualitative, and may benefit from a more quantitative analysis.
Such an analysis would furthermore improve the quality for the paper, since the two real-world experiments rely on expert knowledge for their qualitative interpretation.


### Clarity:
The paper is clearly written, the introduced methods are introduced well and sufficiently.

Figure 4 shows qualitatively how different hyperparameters yield different wavelets.
Beyond the fact that they change, it is not really clear to me how the different wavelets reflect the prediction strategy.
A discussion of the possibly increased intuitive "goodness" of the wavelets for different parameters may increase the clarity of this experiment

Minor:
- l.121-122 may or not -> may or may not

### Significance:
Although a broader application of the method may increase its significance, the results suggest a valid extension for learnable wavelets.



limitations_and_societal_impact: Limitations are not discussed.
While wavelets provide a powerful framework, the authors should discuss in which cases the training may fail.
The problem of the generally loose term of "explanation" as discussed in the introduction.
No negative societal impact is reported.
Maybe at least a potential societal impact more general terms of interpretability may be addressed.
For example, over-confidence in a model caused by over-interpretation of possible red herrings in the wavelets could have a negative impact.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors proposed Adaptive Wavelet Distillation (AWD) for a learnable wavelet transform. It distills the discriminative power of a deep neural network (DNN) by penalizing its feature attributions in the wavelet domain. Specifically, the authors proposed a new loss term consisting of reconstruction loss for invertibility, wavelet loss for yielding a valid transform, and interpretation loss for distilling the information of a trained DNN. Through synthetic, cosmology, and biological datasets, they demonstrated that the proposed method provides SoTA predictive performance and identifies interpretable features that are scientifically meaningful in the corresponding domains. 
main_review: (1) Originality
- The authors proposed to learn wavelet transforms based on the attributions from a trained DNN. To the best of my knowledge, this is a novel approach to use the wavelet transforms and knowledge distillation to interpret a trained DNN model.

(2) Quality
- The authors argue that wavelet models may or may not have better interpretability depending on a problem's context (line 36-38, line 121-123). Can you provide examples where wavelet models have better interpretability? Furthermore, I think explaining why having an orthogonal basis, and sparse representations indicate better interpretability would help understand the benefit of this paper.
- I am not sure the compression rate (line 243-245, 297-299) can show the better interpretability of the proposed method. In addition, while the authors stated that they had close collaboration with domain experts (lines 10-12), I could not find relevant details in the paper. Simply stating that the learned wavelet resembles a canonical pattern does not provide solid grounds.
- The interpretation loss induces the wavelet transform to yield the wavelet whose TRIM attribution is sparse concerning a trained DNN model. Since it largely deviates from the standard knowledge distillation setting where prediction outputs of a DNN are utilized, please provide a more detailed rationale for how the interpretation loss enables "gain information about the predicted outputs."
- It is unclear how critical the interpretation loss is for the proposed learnable wavelet transform. The authors only showed how the learned wavelet function changes in terms of the interpretation penalty (Figure 4). Please provide ablation studies showing how interpretation loss affects the predictive power of AWD. I think AWD without the interpretation loss may provide similar predictive performances and compression rates, which would indicate the distillation of a trained DNN is unnecessary.
- How critical are the initialization using known wavelet filters for the predictive performance? Is it more essential than the interpretation loss? Please compare with the results of using random initialization. Furthermore, please provide more detailed explanations for the different initialization methods (support sizes, Coiflet2, DB5).
- The authors claim that AWD outperforms the SoTA neural networks. Are the compared LSTM, and ResNet models are SoTA models? If not, please provide and adequately compare with the SoTA methods.
- The authors claim that AWD allows us to look at wavelet activation maps (Figure D2, line 307). However, activation maps can also be generated using the feature attribution in the input domain. Please show that whether AWD activation maps provide "better" interpretation. In addition, while the manuscript used a saliency map for attribution scores, Figure D2 uses integrated gradients. The inconsistency may lead to misunderstanding.

(3) Clarity
In general, the manuscript is clearly written. Another round of proofreading would be helpful, considering that there are some typos. (Ex. Line 79 (L_2), Line 308 Fig. C2). 

(4) Significance
Although the paper has its own merits, unfortunately, it seems that both additional quantitative and qualitative analyses are required to claim better interpretability. While I agree that interpretability can be better defined in a particular problem, the manuscript does not seem to define and show the interpretability in the contexts properly. I think it does not fold great significance in the current form.

limitations_and_societal_impact: The main limitation of the paper is that the claimed interpretability is not clearly demonstrated through the experiments. Please refer to the Main Review for more details.
review_title: null

Meta-review: The reviewers agreed that this paper had many strengths as a novel method. Presenting two applications as case studies was a welcome departure from standard NeurIPS submissions. The reviewers have given you a few actionable suggestions which I hope you will take for the final version. The idea of including examples where wavelets are not the best choice in the supplementals is a very good one.
Decision: Accept (Poster)