Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper presents new methods for inference and sampling for Archimax copulas. Archimax copulas are a family of copulas defined through an Archimedian generator and a stable tail dependence function (stdf). In order to discuss the inference and sampling for Archimax copulas, the authors first proposed inferential and sampling methods for Archimedian generator and stdf. Then, combining these methods, the methods for inference and sampling for Archimax copulas are established. In experiments, it is seen that the proposed inferential methods for Archimedian generator and stdf show satisfactory performance. Other experiments are given to illustrate that Archimax copulas outperform, or work as well as, some existing models for a couple of real datasets.
strengths_and_weaknesses: *** STRENGTHS ***

(a) [Originality] The presented inferential and sampling methods for Archimax copulas seem new. These methods are derived mainly by combining existing methods for Archimedian generators and stdfs.
 
(b) [Quality] The paper seems technically sound. Comprehensive experiments are given to assess the performance of the proposed inferential methods and compare submodels of Archimax copulas with some existing models.

(c) [Clarity] The paper is clearly written in general. Section 2 providing the background of the presented theory would be helpful for readers who are not family with copulas.

(d) [Significance] Archimax copulas are flexible models which include Archimedian copulas and extreme-value copulas as special cases. Therefore the proposed inferential and sampling methods for Archimax copulas could be useful in practice when flexible modelling is required.
 

*** WEAKNESSES ***

(e) [Quality] Apart from Archimax copulas, there exist other flexible families of copulas such skew-$t$ copulas (see, e.g., Joe [49], Section 3.17.2). The paper does not sufficiently compare the Archimax copulas with those existing copulas.

(f) [Significance] I am not sure about the popularity of Archimax copulas and the importance of the related theory this paper presents. (Nonetheless I appreciate the results of experiments which suggest the usefulness of the Archimax copulas.)

questions: (g) For the multivariate Archimedean copula (equation (3.4), [49]), all the pairs of variables have the same strength of dependence, and this is considered a drawback in terms of flexibility in general. How about the Archimax copulas? Is it possible to solve this problem by appropriately defining the spectral random variable $W$?

(h) In lines 107 and 108, it is mentioned that $X$ defined in (5) follows an Archimax copula. However it seems to me that $X$ takes values in $[0, \infty )^d$ in general and therefore the distribution of $X$ is not a copula which is defined on $[0,1]^d$. I think a correction is necessary to avoid this misunderstanding.

(i) I would appreciate the authors' responses to my comments (e) and (f).


limitations: The authors have addressed the limitations and potential negative societal impacts of their work in Section 5. Depending on the authors' response to my questions (g) and (i), I might claim that the usefulness of Archimax copulas is limited.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors propose scalable estimation and sampling procedures for Archimax copulas. On simulated and real data, they demonstrate that Archimax copulas fit using their procedure can model complex data with dependencies between extreme values accurately, in comparison to existing deep generative methods.
strengths_and_weaknesses: The paper bridges two important areas of research: copulas and deep generative modeling. It is highly original (the first method of its kind) and technically excellent. It is also potentially very significant in its impact: modeling rare events is critical to managing risk in real world applications, and relying naively on modern deep generative approaches can potentially be very problematic. The paper is overall quite clear, though there are two areas where I struggled: first, an intuitive explanation as to why Archimax copulas are good for modeling dependencies between extreme events would be helpful to the reader; second, I’d appreciate a concise statement of the complete model up front, explaining how the deep generative model determines the stdf. 
questions: See above for suggestions.
limitations: The authors clearly explain some of the method's important limitations.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper proposes novel procedures for both inference and sampling of Archimax copulas. This family of copulas is important due to their ability to represent the bulk and tail of distributions simultaneously which can be suitable for healthcare and hydrology real-world data. The authors propose a 'hybrid' approach mixing copulas and neural networks to allow for more flexible estimation. In experiments, the proposed method is compared to SOTA density modeling techniques and the results suggest that their method can extrapolate to tails and scale to higher dimensions. 
strengths_and_weaknesses: Strengths:
  - Originality - to the best of my knowledge this is the first work to address flexible density estimation of bulk and tail distribution with Archimax copulas.
  - Quality - the authors put a lot of effort into including a high level of technical details and experiments in the paper and appendix 
  - Clarity - since copulas are not a straightforward tool in the machine learning community, I appreciate the background overview and related work mentioned throughout the paper and supplementary. 
  - Significance - considering the tails not only bulk of the data is overlooked problem and can be very significant in many real-world applications

Weaknesses:
  - Originality - although the presented methodology is novel, it does still build on existing work regarding Archimax copulas
  - Quality - I would expect more challenging/motivating experimental results to support the claims of significance and contribution from the introduction
  - Clarity - A running example or one real data motivating example could improve/clarify why Archimax copulas are an appropriate/necessary tool in critical scenarios. This can help bring the paper closer to the ML community, make it more relevant for readers.
  - Significance - since copulas are still not widespread at the ML conferences, I feel like additional motivation for such papers is needed, either to showcase superiority on large-scale real-world datasets or find some new tasks where SOTA models fail. I also appreciate the code submission and effort of implementing everything in Python (rare for statistics methods). 
 



questions: 1. Complexity of the proposed algorithms is not mentioned. Is there any trade-off between flexibility and scalability?
2. Are there any ablation studies regarding data efficiency? Seems like such non-parametric methods can be data-hungry
3. Will this generative network work for non-tabular data? Can there be an example of another NN architecture?
4. How do the sampled marginals compare to the original data?
5. To increase the significance/contribution of the paper, could you add an experiment for out-of-distribution detection for example? (since this is mentioned as a limitation of existing methods) maybe one of the experiments you have already can be used to present some uncertainty scores.
6. related work that is missing:  
Bhatia, Siddharth, Arjit Jain, and Bryan Hooi. "Exgan: Adversarial generation of extreme samples." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. No. 8. 2021.

Minor:
Figure 1 is not referenced in the text

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### Update after authors response:

Thank you for the thorough response and addressing all of the comments in the updated manuscript.
I am happy to accordingly raise the score.
limitations: The authors have addressed the limitations and potential societal negative impact of their work.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors propose an efficient inference and sampling schemes for archimax copulae based on learning of the generator and stable tail dependence functions through deep learning techniques. 
strengths_and_weaknesses: *Originality*. The work is original. To my knowledge, the authors propose a new method to infer and sample from archimax copulas extending the previous work in this area.

*Clarity*. The authors go a long way to make sure the paper is accessible by a larger machine learning community. They provide necessary background info on copulae and their use in machine learning. They also provide extensive derivations in the appendix. That said, the material itself is quite dense. I have not found typos. One minor thing is that Figure 1 does not seem to be referenced anywhere in the text.

*Quality*. The authors build upon a previously developed theory and methods to derive a scheme to infer archimax copulae via the means of deep learning models which also allows for an easy sampling from. The authors provide an analysis of performance of the proposed method comparing it to other state-of-the-art methods. The details of the experiment setups are given in detail in the supplementary material. The authors provide extensive background/related work review in both the main text and the appendix.

*Significance*. Inference of the multivariate distributions from the data is a core statistical/machine learning problem. The authors propose a way to infer multivariate dependencies via archimax copulae representations of which are learned via deep learning models. The method also allows for sampling from the learned distribution. The method serves both the bulk and the tails of a distribution. With that said, a proposed method is of major significance for the field.

questions: As far as I understand when optimizing the presentations of the copula generator and stable tail dependence function the authors start with an initial copula generator $exp(-x)$. Have you considered other initial generators and how would this affect learning of the copula?

Have you considered estimating several copulae as in a hierarchical archimedean copula?

What do you say about modifying the architecture of the generative neural networks?
limitations: The authors faithfully address limitations and possible negative impact of the proposed method.
review_title: null

Meta-review: metareview: The paper proposes a new method for inference and for sampling in archimax copulas. All the reviewers praised the soundess and clarity of the paper, the novetly of the ideas and the experimental results. Copulas might not be one of the core topics of the NeuRIPS community, but the reviewers pointed out that:
1) the authors did a great job at explaining copulas to the ML community, a valuable tool to model extreme events.
2) the method builds a connection between copula and deep generative modeling, and hence opens new research directions.
Hence, they all enthusiastically recommend to accept the paper, and I agree with them.

Some of the reviewers [HCqW, 5Yjr]  also supported the idea to highlight the paper (oral or spotlight presentation).
confidence: Certain
Decision: Accept