From: ICLR.cc/2023/Conference/Program_Chairs
title: Paper Decision
decision: Reject
metareview:_summary,_strengths_and_weaknesses: This paper introduces a generalized additive modeling approach to Poisson processes with the goal of capturing higher-order interaction effects. While the reviewers found the paper to be overall well-written, many questions remained for them about the motivation and effectiveness of key parts of the proposed method, including the discretization step, log-linear specification, hyperparameter choices, and optimization approach. Notably, the main motivation of the GAM approach seems to be scaling to higher dimensions, but I am not sure what high-dimensional applications of Poisson processes the authors have in mind (space+time is usually the limit), and while the authors say that simulations for higher dimensions are computationally costly, I do not believe this to be the case for reasonably sized dimensions (e.g. d = 10 or 20). I would encourage the authors to reflect on the points raised by the reviewers as they revise the paper for a future venue.
justification_for_why_not_higher_score: Reviewers had many concerns that were unaddressed.
justification_for_why_not_lower_score: N/A

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Further Discussion
comment: Thank you for initiating a discussion with us during the discussion period.

It would be appreciated if you could elaborate on which concern was not addressed. We are happy to provide further clarifications to address all your concerns.

From: ICLR.cc/2023/Conference/Paper881/Reviewer_Cwen
title: Keep my score
comment: The authors did not address any of my concerns. Therefore, I would like to keep my score.

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Authors' Reponse to Reviewer 2omy
comment: Thank you for reviewing our paper and for your positive comments. Here is a point-to-point response to each of your concerns.

> a log-linear model is proposed in APP, which is a bit restrictive in its functional form. How easy it is to extend the method to richer class of (non-linear) models?

We model a discrete distribution by a log-linear model, and it is already guaranteed that the log-linear model can represent any distribution. Therefore technically there is no "richer class". Nevertheless, it is interesting for future work to consider different inductive biases by considering non-linear models.

> Is it not obvious to me that such a low-dimensional representation always exists. Related to this, How big of k (i.e., the truncation level of the order) is sufficient? Have you thought about ways to improve/constrain APP such that k needed can be low? On the contrary, as the authors also pointed out, large k may result in overfitting, how should one pick k in practice?
The low-dimensional representation always exists.
In practice, in the majority of real-world applications, we have found that $k$ to be 2 or 3 is usually sufficient. However, this really depends on the application.

Depending on the application, $k$ can be selected by domain expertise or cross-validation.

> The hyperparameters M and h are learned with grid search, which may be expensive in high-dimensional problems.

This is true, however, there are other approaches to select the hyper-parameters, such as Scott’s Rule of Thumb, and from domain expertise depending on your application. So this should not be a problem for any practical application.

> The experimental results are all on synthetic data expect one on the New York taxi dataset. More extensive results will make the paper stronger.

Our current experiment design has already properly evaluated our proposed approach. The synthetic data experiments can effectively demonstrate the capabilities of our proposed approach because we can obtain both the intensity function and the observation. There are additional experimental results in the appendix.

For real-world experiments, it is difficult to obtain both the observation and the intensity function making it difficult to properly evaluate our proposed approach. Adding additional experiments does not provide any additional insights for our proposed approach as we have already properly evaluated our approach using a synthetic dataset.

> the number of parameters grows exponentially with the dimension and there is no structural dependency between the theta's. Will it help to enforce some dependency between them (for examples, theta_{1,2} depends on theta_1 or theta_2)?

There are structural dependencies between the parameters. This is shown in Eq.(7).

> The presentation and clarity of the paper in terms of typos and notations can be improved. For example, g in Theorem 2.1 is not defined; Introduction -> introduction in the first line of page 5; caption in Figure 3 should include first-order Poisson process too. 

$g$ is defined in equation 2 to be a “positive function to guarantee the non-negativity” which we have chosen to be the exponential function.

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Authors' Response to Reviewer x4an
comment: Thank you for reviewing our paper. Here is a point-to-point response to each of your concerns.

> However, the two key points of the paper (the model itself, obtained as an application of generalized additive models, and the convexity of the optimization problem) do not require any deep theoretical or empirical development. This is not necessarily an obstacle but, in this case, I would expect at least some theoretical guarantees on the quality of the estimate obtained by the method, which are lacking.

The theoretical guarantees have been made by projecting the problem onto a Dually Flat Riemannian Manifold. These theoretical guarantees can be found in [Sugiyama et. al., 2017].

> Part of the state-of-the-art in the introduction has been deferred into the supplementary material, which seems to me quite unusual. Moreover, the approaches cited in this appendix are not used as references in the numerical study of the paper. In particular, Poisson factorization appears to be a very relevant baseline.

Although comparison with Possion factorization is relevant, our experiments are still valid and rigorous evaluation of our proposal.

> Selection of hyperparameters should be investigated in more than few lines at the beginning of the numerical section. For example, we can think that hyperparameters should be selected in a coupled way. Moreover, they have a strong effect on the complexity of the algorithm, which should be taken into account.

We agree that hyperparameter selection is a challenging problem. Our work focuses on methodological development, and hyper-parameter selection will vary significantly depending on the application.

> At the beginning of subsection 2.2, the time window is discretized, which avoids to evaluate an intractable integral that appears in the likelihood. I would expect further explanations on this point, like a control of the error induced.

The error induced depends on your grid size and application. It is difficult to discuss this in the general setting.

> I am rather surprised that the largest dimension considered in the numerical part is 4.

Certainly, our approach is generalizable to more than 4 dimensions. However, there are two problems in analyzing the numerical experiments past 4 dimensions (1) it is difficult to present meaningful results that can be critically analyzed beyond 4 dimensions (2) generating the ground truth for the experiment grows exponentially and becomes computationally expensive.

> Page 3, after equation (2): The authors introduce function $g$ then select $g=\exp$. How critical is this choice? For example, it appears in subsection 2.3 that the optimization problem is convex because the model is log-linear.

This is critical. Our proposed approach and theoretical guarantees do not work if g is not chosen to be the exponential function.

> Introduction: There is no Fig. 1b.

Thank you for picking this up. This should be Fig. 2b. Not sure why LaTeX is referencing this incorrectly. We will mainly write Fig. 2b in the camera-ready version.

> Typo: spacial (page 5)

Spacial is British English, we are happy to update this to spatial which is US English to stay consistent with the rest of the text.

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Authors' Response to Reviewer 8qDw (2/2) 
comment: > The authors claim to use natural gradients to solve their proposed optimization problem. However, they never demonstrate the utility of this over standard gradient descent.

Our optimization is convex, so using standard gradient descent and natural gradient will converge to the same solution. Natural gradient is generally faster if you have less than 15,000 parameters. The advantage is that you don’t need to set a learning rate and works better when the features are scaled differently. If you have more than 15,000 parameters, it may be better to use standard gradient descent or an approximation to natural gradient. This was discussed in the last paragraph of Section 2.3.


> "Our model combines the techniques introduced by Luo & Sugiyama (2019) to model higher-order interactions between Poisson processes and by Friedman & Stuetzle (1981) in generalized additive models to learn the joint intensity function using samples in a lower dimensional space" - a lower dimensional space compared to what? Also, this sounds very similar to the approach in the paper. How do they actually differ?

Friedman & Stuetzle (1981) introduces the projection pursuit algorithm for additive models. We have then used the study from Luo & Sugiyama (2019).

Luo & Sugiyama (2019) studies the difference in increasing the representation power of machine learning models using parameters that take into account higher-order interactions or latent representation. Luo & Sugiyama (2019) also introduces the connection between Markov Random Fields (or Boltzmann Machines) and a partial order structure. Their empirical results have demonstrated that for applications with sparse observations (such as Poisson Processes), it is better to have a model which includes parameters that take into account the higher-order interaction effect compared to learning latent features.

An observation in a Poisson process contains several features. The lower dimension refers to a subset of these features.


> "The learning process in our model is formulated as a convex optimization problem to arrive at a unique optimal solution using natural gradient, which minimizes the Kullback-Leibler (KL) divergence from the sample distribution in a lower-dimensional space to the distribution modeled by the learned intensity function. This connection provides..." - the previous sentence is not explaining a connection to anything. What are the authors referring to here?

The connection refers to the connection between GAMs and the statistical manifold proposed in Luo & Sugiyama (2019).


> What do the authors mean by "higher-order intensity function"? Do they mean higher-dimensional? Perhaps they meant to use it analogously to the phrase "higher-order interaction", but I don't think "higher-order intensity function" is standard terminology.

Yes, it is non-standard to mean higher-dimensional intensity function. The phrase higher-order intensity function has the same meaning as higher-order interaction. We have chosen this terminology because the high-dimensional Poisson process could have a different meaning.

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Authors' Response to Reviewer 8qDw (1/2)
comment: Thank you for your detailed review. Here is a point-to-point response to each of your concerns and your questions.

> The authors choose to model the log-intensity function with a GAM, but I do not see how this modelling choice is supposed to constitute any sort of deeper connection. I am hung up on this point because the authors seem to dedicate a significant part of Section 2 to this claimed connection, which appears to only consist of trivial algebraic manipulations.

Section 2.2 shows the connection between the log-intensity function and GAMs and provides background information for our proposed approach in Section 2.3.


> The authors' claim that their method scales better to higher-dimensional problems is dubious. While strictly speaking, their learning process does scale better than other methods, it still scales exponentially in the dimensionality of the space. Hence, their method is limited to moderate-dimensional problems.

This is not entirely correct. The model scales exponentially if you include all parameters in the model. However, this is not the case, as our proposed approach provides a systematic approach to restrict the number of parameters and approximate the higher-order interaction effects making it much more scalable.

> Some of the empirical results on problems that are supposed to be favourable for APP do not seem convincing. A well-known issue of GAMs is that due to their flexibility, they can heavily overfit, which appears to be precisely the case in Figure 4. Especially in the right plot, the APP intensity is wholly concentrated on the data points, while the KDE fit seems far more reasonable.

This is not a correct interpretation of Figure 4. There is no overfitting in Figure 4. The black line is the ground truth. Our proposed approach APP is much closer to the black line compared to KDE, which incorrectly approximates the intensity function. The amount of overfitting or underfitting can be tuned by varying, the grid size ($M$), the bandwidth ($h$), and the number of interactions. This is the same for KDE. We have fairly assessed both APP and KDE by applying the same approach to select $M$ and $h$ and we have shown that APP was able to be much closer to the ground truth.

> Similarly, as shown in Figure 3, KDE outperforms APP on the process with higher-order interactions. The authors note this but claim that "Figures 3a and 3b show that KDE is sensitive to changes in bandwidth, which means that, for any practical implementation of the model, second-order APP with a less sensitive bandwidth is more likely to learn a more accurate intensity function when the ground truth is unknown." - However, I don't think Figure 3 demonstrates that APP would be much less sensitive to the kernel bandwidth than KDE.

You are correct in saying that KDE outperforms APP. However, this is for a very small range of the bandwidth, which is impractical to select in any real-world application. APP has a lower value for KL-divergence for a larger range, making it much more robust and less sensitive to kernel bandwidth choice. Please note, this does not mean that the choice of bandwidth is not important. It is still very important to select the right bandwidth, but APP is much more robust compared to KDE.

> I also think the name for the authors' method is quite misleading. "Additive Poisson process" would suggest that the family of processes described is a subclass of PPs for which some nice additive property holds (e.g. stability under addition). However, this is not the case, and I would like to ask the authors to pick a more fitting name.

We have selected the name “Additive” from Additive Models which have been described in Section 2.


> The authors claim, "Based on the Kolmogorov-Arnold representation theorem, generalized additive models are able to learn the intensity of the higher-order interaction between Poisson processes by using projections into lower dimensional spaces." - This cannot be true, as GAMs use a more restricted family of functions, so the K-A theorem does not necessarily apply. It would be necessary to discuss the approximation properties of the particular family of GAMs they use to make this claim precise.

This statement is not correct. K-A theorem does apply to GAMs, please see our reference [Braun & Griebel 2009] that we have cited in Theorem 1 for the theoretical proof and discussion around this.


> The authors invoke a lot of mathematical machinery, e.g. the poset formulation for the model and information geometry, without seemingly relying on it too much. I think the method and training process could be described in much simpler terms which would heavily improve the paper's readability.

The poset formulation and information geometry provide the theoretical properties for our proposed model. Without it is difficult to make the connection with the theoretical guarantees. The poset formulation also provides a graph structure which is useful in many applications.

From: ICLR.cc/2023/Conference/Paper881/Authors
title: Authors' Response to Reviewer Cwen
comment: Thank you for reviewing our paper again. Here is a point-to-point response to each of your concerns. We hope you can engage in an active discussion with us during the discussion period.

> The major issue I have with this paper is that the proposed approach has long existed in the statistics literature for decades, under the context of generalized additive models. In particular, in chapter 5.6 of [1], the tensor product bases are more flexible than the approach proposed in this paper. In my opinion, the lower-dimensional approximation proposed in this paper is essentially a piecewise constant approximation of the intensity function (in lower dimension), which is a special case of the tensor product basses. If comparisons should be made, they should be between the proposed method and the GAM with tensor product bases, instead of the RKHS formulation.

We disagree with this comment. Our proposed approach has advantages over some of the existing approaches as it can explicitly take the interactions between the features and time into account. Such modeling is not possible in existing approaches. The decomposition into lower dimensional space is meaningful as it represents a Poisson process for each of the features in the interaction effects between features.

> In addition, the partial order in formulation (7) seems to rely on the assumption that all dimensions have the same common support [0, T], which is not the case for the spatial-temporal processes considered in this paper. How is this issue addressed?

This is not correct. All the spatial-temporal processes considered in this paper assume a common support of $[0, T]$. A running example has been explained in paragraph 2 of the introduction. 

> For the proposed method, the most difficult task should be to determine the value of hyper-parameters M and h. If M is too large, it can easily lead to overfitting. If M is too small, there will be a large bias in the estimated intensity functions. Please provide more details on how cross-validation is performed to choose an optimal combination of h and M.

The selection of the number of bins M is application-dependent. It determines the time interval where the events are considered to have interacted with each other. In our synthetic experiment, we have defined that there is an interaction when they are within 0.1sec apart. So a reasonable choice for M in our model is M = T/dt where dt = 0.1sec and T max time. A larger M means that there is a smaller time interval where the events in different stochastic processes are able to interact. Likewise, the rule of thumb or cross-validation is recommended to select M in practical applications.

The bandwidth h in Table 1 is selected through grid search. We tried a range of bandwidths and selected the bandwidth with the lowest KL divergence between KDE and the latent intensity function. We understand this is only possible in synthetic data and not possible to be done in real-world situations since the latent intensity function is unknown. Our experiment is designed to demonstrate the sensitivity of the bandwidth. For practical applications, we suggest using conventional bandwidth approaches such as the rule of thumb or cross-validation depending on the application of the model.

> Another difficult task with the proposed method is how to justify the use of lower-dimensional approximation. For example, is it justifiable to assume that there is no need to consider spatial and temporal interaction in a spatial-temporal point process? In GAM, one can do this by performing an analysis of variance (ANOVA), see, e.g., section 5.6.3 of [1]. Could this be done with the proposed approach?

We have not made the assumption that there are no interactions between spatial and temporal interactions. In fact, spatial-temporal interactions can be included in the model depending on the design of the graph structure. This can certainly be done with our proposed approach.

From: ICLR.cc/2023/Conference/Paper881/Reviewer_Cwen
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: I reviewed this same paper last year for ICLR 2022. By comparing the current manuscript and the manuscript from last year, I did not find significant differences. Therefore, I will reiterate my questions from last year.

This paper proposes a method to estimate the multi-dimensional intensity function of a Poisson process with a lower-dimensional projection. The proposed method is motivated by perspectives of information geometry and generalized additive models in statistics. The performance of the proposed method is extreme through synthetic examples and real data analysis.

 
strength_and_weaknesses: Strength: The paper is overall well written and the presentation is clear. 
Weakness:  The major issue I have with this paper is that the proposed approach has long existed in the statistics literature for decades, under the context of generalized additive models. In particular, in chapter 5.6 of [1], the tensor product bases are more flexible than the approach proposed in this paper. In my opinion, the lower-dimensional approximation proposed in this paper is essentially a piecewise constant approximation of the intensity function (in lower dimension), which is a special case of the tensor product basses. If comparisons should be made, they should be between the proposed method and the GAM with tensor product bases, instead of the RKHS formulation.

In addition, the partial order in formulation (7) seems to rely on the assumption that all dimensions have the same common support [0, T], which is not the case for the spatial-temporal processes considered in this paper. How is this issue addressed?

For the proposed method, the most difficult task should be to determine the value of hyper-parameters M and h. If M is too large, it can easily lead to overfitting. If M is too small, there will be a large bias in the estimated intensity functions. Please provide more details on how cross-validation is performed to choose an optimal combination of h and M.

Another difficult task with the proposed method is how to justify the use of lower-dimensional approximation. For example, is it justifiable to assume that there is no need to consider spatial and temporal interaction in a spatial-temporal point process? In GAM, one can do this by performing an analysis of variance (ANOVA), see, e.g., section 5.6.3 of [1]. Could this be done with the proposed approach?

Minor points:

1. In the motivating example, the time coordinate is the day of the week, which is discrete. Such an example is not coherent with the definition of the Poisson process, which should be with continuous coordinates.

Reference:
[1]. Wood, S.N., 2017. Generalized additive models: an introduction with R. CRC Press. (2nd Edition)
clarity,_quality,_novelty_and_reproducibility: I think the novelty of the paper is limited.
summary_of_the_review: I think the paper is well written, but I do not see the methodology proposed to be better than the existing ones.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper881/Reviewer_8qDw
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: The authors consider the problem of learning the intensity function of an inhomogeneous Poisson process (PP) over higher dimensional spaces:
 1. The authors propose a certain family of generalized additive models (GAM) to model the log-intensity function of the PP and call their method the additive Poisson process (APP).
 2. They propose to learn the parameters of the GAM by minimizing the Kullback-Leibler (KL) divergence between a tractable kernel density estimate (KDE) approximation to the empirical intensity and the intensity given by the GAM.
 3. The authors test their method on two synthetic and one real-world dataset.
strength_and_weaknesses: ## Strengths
The authors consider a relevant problem, and their proposed solution is sound and holds some promise for estimating the intensity functions of low- to medium-dimensional PPs.

## Weaknesses
While the paper has some good ideas, some of the presented theoretical connections are dubious, and I am not convinced that the empirical results bear out all of the authors' claims. Concretely:
 - The authors choose to model the log-intensity function with a GAM, but I do not see how this modelling choice is supposed to constitute any sort of deeper connection. I am hung up on this point because the authors seem to dedicate a significant part of Section 2 to this claimed connection, which appears to only consist of trivial algebraic manipulations.
 - The authors' claim that their method scales better to higher-dimensional problems is dubious. While strictly speaking, their learning process does scale better than other methods, it still scales exponentially in the dimensionality of the space. Hence, their method is limited to moderate-dimensional problems.
 - Some of the empirical results on problems that are supposed to be favourable for APP do not seem convincing. A well-known issue of GAMs is that due to their flexibility, they can heavily overfit, which appears to be precisely the case in Figure 4. Especially in the right plot, the APP intensity is wholly concentrated on the data points, while the KDE fit seems far more reasonable.
 - Similarly, as shown in Figure 3, KDE outperforms APP on the process with higher-order interactions. The authors note this but claim that "Figures 3a and 3b show that KDE is sensitive to changes in bandwidth, which means that, for any practical implementation of the model, second-order APP with a less sensitive bandwidth is more likely to learn a more accurate intensity function when the ground truth is unknown." - However, I don't think Figure 3 demonstrates that APP would be much less sensitive to the kernel bandwidth than KDE. 
 - I also think the name for the authors' method is quite misleading. "Additive Poisson process" would suggest that the family of processes described is a subclass of PPs for which some nice additive property holds (e.g. stability under addition). However, this is not the case, and I would like to ask the authors to pick a more fitting name.
 - The authors claim, "Based on the Kolmogorov-Arnold representation theorem, generalized additive models are able to learn the intensity of the higher-order interaction between Poisson processes by using projections into lower dimensional spaces." - This cannot be true, as GAMs use a more restricted family of functions, so the K-A theorem does not necessarily apply. It would be necessary to discuss the approximation properties of the particular family of GAMs they use to make this claim precise.
 - The experimental methodology is not properly described in the case of synthetic data experiments. It is unclear what data the methods were tested from which the plots were generated.
 - The authors invoke a lot of mathematical machinery, e.g. the poset formulation for the model and information geometry, without seemingly relying on it too much. I think the method and training process could be described in much simpler terms which would heavily improve the paper's readability.
 - The authors claim to use natural gradients to solve their proposed optimization problem. However, they never demonstrate the utility of this over standard gradient descent.

## Questions
 - "Our model combines the techniques introduced by Luo & Sugiyama (2019) to model higher-order interactions between Poisson processes and by Friedman & Stuetzle (1981) in generalized additive models to learn the joint intensity function using samples in a lower dimensional space" - a lower dimensional space compared to what? Also, this sounds very similar to the approach in the paper. How do they actually differ?
 - "The learning process in our model is formulated as a convex optimization problem to arrive at a unique optimal solution using natural gradient, which minimizes the Kullback-Leibler (KL) divergence from the sample distribution in a lower-dimensional space to the distribution modeled by the learned intensity function. This connection provides..." - the previous sentence is not explaining a connection to anything. What are the authors referring to here?
 - What do the authors mean by "higher-order intensity function"? Do they mean higher-dimensional? Perhaps they meant to use it analogously to the phrase "higher-order interaction", but I don't think "higher-order intensity function" is standard terminology.
clarity,_quality,_novelty_and_reproducibility: The text contains some very long and hard-to-parse sentences and questionable stylistic and typographic choices. For example:
 - After reading the introduction, it is hard to ascertain what the authors are claiming as their contributions. Therefore, I would suggest they list their claimed contributions at the end.
 - I discourage using $\mathbb{N}$ for the counting process, as this symbol is usually reserved for natural numbers. Perhaps use $\mathbf{N}$ instead?
 - "Each J ⊆ [D] determines the condition of the occurrence of the event." - what does this mean?
 - "To flexibly consider any combination of possible states, [D] is composed of possible states across all dimensions." - what does this mean?
 - "For example, in the taxi pick up example in Introduction, domains of x, y, W are discretized to disjoint sets as ..." - this is very confusing. I thought these three variates were each one-dimensional. It is unclear whether the authors mean to say to quantize/ bin the values of continuous variates or simply that multivariate variates can be concatenated to form one big multivariate state. In the latter case, using the phrase "discretizing domains" is incorrect.
 - "We define the functional prior" - This is misleading, as the authors only set the functional form but do not define a prior distribution over log-intensities.
 - "replacing the range with $I \subseteq J$" - this is not what the authors actually mean. They mean that the range becomes the power set of $J$.
 - Sections 2.2 and 2.3, containing the authors' methodology, should be separated from the earlier subsections describing the technical background.
 - The definition of $Z$ in Eq 10 is incorrect. As it is currently defined, $Z$ would necessarily equal 1. The authors should either define $\hat{p}$ as an unnormalized probability or define $Z$ as a sum of the $\sigma_I$ terms.
 - The size of the plots in Figures 3-5, but especially the font size of their labels, need to be increased as they are currently close to unreadable.
 - Figure 4 is cluttered and hard to read.
summary_of_the_review: The authors present a fine idea, but the presentation quality is low, and the empirical results seem weak.
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper881/Reviewer_x4an
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: This paper introduces an inference model for learning the intensity function of a multidimensional Poisson process. The approach is presented as an application of generalized additive models to Poisson processes. One key point comes from information geometry for the log-linear model which shows that the likelihood function of the model at stake is convex.
strength_and_weaknesses: This paper makes an interesting connection between multidimensional Poisson processes, generalized additive models, and information geometry. The approach seems to me original and deserves to be studied.

However, the two key points of the paper (the model itself, obtained as an application of generalized additive models, and the convexity of the optimization problem) do not require any deep theoretical or empirical development. This is not necessarily an obstacle but, in this case, I would expect at least some theoretical guarantees on the quality of the estimate obtained by the method, which are lacking.

clarity,_quality,_novelty_and_reproducibility: The paper is quite well written and organized, but I have the following comments, in order of importance:

- Part of the state-of-the-art in the introduction has been deferred into the supplementary material, which seems to me quite unusual. Moreover, the approaches cited in this appendix are not used as references in the numerical study of the paper. In particular, Poisson factorization appears to be a very relevant baseline.

- Selection of hyperparameters should be investigated in more than few lines at the beginning of the numerical section. For example, we can think that hyperparameters should be selected in a coupled way. Moreover, they have a strong effect on the complexity of the algorithm, which should be taken into account.

- At the beginning of subsection 2.2, the time window is discretized, which avoids to evaluate an intractable integral that appears in the likelihood. I would expect further explanations on this point, like a control of the error induced.

- I am rather surprised that the largest dimension considered in the numerical part is 4.

- Page 3, after equation (2): The authors introduce function $g$ then select $g=\exp$. How critical is this choice? For example, it appears in subsection 2.3 that the optimization problem is convex because the model is log-linear.

- Introduction: There is no Fig. 1b.

- Typo: spacial (page 5)
summary_of_the_review: The approach followed in this paper is in my opinion interesting but requires further developments, both theoretical and numerical, to be published in this conference.
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper881/Reviewer_2omy
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: The paper proposed Additive Poisson Process (APP), which models higher-order interaction effects of the intensity functions in Poisson processes using projections into lower-dimensional spce. The intensity function is modelled using a log-linear model, whose parameters are learned by minimizing the KL divergence between a sample distribution in lower-dimensional projections and the distribution with an intensity function. The empirical experiments showed promising results on synthetic datasets and a real world dataset. 

strength_and_weaknesses: Strength:
* The idea is interesting. The problem of modelling the intensity function in multi-dimensional Poisson processes is challenging due to sparse observations. Approximating the high-dimensional intensity function using low-dimensional projections and framing it in an GAM form is a nice idea.

Weakness:
* a log-linear model is proposed in APP, which is a bit restrictive in its functional form. How easy it is to extend the method to richer class of (non-linear) models? 
* Is it not obvious to me that such a low-dimensional representation always exists. Related to this, How big of k (i.e., the truncation level of the order) is sufficient? Have you thought about ways to improve/constrain APP such that k needed can be low? On the contrary, as the authors also pointed out, large k may result in overfitting, how should one pick k in practice? 
* The hyperparameters M and h are learned with grid search, which may be expensive in high-dimensional problems. 
* The experimental results are all on synthetic data expect one on the New York taxi dataset. More extensive results will make the paper stronger. 
* the number of parameters grows exponentially with the dimension and there is no structural dependency between the theta's. Will it help to enforce some dependency between them (for examples, theta_{1,2} depends on theta_1 or theta_2)? 


clarity,_quality,_novelty_and_reproducibility: The presentation and clarity of the paper in terms of typos and notations can be improved. For example, g in Theorem 2.1 is not defined; Introduction -> introduction in the first line of page 5; caption in Figure 3 should include first-order Poisson process too. 

Authors included code in the supplementary material.

summary_of_the_review: The paper presents a framework for modelling the intensity function in high-dimensional Poisson processes. There are potential improvement area in terms of the presentation, more thoroughly analysis/experiments and discussion, but overall I think it is an interesting idea. 

correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 6: marginally above the acceptance threshold