Venue: ICLR 2021 Conference

Reviews:

final_score: rating: 3: Clear rejection
review_content: This paper proposes a generalized additive model to learn joint intensity functions for multiple Poisson processes. The main contribution is to consider the partial order of high-order interactions when parameterizing the model.

The proposed method is a reasonable solution, but there are several points in the paper that are particularly weak:

First, the paper does not find a proper scenario to apply the additive Poisson process model. The experimental part lacks real data analysis. In Section 3.3, they only show a quantitative result regarding two Poisson processes [T, W]. It is hard to summarize that the performance gain comes from high-order modeling without explaining it in detail. For example:
i) Why are KDE and RKHS predictions inconsistent across January through March?
ii) How do you define M?
iii) How sparse is the data?

Second, jumping from section 2.1 to 2.2 is confusing, since section 2.2 seems to have nothing to do with the Poisson process. So my first question is:
i) Can you define the likelihood in a single Poisson process model in order to consider all the interactions you mentioned in section 2.2?
After carefully reading the setup in section 2.2, I feel that the entire method is based on a crude approximation of Poisson processes. For example, I’m curious about:
ii) Does your method really depend on discretizing the time window? What if you let M goes to infinity? I’m particularly interested in this question because the original Poisson process likelihood model does not require discretization.

To conclude, this paper proposes a reasonable method to model high-order correlations. However, its experimental result is weak and the model construction is of a question to me.

review_title: This paper proposes a reasonable method to model high-order correlations. However, its experimental result is weak and the model construction is of a question to me.

final_score: rating: 4: Ok but not good enough - rejection
review_content: ## Summary

This work claims to address the problem of learning the structure of interactions between multiple point processes. This can, e.g., be used to predict co-occurrences between different types of events. The authors develop a method that (a) is theoretically principled and (b) scales to large datasets.

## Reasons for score

The overall problem is clearly relevant, and some aspects of the contribution appear to be promising (connection to GAMs, approach to inference). However, in its current form, the paper is excessively unclear, to the point where I am not sure that I understand the precise problem the authors are trying to address (despite my familiarity with the area).

## Pros

- if the problem is that of understanding correlations between multiple point processes, then that is an important problem that is clearly relevant to the ML community.
- The approach builds on generalized additive models, which gives their approach a solid theoretical foundation
- The inference algorithm based on minimizing KL-divergence appears to be interesting.

## Cons

- the paper lacks clarity throughout, making it difficult to understand the problem, let alone the contribution itself (examples below).
- the experiments are not explained in sufficient details (NYC taxi dataset for example), and as such they are not supporting the contribution.

## Comments

Referring to "stochastic processes" is too general—maybe talking about "point processes" would be better. For example, a Gaussian process is a stochastic process, but doesn't have anything to do with the processes considered in this work.

Section 2, "Formulation" is confusing. Usually, a multi-dimensional / spatial poisson process is a *single* point process defined on $\mathbf{R}^D$, and a realization of that process is a collection of points in D-dimensional space. This is what, e.g., Flaxman et al. (2017) consider.

In this paper, the authors seem to assume that every dimension is a different process, and somehow events in each dimension are grouped together based on their arrival order ("Each $t_i$ is the time of occurrence for the $i$-th event across $D$ processes and $T$ is the observation duration"). I am not even sure this is always possible, since it seems to assume that every process / dimension has exactly $N$ events during the interval $[0, T]$. Thus, I fail to understand even the basic foundation on which their work builds.

It seems to me that the authors are really considering a *multivariate* Poisson process, where the intensity function of each process are dependent. In that case, the likelihood would be different from (1). Can the authors comment on this?

Experiments:

- "the sample size is randomly chosen by the mixture of Gaussians" - what does this mean? a GMM does not define a distribution over number of samples.
- where is the notion of timestamps in the synthetic GMM experiments?
- Figure 3 does not seem to be referenced in the text. How can the intensity function be 1D for if the experiments are with 2D Gaussians?
- NYC taxi dataset: how is space taken into account? Defining a "common pick-up" as one that happened within at 1 min interval but at opposite ends of the city seems pointless.

Other comments / questions:

- How does your work compare / contrast to multivariate Hawkes processes?
- In what way is Flaxman et al. 2017 non-convex?
- " ...where the different events are marked so that a particular subset of events can be used to generate the intensity function" what does this mean?

## POST-REBUTTAL

Thank you for rebuttal which has helped me, to some extent, to understand your model. I still believe the formulation needs to be clarified before the contributions can be assessed objectively.

For example, on page 2:

> Given a realization of timestamps $\mathbf{t}_1, \mathbf{t}_2, \ldots ,\mathbf{t}_N$ with $\mathbf{t}_i \in [0, T]^D$ from an inhomogeneous (multi-dimensional) Poisson process with the intensity $\lambda$. Each $\mathbf{t}_i$ is the time of occurrence for the $i$-th event across $D$ processes and $T$ is the observation duration.

This is confusing: only a single one of the dimensions will likely represent "time". Other dimensions will be space, etc. Furthermore, why would every dimension need to be within $[0, T]$? Different dimensions might use different units and be in different ranges.

I am upgrading my score slightly, because I sense that your contributions might be interesting once properly explained, but in the present state I believe the paper is not ready for publication.
review_title: Paper too unclear to understand & assess contributions

final_score: rating: 6: Marginally above acceptance threshold
review_content: The paper under review proposes a new model for multi-dimensional temporal Point processes, allowing efficient estimation of high order interactions. This new model, called additive Poisson process, relies on a log-linear structure of the intensity function that is motivated thanks to the Kolmogorov-Arnold theorem. Such structure is then linked to generalized additive models, a result that is used to devise an efficient estimation procedure with formal guarantees  of convergence.

The paper is well written but some part lacks of details making the exposition sometime difficult to follow. Overall the contribution is original and promising, but in its current state suffers from imprecision making difficult to assess its quality. The following points summarizes these remarks:


## Major points

* The experimental design is unclear: what is the structure of the intensity function? Is it log-linear? If yes, why APP, which is designed for this case is outperformed by KDD? Why is the interaction between processes 1 and 2, is summarized by a univariate function ? Does this imply that only simultaneous interactions are considered in this work, as suggested by the second sentence of Section 3.1? 
* How would the method compare to others when the log-linearity is not satisfied, e.g., $\lambda(t_1,t_2) = \lambda_1(t_1)\lambda_2(t_2)$? Such cases, if not studied in detail, should at least be mentioned and discussed as a limitation of the current model.
* The application is used to showcase the usefulness of the new model, but model comparison relies on a negative log-likelihood criteria which does not account for the complexity of the models. If possible, some more general criteria such as AIC should be considered. If such calculations are not possible/does not apply, then differences in model's complexity should be at least discussed.

## Minor points

* p.1: the first sentence is really unclear. Not all stochastic process have an 'arrival time'. Note that correlation does not mean simultaneity of the events.
* p.1: 'intensity function is multiplied ...'
* p.1 : twice 'the'.
* p,1 'Differently' sounds awkward.
* p.1: 'if observations are sparse'.
* p.2: 'intensity is often low' would be better to say 'lower than lower-orders'.
* p.2: of the model 'are'.
* p.2: 'the model is robust to' this is not clear and never discussed in the paper. No misspecified experiment is presented.
* p.2 'arrival intensity'?
* p.2: 'for time t > 0, ...' is this sentence needed? Why not simply giving the classical setting with integration over a non-negligible region?
* p.3: the notation $g_{p,q}$ is never defined.
* p.3: the link between Theorem 1 and the log-linear model is unclear: how the two summations become only one? The summation in the Theorem uses the number of sample $n$ while the summation in (5) depends on the dimensionality of the subset. This is particularly unclear.
* p.3: notation $\perp$ is not defined.
* p.4: the variable $Z$ is defined but never used.
* p5.: 'We can always use the natural gradient ...' this sentence is unclear.
* p.5: 'uniform distribution' on $[0,1]$? What are the 'expected number of samples' that are used?
* p.8: Table b: What are $T$ and $W$? How is it possible to have a joint process for APP-1?

review_title: The paper is well written but some part lacks of details making the exposition sometime difficult to follow. Overall the contribution is oroginal and promising, but in its current state suffers from imprecisions making difficult to assess its quality.

Meta-review: This paper proposes a method for modeling higher-order interactions in Poisson processes. Unfortunately, the reviewers do not feel that the paper, in its current state, meets the bar for ICLR. In particular, reviewers found the descriptions unclear and the justifications lacking. While the responses did aid the reviewers understanding, the paper would benefit from rewriting and more careful thought given to the experimental design.
Decision: Reject