Response

We would like to thank the reviewer for their thoughtful and constructive feedback. We appreciate the points raised and hope the following addresses the questions and suggestions.

W1, Q1 & W3: Computational complexity

We have added the average training runtimes to the appendix (see A.8), demonstrating that our model is comparatively fast to train. Notably, our model has the fewest learnable parameters—up to two orders of magnitude fewer than the baselines—further allowing to reduce the effective compute time by running multiple models in parallel on a single GPU. Furthermore, as shown in Figure 6 and discussed in Section 4.4, the computational complexity with respect to the number of points in a set remains nearly constant for our model. While the full attention in our encoder could eventually impact scaling, this could then be mitigated by limiting the attention to a fixed context window, a practice already leveraged by the baselines to keep them tractable.

W2 & Q2: Hyperparameter selection and sensitivity

We generally find our model to be robust across different hyperparameter values, allowing us to use the same hyperparameters for all datasets. We added the results for different numbers of diffusion steps $T$ to the appendix (see A.9) and have discussed them in our response to reviewer r6U8; for further discussion of the noise schedule, please also refer to our response to reviewer mmd5.

Q3: Interpretability

Without ordering, it is generally not possible to effectively model or sample the conditional Janossy intensity (Eq. 2) for point processes on general metric spaces. To address this, our proposed method learns the joint density of point sets through a diffusion-based latent variable model. While this enables modeling point processes on general metric spaces, supports efficient parallel sampling, and allows for flexible generation in complex conditional tasks, it does not permit evaluating or interpreting the conditional intensity or its parameters. Therefore, for applications involving ordered point processes (STPP, TPP) that require evaluation of the conditional intensity—such as estimating the likelihood of the next point given the past—point process models that directly approximate the conditional intensity are better suited. In contrast, our method prioritizes generality, scalability, and flexibility, excelling in unconditional and conditional generation tasks and addressing key limitations of intensity-based models. We have added a small discussion of this limitation to the last paragraph of our conclusion.

Response

We appreciate the reviewer’s detailed evaluation and feedback on our work and respond to the comments below.

W2: Definition of $q(X_{t-1}|X_0^c)$

$q(X_{t-1}|X_0^c)$ in Algorithm 1 refers to the Markov Chain of the forward process (Section 3.1), which intuitively noises the condition by thinning and adding noise on the whole domain. Then by applying the conditioning mask (line 6), we obtain the noised condition. We agree that the brevity of this notation might be hard to follow, and we annotated the lines in the algorithm accordingly.

Typos

Thanks for pointing out the typos; we have adjusted the manuscript accordingly.

W1 & Q2: Connection and performance difference to ADD-THIN

First, ADD-THIN also leverages the thinning and superposition properties to define a diffusion process for TPPs, where they mix the thinning and superposition in their noising process so that it consists of the superposition of $T+1$ point sets with different intensity functions, where even added points can be removed again, which significantly complicates the posterior and introduces redundant steps. In contrast, our model disentangles the superposition and thinning to attain two independent processes to allow for more explicit control and define the diffusion model independent of the intensity function as a stochastic interpolation of two point sets. Further, the parametrization of ADD-THIN is specific to TPPs and directly leverages the ordering of points (temporal embeddings, inter-event times, convolutional layers), while POINT SET DIFFUSION is agnostic to the ordering of points, making it applicable for modeling the general class of point processes on any metric space, including for example SPPs. Lastly, ADD-THIN needs to be explicitly trained for specific conditioning tasks, while we show how, after training, our unconditional POINT SET DIFFUSION model can be conditioned for arbitrary and unknown conditioning tasks on the metric space. We discuss these connections in our related work section in more generality.

Second, to show the performance difference, we ran all TPP experiments from ADD-THIN with our model and report them in appendix A7. As can be seen, we are able to match the SOTA performance of ADD-THIN on TPPs with our more general setting. It is especially noteworthy that our unconditionally trained model can surpass the conditionally trained ADD-THIN model on their forecasting task, which shows that our model can effectively capture the interaction between points by directly modeling the joint density.

Q1: Noise schedule

$T$ is set to 100 steps, where we refer the reviewer to our discussion of the impact of $T$ in our answer to reviewer r6U8. Regarding $\alpha$ and $\beta$, we have found $\bar{\alpha}_t = 1-\bar{\beta}_t$ to be effective since it ensures a direct interpolation between the two point sets, which if $\int_A \lambda^{\epsilon} = E[N(A)]$ ensures a constant expected number of points throughout the process. Regarding the choice of a noise schedule, we initially experimented with a linear and a cosine schedule, where we found the cosine schedule to work marginally better. However, the specifics of the noise schedule are an interesting direction for future work to explore, especially since most (continuous) noise schedules were designed for continuous Gaussian diffusion models focusing on the noise-to-signal ratio of a continuous variable. In contrast, our diffusion process is distinctively more discrete (i.e., thinning and superposition process), possibly opening a new pathway for future work.

Response

We want to thank the reviewer for their feedback and appreciation of our work and address their two points.

W1: SPP baselines

We agree that while the very popular (Log-Gaussian) Cox process and the Regularized Model proposed at NeurIPS 2019 are highly relevant baselines, comparisons to additional SPP baselines would be valuable. However, the unordered nature of SPPs does not permit direct and effective modeling or sampling of the conditional Janossy intensity, thereby precluding the application of traditional point process models designed for ordered spaces. To the best of our knowledge, there exists no other SPP model capable of capturing point-to-point interactions that underlie the unconditional or conditional SPP tasks. We hope this explanation clarifies our choice of baselines and emphasizes the challenges in identifying comparable methods within the scope of this work.

Q1: Trade-off between sampling time and quality $(T)$

We have used $T=100$ for all experiments and added a sentence to the hyperparameter paragraph in the Appendix.

The sampling time scales linearly with the number of diffusion steps $T$. To provide insight into how the number of steps affects sample quality, we have run a hyperparameter study for the unconditional STPP experiment on the validation set of the Earthquake dataset, evaluating $T \in \{ 20, 50, 100, 200 \}$, averaged over three random seeds. Our findings indicate that while fewer diffusion steps result in reduced sample quality, $T = 100$ strikes a good balance, already matching and even surpassing the quality observed at $T = 200$. Although this result may seem counterintuitive to those familiar with standard Gaussian diffusion models, it highlights a key distinction of our approach: unlike Gaussian diffusion processes, our model employs inherently discrete Markov steps—specifically, the superposition and thinning of point sets with fixed cardinality. As a result, only a limited number of points can be added or removed over $T$ steps, imposing a natural ceiling on how much additional steps can improve sample quality.

Response

We thank the reviewer for their appreciation of our work and their feedback.

As the raised points seem rooted in one misunderstanding of our model, we clarify it before addressing the specific comments.

Our model captures the joint density of point sets

When modeling Point Processes, we are interested in capturing the complex interactions between all points, which can be expressed as the conditional Janossy intensity (see Eq. 2). However, as explained in the subsequent paragraph, parameterizing and sampling this intensity is not feasible in general for non-ordered spaces. For ordered point processes (TPP, STPP), most models rely on the history-dependent intensity to parameterize the density of the point process, i.e., $p(X)= \prod^N_i \lambda(x_i|H_{x_i}) e^{-\int \lambda(x|H_{x})}$, where $H_{x}$ is the conditional history up to point $x$. This introduces a factorization across time, enabling autoregressive sampling but limiting conditioning tasks to simple forecasting and enforcing sequential sampling. In contrast, our approach leverages the thinning and superposition properties to sample from any conditional Janossy intensity without explicitly parameterizing it. Intuitively, one can think of it as a different factorization of the Point Process density $p(X)$, not across time, but across a latent variable process -- Point Set Diffusion. Thus, our model is neither restricted to inhomogeneous Point Processes, i.e., $\lambda(x_i|H_{x_i})= \lambda(x_i)$ nor first-order statistics but generalizes to point processes with arbitrary interactions.

W1: Point set diffusion is not limited to "first-order-statistics"

As noted above, our model is not limited to first-order statistics or inhomogeneous intensities. It generalizes beyond conditional intensities of ordered processes, capturing any interaction between points on the metric space. For instance, it can predict the history given the future or a time window given past and future points. In short, our model supports a broader range of intensity functions than the conditional intensities captured by standard STPP or TPP models.

W2: Why use the conditional sampling method (Algorithm 3.1)

Our model is trained unconditionally to learn the joint density of point sets, so it is non-trivial to solve conditioning tasks by leveraging our joint density parametrization. With Algorithm 3.1, we demonstrate the flexibility of our model and show how to condition our unconditionally-trained model for conditioning tasks on the metric space, subsuming forecasting, history prediction, and more complex conditioning tasks. This algorithm is used for all conditional tasks in the paper: spatial conditioning (Sections 4.3, 4.5), temporal forecasting (Section 4.4, Appendix A.7.2, A.10).

Clarifying definition of $q(X_{t-1}|X_0^c)$

$q(X_{t-1}|X_0^c)$ refers to the Markov Chain of the forward process (Sec. 3.1), which noises the condition by thinning and adding noise across the domain. Applying the conditioning mask (Algorithm 3.1, line 6) yields the noised condition. We agree that this notation may be unclear and have annotated the algorithm for clarity.

W2, W3, W4: Conditional generation for (ordered) point processes

Visualization

While Figure 7’s density plots may appear similar, they are distinct. The spatial similarities are common for S(T)PPs, such as Earthquakes (Figure 7), and reflect a "shared" spatial pattern along tectonic plates. To better demonstrate our conditional generation, we added plots (Appendix A.10) showing the spatial density at different time points for the Earthquake dataset. Since, unlike other STPP or TPP models, our approach does not parameterize the intensity of the next point given the past but models all following points, we cannot show the conditional density plots requested in W3 and W4. However, in the plot, we show sliding forecast windows (e.g., forecasting 1/6th of the time domain at different time points, (0, 1/6, 2/6,...,5/6)), revealing that our conditional densities change in time and are influenced by prior events (e.g., earthquake aftershocks).

Conditional results

Sections 4.3 and 4.4 report state-of-the-art results for conditional tasks on SPPs and STPPs. Furthermore, Appendix A.7.2 compares our model to ADD-THIN [3] on their TPP forecasting task, where they recently showed state-of-the-art results. Note that our model, unlike ADD-THIN, is not specifically trained for this task. Still, our model closely matches or even outperforms ADD-THIN on all datasets.

W5: Log-likelihood metric

The log-likelihood (LL) measures the likelihood of the next event given its history, typically parameterized through the conditional intensity. Since our model does not parameterize this conditional intensity—unlike the diffusion models cited in the review—reporting the LL is not applicable.

That said, we would like to respectfully challenge the notion of the LL as the gold standard for evaluation. The LL is computed as $\sum^N_i\lambda(x_i|H_{x_i}) - \int \lambda(x_i|H_{x_i})$, with the integral spanning space and time for STPPs. While the LL is a natural metric for STPP baselines trained to optimize it, it is evaluated using the ground truth history, primarily reflecting the next-event prediction but not the quality of generated samples or forecasts in real-world applications. Further, the LL has known failure modes, and models with high LL can still produce samples significantly different from the training data [5], an issue worsened by error accumulation in autoregressive sampling. This issue has been raised by different PP papers [1][2][3], with [1] even stating that "the NLL is mostly irrelevant as a measure of error in real-world applications." Similarly, probabilistic forecasting has replaced LL as the "gold standard" in time series [4][6]. Additionally, the reported LL directly depends on model-specific approximations and parametrizations, complicating and making a fair comparison across (S)TPP models error-prone due to differing approximations and implementations.

In contrast, our experiments follow [2] and [3] by comparing the distributions of point set samples for different tasks, independent of the implementation of each model. It is very important to state that the applied metrics capture more than just the first-order characteristics of the data. For the unconditional task, we report the Wasserstein distance between the count distributions and the maximum mean discrepancy. This kernel-based statistic test compares the two distributions based on a sample-based distance metric (CD) (for further details on the MMD for point processes, please also refer to Appendix E.2 of [2]). Given the ground truth target in the conditional tasks, we do not need to compare distributions of point processes. Hence, we leverage the MAE for the difference in the number of points and the Point Process Wasserstein Distance, again a distance between two distributions as each instance of a point process is a stochastic process.

Lastly, restricting evaluations to one-step-ahead LL would limit model design to parameterizations of a conditional intensity function for which we can efficiently or approximately compute the integral–in our opinion, a significant restriction for the field of point processes.

[1] Shchur, O., Türkmen, A. C., Januschowski, T., & Günnemann, S. "Neural temporal point processes: A review." IJCAI (2021)

[2] Shchur, O., Gao, N., Biloš, M., & Günnemann, S. "Fast and flexible temporal point processes with triangular maps." NeurIPS (2020)

[3] Lüdke, D., Biloš, M., Shchur, O., Lienen, M., & Günnemann, S. Add and thin: Diffusion for temporal point processes. NeurIPS, (2023)

[4] Tilmann Gneiting and Matthias Katzfuss. Probabilistic forecasting. Annual Review of Statistics and Its Application, (2014)

[5] Theis, Lucas, Aäron van den Oord, and Matthias Bethge. "A note on the evaluation of generative models." ICLR (2016)

[6] Alexandrov et al. GluonTS: Probabilistic and neural time series modeling in Python. JMLR 2020

I appreciate the response from the authors. In general, some of my concerns have been addressed, but others have not, and I am not fully convinced in a few parts of the author's response.

My confusion about "the model focuses on the first-order property of the point process" has been addressed, and I can agree with the logic behind the modeling of the Janossy density. I understand the necessity of a conditional sampling algorithm, and the newly added example of Appendix A.10 did demonstrate that future events' distributions are influenced by prior events. I give credit to the authors for helping address my above concerns.

However, what I am concerned with in my previous review is the effectiveness or accuracy of the conditional sampling instead of the necessity. An easy way to prove this is via simulation: given an observed history generated from a true STPP model $m^*$, the authors can first use $m^*$ to generate enough samples of sequences in the future time frame and calculate the density of points as "distribution of future events". Then, the authors can use their model to generate future sequences and compare the distribution of those events with the true distribution of future events obtained from the $m^*$. A synthetic experiment like this can clearly prove whether the conditional sampling really captures the ground truth.

#######

The following are my comments on the discussion of the log-likelihood metric. With respect to the authors, I am afraid I cannot agree with their opinions on the log-likelihood metric:

1. First, the computation of the likelihood of a point process does not require the parametrization of the conditional intensity function. The intensity-based computation listed by the authors is only one of the approaches to compute the likelihood when the intensity function is available.

2. In fact, the derivation of the point process likelihood is closely connected with the Janossy density (see the derivation in section 2.4 in [1], or in Section 5.3/Definition 7.1.II in [2]). The likelihood is computed by a series of density functions (which is also the very original definition of any likelihood), seeing equation 6 in [1].

   • A few examples of calculating likelihood without parametrizing the conditional intensity function can be found in [3][4]. These pp studies also use diffusion models to sample events (although I admit that they are using diffusion in an autoregressive way, and this paper contributes to it by extending the modeling beyond autoregression by sampling a few points in parallel), and the likelihood can be computed by sampling candidate points and calculating the density of observed ground truth.

3. If the authors are claiming that their model can learn the Janossy density of the point processes, I would also expect the model to have a good likelihood of the data. I can even think of a way for the authors to calculate the likelihood of a sequence: giving the observed history, generate multiple sequences, keep the first event in each sequence, and calculate the density. This is the density of the next event based on the history. The likelihood of an entire sequence is computed by iterating over all the events.

   • I believe this evaluation procedure would cost linear complexity as others since the Point Set Diffusion is generating one sequence at a time.

In summary, I hold my opinion about the golden standard of likelihood in point processes and cannot agree with the authors' opinion that it is the significant restriction for point processes. Again, it does not require the parameterization of the conditional intensity.

#######

Still, I understand that the computation of likelihood will not be a main flaw of the proposed Point Set Diffusion. I can now see the value of the proposed method to the fields of point processes. Meanwhile, I still hope the authors can consider adding the synthetic data experiments and the possible evaluation of the likelihood. This would improve the quality of the paper and make it look more convincing.

Based on all the assessments above, I decided to raise my score.

Do the authors plan to release the code/implementation?

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[1] Reinhart, Alex. "A review of self-exciting spatio-temporal point processes and their applications." Statistical Science 33.3 (2018): 299-318.

[2] Daryl J Daley, David Vere-Jones, et al. An introduction to the theory of point processes: volume I: elementary theory and methods. Springer, 2003

[3] Dong, Zheng, Zekai Fan, and Shixiang Zhu. "Conditional Generative Modeling for High-dimensional Marked Temporal Point Processes." arXiv preprint arXiv:2305.12569 (2023).

[4] Yuan, Yuan, et al. "Spatio-temporal diffusion point processes." Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2023.