Unlocking Point Processes through Point Set Diffusion
Authors: David Lüdke
Enric Rabasseda Raventós
Marcel Kollovieh
Stephan Günnemann
Venue: ICLR 2025 Poster (2025)
Keywords: Generative Model
Diffusion Model
Set Model
Point Sets
Forecasting
Density Estimation
Spatial
Temporal
Probabilistic Models
Primary area: generative models

Abstract: Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics.
Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between efficiency and flexibility.
In this paper, we introduce Point Set Diffusion, a diffusion-based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function.
By directly learning to stochastically interpolate between noise and data point sets, our approach effectively captures the distribution of point processes and enables efficient, parallel sampling and flexible generation for complex conditional tasks.
Experiments on synthetic and real-world datasets demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling.

Published as a conference paper at ICLR 2025
UNLOCKING POINT PROCESSES THROUGH POINT SET DIFFUSION
David Lüdke; Enric Rabasseda Raventós; Marcel Kollovieh, Stephan Günnemann Department of Informatics & Munich Data Science Institute Technical University of Munich, Germany (d. le.lue e. ra rabasseda m. lowoooiieeehheooooooooeeeoaawele s guennemannn @mm dee
ABSTRACT
Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications fields like seis- mology, neuroscience, and economics. Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between ef- ficiency and flexibility. In this paper, we introduce POINT SET DIFFUSION, a diffusion- based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function. By directly learning to stochastically interpolate between noise and data point sets, ap- proach effectively captures the distribution of point processes and enables effi- cient, parallel sampling and flexible generation for complex conditional tasks. Ex- periments on synthetic and real-world datasets demonstrate that POINT SET DIF- FUSION achieves state-of-the-ard performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling.
INTRODUCTION
Forward Process
Reverse Process
Point processes describe the distribution of point sets in mathematical space where the location and number of points are random. On Euclidean spaces, point processes (c.g., spatial and/or temporal; SPP, STPP, TPP) have been widely used to model events and entities in space and time, such as earthquakes, neural activity, transactions, and social media posts.
X Paata (X)
(X,
ol X
Xr Proise (X)
Point processes exhibit complex inter- actions between points, leading to corre- lations that are hard capture effectively (Daley & Vere-Jones, 2007). The distri- bution of points is typically characterized by non-negative intensity function, rep- resenting the expected number of events bounded region of space (Daley et al., 2003). common approach to modeling point processes on general metric spaces is to parameterize an inhomogeneous in- tensity as function of space. However, this approach assumes independence between points, which restricts its ability to model complex interactions and hinders generalization across different point sets (Daley al., 2003; Daley & Vere-Jones, 2007).
Figure 1: Illustration of POINT SET DIFFUSION for earthquakes Japan. The forward process stochasti- cally interpolates between the original data point set Xo and a noise point set Xr, progressively remov- ing data points and adding noise points. To generate new samples from the data distribution, we approxi- mate the reverse posterior (XX, X(+1) and add ap- proximate data points and remove noise points.
"Equal contribution
Code available https:///wwwtt of www. /poi -t-nst-ddi ffusion

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For ordered spaces like time (STPP, TPP), the predominant approach is to model the conditional intensity autoregressively, conditioning each point on its past, allowing temporal causal dependen- cies. which can be conveniently captured by state-of-the-an machine learning models (Shchur al., 2021). While this enables point interactions, these models rely on rikelihood-based training and se- quential sampling, which require integrating the intensity function over the entire space. Ultimately, this constrains models, as it either necessitates oversimplified parameterizations that restrict point dependencies and introduce smoothness (Ozaki, 1979; Ogata, 1998; Zhou & Yu, 2023), or approx- imations using amortized inference (Zhou et al., 2022), numerical (Chen et al., 2020), or Monte Carlo methods (Hong& & Shelton, 2022). Thus, capturing complex point dependencies and sampling from point processes, particularly on general metric spaces, remains an open research challenge.
Lüdke et al. (2023) overcame the limitations of the conditional intensity function for temporal point processes by proposing ADD-THIN, diffusion model for TPPs based on the thinning and super- position property of TPPs directly modeling entire event sequences. In this paper, we generalize this idea to point processes on general metric spaces and derive a diffusion-based latent variable model, POINT SET DIFFUSION, that directly learns to model the stochastic interpolation between data point set and samples from any noise point process (see Figure 1). Furthermore, we show how to generate conditional samples with our unconditional POINT SET DIFFUSION model to solve arbitrary conditioning tasks on general metric spaces. Our experiments demonstrate that POINT SET DIFFUSION achieves state-of-the-art results on conditional and unconditional tasks for SPPs, TPPs and STPPs. Our contributions can be summarized as follows:
We derive a diffusion- based latent variable model that captures the complex distribu- tion of point processes on general metric spaces by learning stochastic interpolations between data and noise point sets.
Our model enables efficient and parallel sampling of point sets while supporting flex- ible conditioning through binary functions on the metric space.
We propose model -agnosic evaluation framework for assessing generative point pro- cess models on Euclidean spaces.
Our method achieves state-of-the-arr results for conditional and unconditional genera- tion of SPPs, TPPs, and STPPs while offering orders of magnitude faster sampling.
BACKGROUND
2.1 POINT PROCESSES
A point process (Daley et al., 2003) is a stochastic process where realizations consist of finite sets of points randomly located in a mathematical space. More formally, let (D, d) be complete, separable metric space equipped with its Borel o-algebra B. A point process on D is mapping X from probability space (9,, A. P) into N the set of all possible point configurations, such that for any bounded Borel set A C D, the number of points in A, denoted by N (A), is a finite random variable.
Given a realization of the point process X = {ar; E Dhisinn where n is the number of points, the number of points in region is expressed as the counting measure ((A) E 1( E A). Here, we assume the point process is simple, i.c., almost surely N ((xx)) & for all æ, E D, meaning no two points coincide. Point processes are commonly characterized by their intensity function, which is defined through the following random measure:
A - ((() = E(N(A)) - J1 X(x) da,
(1)
where ((A) represents the expected number of points in region A. Then, point process is said to have intensity À if the measure above has a density À with respect to the Lebesgue measure (u(A). Thus, the intensity function A(x) gives the expected number of points per unit volume in small region of the Borel set A C D.
The points in a realization X can exhibit complex correlations, so the intensity function is non- trivial to parameterize. On a Euclidean space IR we can specify the Papangelou intensity (Daley al., 2003):
x(x)= = lim ([N(Bs(z)) = 1RC(N(R) Bs(x))))
(2)
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[s(x)]
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where (5(x) is the ball centered at with radius of 8, and C(N(R) \ Bs(z))) represents the information about the point process outside the ball. If the Euclidean space is ordered, for instance, representing time, the conditioning term would represent the history of all points prior to
In general, effectively modeling and sampling from the conditional intensity (or related measures, e.g., hazard function conditional density), for arbitrary metric spaces is fundamental problem (Daley et al., 2003; Daley & Vere-Jones, 2007). This difficulty has led to a variety of simplified parametrizations that restrict the captured point interactions (Ozaki, 1979; Zhou & Yu, 2023; Daley et al., 2003; Daley & Vere-Jones, 2007); discretizations of the space (Ogata, 1998; Osama et al., 2019); and numerical or Monte Carlo approximations (Chen et al., 2020; Hong & Shelton, 2022).
In contrast, we propose a method that bypasses the abstract concept of (conditional) intensity function by directly manipulating point sets through a latent variable model. Our approach leverages the following point process properties:
Superposition:
Given two point processes N1 and N2 with intensities \ and X2 respectively, we define the superposition of the point processes as N = + N2 or equivalently X1 Ux, Then, the Å +22 Independent thinning: Given point process N with intensity X, randomly removing each point with probability is equivalent to sampling points from a point process with intensity (1 - pp)
resulting point process has intensity
2.2 DIFFUSION MODELS
Ho et al. (2020) and Sohl-Dickstein et al. (2015) introduced new class of generative latent variable models- - probabilistic denoising diffusion models. Conceptually, these models learn to reverse probabilistic nosing process to generate new data and consist of three main compo- nents: a noising process, a denoising process, and a sampling procedure. The noising process is defined a forward Markov chain q(X++1 Xt), which progressively noises data sample Xo ~ Pdata (X) over T steps, eventually transforming it into a sample from a stationary noise distribution XT Pnoise(X). Then, the denoising process learned process by approximating the posterior d(X with a model pa (x,](Xe++)) Finally, the sampling procedure shows how to generate samples from the learned data distribution pe (X) = Puoise
reverse the noising
(Xa) (Xr) I Po(X, .dX...
3 POINT SET DIFFUSION
In this section, we derive diffusion-based latent variable model for point sets on general metric spaces by systematically applying the thinning and superposition properties of random sets. This approach allows direct manipulation of random point sets, avoiding the need for the abstract concept of an intensity function. We begin by outlining the forward noising process in Section 1. which stochastically interpolates between point sets from the generating process and those from a noise dis- tribution. Subsequently, we demonstrate how to learn reverse this noising process generate new random point sets in Section 3.2. Finally, in Section 3.3, we show how to sample from our uncondi- tional model and generate conditional samples for general conditioning tasks on the metric space.
3.1 FORWARD PROCESS
Let X, ~ (X)ata()) be an i.i.d. sample from the generating point process, and let Xy ~ Pnoise (X) represent a sample from a noise point process. We define the forward process as stochastic interpolation between the point sets Xo and X2 over T' steps. This process is modeled as a Markov chain ((xe+ where X, is the superposition of two random subsets: Xthin S Xo and X3 S Xy. Specifically, Vt X, Xthin UX,, where X-hin and X; are independent samples from a thinning and noise process, respectively. We define the thinning and noise processes given two noise schedules {a, E (0, 1)]=" and (3e E (0, 1) as follows:
Thinning Process: This process progressively thins points in Xthin Xo. removing signal over time. At every step + 1, each point æ E Xthin is independently thinned with probability 1 - 0:41:
q(x
Xthin | E Xthin) 0441-
(3)
We provide proof of both properties for general Borel sets in A.I.
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q(x++1pp)
X
xT
Thinning Process
Noise
Process
The forward process is Markov Chain g(Xx+11 1 (Xe), that stochastically interpolates data sample Xo with noise point set Xr over I steps by applying thinning and a noise process.
Figure 2:
Consequently, the thinning defines n independent Bernoulli chains, and the probability of any point Xo remaining in X is:
g E X:hhn ae Xo) = Ct.
(4)
where dt hh= 0r Equivalently, the intensity of the thinned points step is given by Xthin = Ao and the number of remaining points follows Binomial distribution: Pr(ms|Xphin Binomial([Xo) āz), where nhe = |
Noise Process: This process adds random points X4. ~ Puoise sampled from a noise point process with intensity X At step t + 1, express X X1+1 as:
X = X U where X ~ Be+115.
(5)
By the superposition property, the intensity of X is XE B&X where 3, = 3 and 3r
[0. 1]. Alternatively, we can view the noise process a reversed thinning process: we sample
X7. Pnoise (X) and thin it by B, obtain X₁. Given noise sample X;, we then find that:
a(x: E X1 E Xf) = B..
(6)
Notably, this process is independent of the random point set Xo. ie., Vt ((x = (X;).
We present visual depiction of the two forward processes in Figure 2. Finally, given that Vt X,
X.hin U-F it follows that for lim ā₃ = and lim, Bg 1 the stationary distribution is ((T) Xo) Proise (X)), which can be seen by applying the superposition property and finding the intensity of x, to be 04 + BeA*. To summarize, the forward process gradually removes points from the original point set Xo Pdata while progressively adding points of noise point set XT ~ Proiso (X), stochastically interpolating between data and noise.
3.2 REVERSE PROCESS
To generate samples from our diffusion model, i.e., Xy - ... - Xo. we need to learn how to reverse the forward process by approximating the posterior q(X, X X241) with model pa
We will start by deriving the posterior (o,,)oo Xx+11) from the forward process (X+41 Xx) and then show how to parameterize and train po( to approximate the posterior.
Since the forward process consists two independent processes (thinning and noise) and noticing that Xthin Xo 0 and X Xx+1 Xo. the posterior can be derived in two parts:
Thinning posterior: Since all points in Xthin have been retained from 0, follows that X C X/hin_ Then for each point in æ E X, ) Xx we derive then posterior using Bayes' theorem, applying Equation 3, Equation 4 and the Markov property:
q(x & Xthin w XXthin) e Xthin| Xo)
q E Xthiin l & Xthin x E Xp)
(7)
glx & Xthim æ E Xn)
04+1 )a
ā - c++1
(8)
(1 (++1)
1 0++1
Thus, sample Xthin by superposition of Xthin and thinning Xo X with Equation
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q (xxxxxx
X
Thinning Process
Noise
Process
Figure 3:
The posterior reverses the stochastic interpolation of Xo XT of the forward process by adding back thinned points from the thinning process and thinning point added in the noise process.
Noise posterior: Following the reverse thinning interpretation of the noise process, each point in x must have been in both Xi+1 and X7. Hence, we derive the posterior for each point in Xi+1 to still be in X; by following Equation 6, along with the fact that X; is independent from Xo:
((x E XiIæ E xi+1, x & Xo) q( E X2+ x E X()((x X;)
(xx E x14)
1 Be Be
(10)
(Be+1)
3r+1
Thus, can sample Xi by thinning Xi+1 with probability
3,
Be+1
Parametrization: Given Xg and Xy, the derived posterior can reverse the noising pro- cess to generate Xo- However, generate new approximate sample Xo PX).a(X). we need to be able to sample from the posterior [X,,|Xn, Xx+1) without knowing Xo- For this reason we approximate the posterior with model po (x(o(XXee=1.. where we choose (X,, (x+1) = f (Xx,/oo, Xp (Xoo|e+11) dXo and training neural network pa (X() to approximate Xol X for each +
To effectively train this model, we have to condition our model pa (X) on X,. We propose to embed the points apply a sinusoidal embedding to embed n = XX, and t. Then, to probilistically predict Xo) X, we make use of the following distinction for Xthin Xy n X, and X, X,:
X, permutation invariant with Transformer encoder with full attention and
First, predicting the retained points in Xt, i.e., the intersection of Xo and X,, is a binary classifica- tion task for which train prrreppo (MLP) go (xr E X.hinn [X,, () with binary cross entropy loss LACE. Second, the thinned points in Xo. i.e., Xo X,, is a point set N, which can be represented by its counting measure, as mixture of Dirac measures:
dx.-
(11)
In A.2, we prove that any finite mixture of Dirac deltas, such as N, can be approximated by an L2 function in (D(, μ1) for any metric space D. In Euclidean spaces, we approximate the Dirac measure with a mixture of multivariate Gaussian distributions with diagonal covariance matrices. Note that the multivariate Gaussian density function is standard approximation of the Dirac delta function and, as the determinant of a diagonal covariance matrix Σ == of approaches zero, the Gaussian increasingly resembles the Dirac delta (See Equation A.2). We parameterize the number of points to sample ng and the components the mixture weights wa, mean μ4 and diagonal covariance matrix Σg with MLP fo and train it with the negative log likelihood LNLL-
Lastly, to ensure the expected number of points at any time t throughout the diffusion process is constant, we ar
3. and noise process with constant intensity such that SΛ X (A)( for the bounded Borel set A that represents our domain.
3.3 SAMPLING PROCEDURE
Unconditional sampling: Starting from a sample Xy of the noise distribution, we apply our POINT SET DIFFUSION model to sample Xo over T steps. We start by sampling XT X and
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then for all E (T, ...... 1) sample Xo (X X2) to subsequently apply the denoising posterior
x X2) and attain X,-1. Finally, at step 1 we sample Xo ~ po po(XoooXi) We present the extended sampling algorithm in Algorithm 2.
Conditional sampling: Let C D + {0,1} be conditioning mask on metric space D. where we define the masking of a subset X S D as C(X) for (())))
1} and its comple- ment as (X)) = {æ E x((()) 0}. Then, we can leverage our POINT SET DIFFUSION model conditionally generate random point sets outside the conditioning mask by applying Algorithm 1:
Algorithm Conditional sampling
Require: X8 = C()))
1: X7 A₂
2: for = =T, -T...., 1 do
3:
Xo ~ pxx (x.)
4:
X,-1 9(x,_1 Xi)
(reverse 3.2) (forward 3.1)
5:
Xf-1 ~ q(x X()
6:
X,-1 = (X
7: end for
8: return C'(X))
Examples of conditioning masks for Ryo and R.2
Figure 4:
Thus, following this sampling procedure, we can generate conditional samples for any conditioning mask C, where we represent some illustrative conditioning masks for bounded sets on 2go and g² depicting temporal forecasting, history prediction and general imputation tasks in Figure 4.
4 RELATED WORK
Since large parts of the real-world can be effectively captured by Euclidean spaces, point processes have mainly been defined on spatial and temporal dimensions, represented by an Euclidean space. Hence, for this discussion of the related work, we will focus on unordered and ordered point pro- cesses on Fuclidean spaces, mainly SPPs, TPPs and STPPs. For completeness, we want to mention traditional parametric point processes defined on manifolds, such as determential point processes (Berman, 2008; Katori & Shirai, 2022) and cluster point processes (Bogachev & Daletskii, 2013).
Unordered Point Processes (SPP): Modeling permutation-invarian intensity for unordered point sets that captures complex interactions while remaining efficient for sampling is challenging (Daley & Vere-Jones, 2007), seemingly limiting the development of machine- becchin-learning.. based models for SPPs. Classical models like the Poisson Point Process (Kingman, 1992) use either homogeneous inhomogeneous intensity functions across space. More flexible models, such Cox processes (Cox, 1955), and specifically the popular Log-Gaussian Cox Process (Jesper Møller, 1998), extend this by modeling the intensity function through doubly stochastic process, allowing for flexible spatial inhomogeneity. A recent approach, the Regularized Method by Osama et al. (2019), parame- terizes spatial Poisson process on hexagonal grid with splines, offering out-of-sample guarantees. However, these methods often rely on spatial discretization and simple parametric forms and some require separate intensity estimates for each point set, limiting their ability capture the underlying distribution across different samples (Daley & Vere-Jones, 2007; Osama et al., 2019).
The causal ordering of time enables the parametriza- tion of conditional intensity, which classically is being modeled with parametric functions, where the Hawkes Process (Hawkes, 1971) is the most widely used model and captures point interaction patterns like self-excitation. Given the sequential nature of ordered point process, variety of Ma- chine Learning based approaches for TPPs and STPPs have been proposed (see Shchur et al. (2021) for review neural TPPs). Where recurrent neural network- (Du al., 2016; Shchur et al., 2020a) and transformer- based encoders (Zhang et al., 2020a; Zuo et al., 2020; Chen et al., 2020) leveraged to encode the history and neurally parameterized Hawkes (Zhou & Yu, 2023; Zhang
Ordered point processes (TPP and STPP):

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et al., 2020a; Zuo et al., 2020), parametric density functions (Du et al., 2016; Shchur et al., 2020a), mixtures of kernels (Okawa et al., 2019; Soen et al., 2021; Zhang et al., 2020b; Zhou et al., 2022), neural networks (Omi et al., 2019; Zhou & Yu, 2023), Gaussian diffusion (Lin et al., 2022; Yuan et al., 2023) and normalizing flows (Chen et al., 2020; Shchur et al., 2020b) have been proposed to (1ram-parametrically decode the conditional density intensity of the next event.
Differences to ADD-THIN (Lüdke et al., 2023): Since our method is closely related to ADD-THIP we want highlight their key methodological differences. While ADD-TIIIN proposed leverage the thinning and superposition properties to define a diffusion process for TPPs, POINT SET DIF- FUSION generalizes this idea define diffusion- -ased latent variable model for point processes on general metric spaces. In doing so, we disentangle 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 stochastic interpolation of point sets. Furthermore, ADD-THIN has to be trained for specific conditioning tasks, while we show how to condition our unconditional POINT SET DIFFUSION model for arbitrary conditioning tasks on the metric space. Lastly, POINT SET DIFFUSION and its parametrization are agnostic to the ordering of points, making it applicable to model the general class of point processes on any metric space, including, for example, SPPs.
5 EXPERIMENTS
Although point processes are fundamentally generative models, the standard evaluation method re- lies on reporting the negative log-likelihood (NLL) on a hold-out test set, effectively reducing the evaluation to single-event predictions for STPPs and TPPS. However, this approach presents two key issues. First, computing the NLL depends on the specific implementation and parameterization of the (conditional) intensity function and is intractable for many models. necessitating approxima- tions using Monte Carlo methods, numerical integration, or the evidence lower bound (ELBO). This complicates fair comparisons between models. Second, evaluating the likelihood conditioned on ground-truth history, does not necessarily reflect how well a model captures the data distribution or its ability to perform on complex conditional generation tasks (Shchur et al., 2021). To overcome these limitations, we evaluate the generative capabilities of our proposed POINT SET DIFFUSION model by benchmarking it on a range of unconditional and conditional generation tasks for SPP and STPP. Further, we compare our model's performance with the state-of-the art TPP model ADD- THIN in A.6. Details of our model's 's training and the hyperparameters are in A.4, while all baselines trained reproducing their reported NLL using their proposed hyperparameters and code.
5.1
DATA
We follow Chen et al. (2021) and evaluate our model on four benchmark datasets with their proposed pre- mre-processing and splits: three real-world datasets Japan Earthquakes (U.S. Geological Survey, 2024), New Jersey COVID-19 Cases (The New York Times, 2024), and Citibike Pickups (Citi Bike, 2024) and one synthetic dataset, Pinwheel, based on a multivariate Hawkes process (Soni, 2019).
5.2 METRICS
To evaluate both unconditional and conditional tasks, compute distances between point process distributions and individual point sets, assuming the space is normed, and all points are bounded, i.e., Vi, ,F 1,, We use the following metrics in our evaluation:
Sequence Length (SL): To compare the length distribution of point sets, we report the Wasserstein distance between the two categorical distributions. For conditional tasks, compare the length of the generated point set to the ground truth by reporting the Mean Absolute Error (MAE).
Counting Distance (CD): Xiao al. (2017) introduced a Wasserstein distance for ordered TPPs based on Birkhoff's theorem. We generalize this counting distance to higher-dimensien ordered Euclidean spaces (e.g., STPPs) using the L₁ distance:
CD(X,Y) 1 Σ
a villi
Σ j=k-1
IU y
(12)
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Table 1:
Density estimation results on the hold-out test set for SPPs, averaged over three random
seeds (bold best and underline second best).
Earthquakes
Covid NJ
Citybike
Pinwheel
SL(L)
MMD(C)
SL(()
MMD(1)
SL(J)
MMD(L)
SL(L)
MMD(J)
GAG-GAUS COx
0.047
0.214
0.209
0.340
0.104
0.336
0.017
0.285
REGULARIZZDD METHOD
2.361
0.391
0.255
0.411
0.097
0.342
0.039
0.411
POINT SET DIFFUSION
0.038
0.173
0,199
0.268
0.056
0.092
0.017
0.099
Table 2: Conditional generation results on the hold-out test set for SPP, averaged over three random seeds (bold best).
Earthquakes
Covid NJ
Citybike
Pinwheel
M
WD())
MAE())
WD(L)
MAE())
WD(L)
MAE())
WD(J)
REGULARIZED METHOD
30.419
0.162
16.075
0.148
7,740
0.115
3.547
0.150
POINT SET DIFFUSION
4.651
0.106
5.056
0.119
3.498
0.085
2.256
0.122
where X = { and Y = (mi)i= are two ordered samples from a point process on metric space of dimensionality d, i.e. _ IR. Further, U == (u1,.... U,z) represents the upper bounds of the metric space D along each dimension and we assume, without loss of generality, 2 k.
Wasserstein Distance (WD): An instance a Point Process itself a stochastic process points in space. Hence, we can compute a distance between two point sets based on the Wasserstein distance on the metric space C Rd between the two sets of points.
Maximum Mean Discrepancy (MMD) (Gretton et al., 2012): The kernel-based statistic test com- pares two distributions based on a distance metrie; we use the WD for SPPs and CD for STPP.
5.3 SPATIAL POINT PROCESSES
We evaluate our model's ability to capture the distribution of spa- tial point processes (SPP) by benchmarking it against two methods. The first is the widely used LOG-GAUSSIAN Cox PROCESS (Jes- per Moller, 1998), doubly stochastic model that parameterizes the intensity function using a Gaussian process. The second is the REG- ULARIZED METHOD (Osama al., 2019), leveraging a regularized criterion to infer predictive intensity intervals, offering out-of-sample prediction guarantces and enabling conditional generation.
Unconditional Generation (Density Estimation): In this experi- ment, we generate 1,000 unconditional samples from each model and compare their distribution to hold-out test set using the WD-SL SL and WD-MMD metrics. As shown in Table 1, our POINT SET DIFFUSION model consistently generates samples most closely matching the data distribution across all datasets. While the baseline models perform reasonably well in capturing the count distributions for most datasets, their reliance on spatial discretization and smoothness properties of the intensity function limit their ability capture the complex spatial patterns in the data, as reflected by higher WD-MMD scores.
Figure 5: SPP condi- tioning task: top ground truth, middle REGULAR- IZED METHOD and bottom POINT SET DIFFUSION.
Conditional Generation: To assess POINT SET DIFFUSION's ability to solve spatial conditioning tasks, we sample 50 random bounding boxes (with widths uniformly sampled between 1/8 and 3/8 of the metric space) for imputation on the hold-out test set, and report the results in Table 2. The REGULARIZED METHOD fits a spatial Pois- son model with out-of-sample accuracy guarantees and has been shown by Osama et al. (2019) to outperform the LOG-GAUSSIAN Cox PROCESS on interpolation and extrapolation tasks. However, find that the REGULARIZED METHOD'S 's reliance on predicting smooth and discretized inten- sity function conditioned on neighboring areas leads to inaccurate imputations when the adjacent regions contain significantly different numbers of points (see the hexagonal discretization structure
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Table 3: Density estimation results on the hold-out test set for STPP, averaged over three random
seeds (bold best and underline second best).
Earthquakes
Covid NJ
Citybike
Pinwheel
SL(()
MMD())
SL(J)
MMD(J)
SL(L)
MMD(L)
SL(J)
MMD())
DEEPSTPP
0.105
0.266
0.169
0.166
3.257
0.677
1.067
0.197
DIPESTP
0.088
0,064
0.332
0.146
0.560
0.611
0.196
0.055
AUTOSTPP
0.073
0.062
0.364
0.280
0.598
0.331
0.127
0.147
POINT SET DIFFUSION
0.042
0.023
0.189
0.043
0.032
0.020
0.023
0.020
Table 4: Forecasting results on the hold-out test set for STPP, averaged over three random seeds
(bold best and underline second best).
Earthquakes
Covid NJ
Citybike
Pinwheel
MAE(J)
CD(J)
MAE(.)
CD(L)
MAE(J)
CD(J)
MAE())
CD())
DEEPSTPP DIFFSTPP POINT SET DIFFUSION
10.154
11.211
6.264
8,492
127.968
125.747
18.651
15.792
16.027
17.466
18.822
14.302
7.516
8,460
14.461
13.062
7,407
10.458
7,293
10.865
5.928
7.225
6.341
6.437
and smoothness in Figure 5). This issue is exacerbated by not capturing a shared intensity func- tion across point sets, making it difficult for the REGULARIZED METHOD to handle non-smooth spatial patterns, such as varying inhomogeneous intensities shared across multiple point sets. This highlights a core limitation SPP models that rely on instance-specific intensity functions.
5.4 SPATIO-TEMPORAL POINT PROCESSES
For STPPs, we compare our model to three state-of oofte--ffthea-tt the-arrt STPP models, which parameterize an au- toregressive intensity function, to assess the model's ability to capture the point process distribution. DEEPSTPP (Zhou et al., 2022) uses a la- tent variable framework non-parametrically model the conditional intensity based on ker- nels. DIFFSTPP (Yuan al., 2023) based on diffusion model approximating the con- ditional intensity. Lastly, AUTOSTPP (Zhou
1000
PEI DIFFUSION
Sample runtime
DEEPSTPP
(seconds)
100
DifrSTPP
AUTOSTPP
& Yu, 2023) uses the automatic integration for neural point processes, presented by Lindell et al. (2021), to parameterize generalized spa- tiotemporal Hawkes model.
10
Sampling Runtime: We report the median
5
10
20
50
100
200
sampling runtime over ten runs generating ten point sets of length PCIE-40GB for all STPP models in Figure 6. POINT SET DIFFUSION maintains nearly con- stant runtime for all point set lengths as it generates all points in parallel, whereas autoregressive baselines, due their sequential sampling, exhibit a linear relationship between runtime and n.
Point set length (n)
NVIDIA A100-
Figure 6: STPP runtime for sampling n points.
Unconditional Generation (Density Estimation): We evaluate the performance of each model by comparing the WD-SL and CS-MMD between the hold-out test set and 1,000 samples generated by the trained models, as shown in Table 3. Again, the POINT SET DIFFUSION model best captures the distribution of the point process distribution for all datasets. The autoregressive intensity functions of the baseline models fail to generate point sets that align closely with the data distribution for most datasets, as reflected the differences in the WD-SI. and CD-MMD metrics compared to POINT SET DIFFUSION. While these baselines are trained to predict the next event given history window, they struggle to unconditionally sample realistic point sets when starting from an empty sequence. Consequently, this highlights argument that the standard evaluation based on NLL is insufficient to assess the true generative capacity of point process models.
Conditional Generation (Forecasting): Forecasting future events based on historical data is chal- lenging and a fundamental task for STPP models. To evaluate this capability, we uniformly sampled 50 random starting times from the interval [ Utime / ] where Unme is the maximum time, for

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Figure 7: Complex spatial conditioning tasks solved with POINT SET DIFFUSION: Top condition and ground truth data, bottom density plots for predictions.
each point set in the hold-out test set. The results are detailed in Table 4.2 The autoregressive base- lines, trained predict the next event based on history, achieve good forecasting results for most datasets, one even surpassing POINT SET DIFFUSION on the Covid NJ dataset. Still, our uncondi- tional model outperforms the autoregressive baselines across all other datasets.
5.5 OTHER CONDITIONING TASK
Since the STPP baselines autoregressive models, they are limited to forecasting tasks. However, our model can generate conditional samples for any conditioning mask C' on our metric space. To showcase this feature, present few visual examples of complex conditioning tasks in Figure 7.
CONCLUSION
To model general point processes metric spaces, present POINT SET DIFFUSION, novel diffusion-based latent variable model. We derive POINT SET DIFFUSION as a stochastic interpola- tion between data point sets and noise point sets governed by the thinning and superposition prop- erties of random point sets. Thereby, we attain a very flexible, unconditional Point Process model that be conditioned for arbitrary condition masks on the metric space and allows for efficient and parallel sampling of entire point sets without relying on the (conditional) intensity function. In conditional and unconditional experiments on synthetic and real-world SPP, TPP and STPP data, we demonstrate that POINT SET DIFFUSION achieves state-of-the ofte-----thhtrtt performance while allowing for up to orders of magnitude faster sampling.
We have introduced generative model for point processes on general metric spaces, prioritizing generality, scalability, and flexibility address key limitations of intensity- based models. While this enables unconditional modeling and flexible generation for arbitrary conditional tasks on any metric space, does not permit interpreting the conditional intensity or its parameters. Thus, for inference applications of STPPs or TPPs that require estimating the conditional intensity of the next event, point process models that directly approximate this conditional intensity are better suited. Ultimately, with POINT SET DIFFUSION, would be interested to see how future work explores its limitations on other (high-dimensional) metric (e.g., Riemannian manifolds), topological and discrete spaces with potential applications extending beyond traditional point sets including but not limited to natural language and graphs
have presented novel set modeling approach and
2 AutoSTPP is not included in this analysis due to its prohibitively slow sampling speed (see Figure 6 and the limitations discussed in Zhou & Yu (2023)), which made it impractical to sample the 50 forecast windows for all instances in the test set within reasonable timeframe.
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ACKNOWLEDGEMENT
This research was supported by the German Research Foundation, grant GU 1409/3-1.
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A APPENDIX
A.1 POINT PROCESS PROPERTIES
The thinning and superposition properties have been proved by other works for different versions of point processes. For completeness and generality, we prove them for general Borel set A. To apply these proof' for SPPs consider A C S, where a metric space in IR and for STPPs consider A S [0, 7 X S, where T > 0.
Superposition: Proof. It is straightforward to obtain the superposition expectation measure from Equation 1:
(AA) (IN = (A) (2)) [[N (1 (2(A). (13)
Then, every point process has intensity of X and X₂ for each of the expectation measures μl and P2, respectively. Therefore, taking the right-hand side of Equation 1, we obtain the following intensity function for the superposition of point processes:
((A) (1( + (2(A)= / (z(2)) + de(x)do== \1(x) + )2(x)dr.
(14)
This states that the density function for expectation measure == A + X2, and concludes the proof for the superposition property of intensities for point processes.
Thinning: Proof.
For this property, we need assume that the singletons are simple, so we can only have one point at each position: N ({+3)) < I; these point processes are called simple. Simple point processes can be represented as a sum of Dirac measures at the random points X, E S:
N = Σ -=>ox:
(15)
The previous assumption on singletons makes the sum above a finite sum If Zg E {0, 1] are Bernoulli random variables with success probability p we can define random thinning process as the superposition of the following point processes:
N1 Σ Z,0x.
(16)
No Σ (1 - (1=2,)
(17)
Since there are only two options for the Bernoulli random variable, holds that the superposition of the point processes defined in Equations 16 and 17 are equivalent the original point process, ie., N = Ni + N2.
Given that all Z; ~ Bern(p) are i..dd,, we obtain conditional probability distribution on the thinned point process N, (A)) n (inom p). And by the law of total expectation, we derive:
(A) = B (A) F. [ [E] = E[N(()p) = p(A)p.
(18)
We can write the terms of the equation before in terms of the intensity measure of the point process:
(A) ((A)
(xx)dz = pd(x)dr.
(19)
Hence, the equation above implies that the intensity of the new point process N1, which keeps the points of the original point process with probability p. pd.
By the property of superposition, since fore, the intensity of the point process N2, containing the thinned points, is (1 - >)
N₁ + No. then À = pd + (1- p)d. There-
This proves that, in the opposite case, when removing points with probability p from given point process with intensity X, the intensity of the point process with the points kept after thinning (1 X)
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A.2 APPROXIMATION OF MIXTURE DIRAC DELTA FUNCTIONS BY L C2-FUNCTIONS
Definition (Dirac delta function) Let (D, d. p) be a general metric space equipped with a mea- sure μ.. A Dirac delta function 6n at a point x E is defined as a distribution such that for any test function f:
J. f(y)
f(z).
(20)
Theorem Let (y) be finite mixture of Dirac deltas:
fM Σ wrdx. (z),
(21)
where 2I,...,2,, each Dirac delta function. Then, this finite mixture of Dirac deltas f'm can be approximated by L functions in (D, p.).
In E D are points in the metric space, and wi E IR are weights associated with
Proof. We use sequence of smooth functions that approximate each Dirac delta in the mixture and then show that this approximation converges in the norm..r.
Firstly, we show how to approximate Dirac delta functions. Let us consider a family of smooth functions &. (a) (such as bump functions mollifiers) that approximate the Dirac delta function 6 as + O. These functions Φe 2;) are supported near æ; and satisfy:
lim de(x xxi) 6n, (æ). c00
(22)
In particular, for any test function f, we have:
J f(u))e(u - xai--(()))))))))) + f(x;) as c c 0.
(23)
Hence, 6x 2;) has similar property the one of Dirac deltas given in Equation 20 and serves as an approximation of the Dirac delta 6z, (x) for small e.
Secondly, approximate the mixture of Dirac deltas far by a function in L²(D, (1) using the same &e ( -base approximation for each Dirac delta, defining:
f( Σ w,oh x.).
(24)
Each term o. - x,) is a smooth approximation of the corresponding Dirac delta 8z, (3). and the sum represents the approximation of the entire mixture of Diracs.
Thirdly, we show that the sequence fe converges to fM the n-2-oorm, i.e., that:
lim Il fMIlL>(DA)) = 0. c-0
(25)
Since fM is sum of Dirac deltas, is not directly in LD),,,4), but its approximation f. is because each de is a smooth function and smooth functions with compact support are L²),,
We compute now the squared 7,2 norm of the difference in Equation 25:
Ilfe fMlliA = (f((z)) fM)( du(y).
(26)
Note that the squared difference of fe and fas have quadratic and crossed terms. However, we can neglect the crossed terms: Eiey Wyyy SD (she (y ax)) - 8z, (y)) (oe - az)) - 8n, (v)) du(y), since every smooth function de(y - x;) is concentrated near x; and terms involving different indices do not contribute to the limit.
the above equation will
Hence, we can simplify the norm in Equation 26 into the sum of the individual terms:
" Ilfe - fM 12 = -
w (0. - x.) -c. -
(27)
1=1
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For every i, the term SD ( x.) ôæ; dy))) becomes small - 0, because by struction (y 2;) - 8n, (y) in the sense of distributions. Thus, by the properties of de, we conclude that:
lim Ilfe fMIlLP(D =0. e
(28)
Lemma Let p(a;; pr, Σ) be the probability density function (PDF) of multivariate Gaussian distribution. Then p E (R*).
Proof. The PDF of multivariate Gaussian distribution in I& with mean vector μ Rd and
covariance matrix E (which is positive definite) is given by:
(- 1)
11) 11)).
pla: n. Σ)
1 exp
(29)
(2
where E R Σ is the determinant of the covariance matrix E, and E is the inverse of the covariance matrix. We show that ||p|I = (Jzu [p(x)]2 dx) is finite.
We need compute the following integral:
p(x)2 dx
exp (-(x - (000-1))30 E 11) dx.
(30)
(2x)
Rd
Rd
To simplify the calculation, we perform a change of variables: y Σ (12 u)) Under this transformation: (æ: 11)² Σ
1 (æ: (u) y" y |/yl| and the differential da transforms as: dx = =/1// dy E1// dy. Substituting these into the integral, we get:
/ exp (-(x - 1T) I
x 1x 14) dx = |////2 [
expl(Ilbll's dy = n/2
(31)
since the remaining integral is a standard Gaussian integral. Thus, the 1 norm integral becomes:
[ p(z)² dx
1
1
/12/2 2/ 1/2
(32)
(2x)
Since the integral is a finite constant, we conclude that the PDF belongs to L (RR)).
Corollary Given an Euclidean space D C Idd a finite sum of Dirac deltas can be approximated with mixture of mallivariate Gaussian distributions:
PM (M = Σ wi - NN(:; p4, E.).
(33)
Proof. Note that we do not show that a mixture of multivariate Gaussian distributions is the best candidate to approximate a finite sum of Dirac deltas. However, note that multivariate Gaussian distribution is a standard approximation of a Dirac delta function and can, in the limit of small covariance matrix, i.e. | << approximate it.
The aim of this proof is to show that the mixture of Gaussians pM can be a candidate to approximate the Dirac deltas. From Theorem 1, this is equivalent showing that PM is L function.
To prove this, we need to integrate:
J. Pm(
Σ ? S N( E,) + ② waw; / N( E,W(a; Dj)da. 1=1
(34)
On the one hand, Lemma shows that the integrals of the first sum are finite constants. On the other hand, the integrals the second cannot be computed in closed form, but it is well known that the product decays exponentially as ||>||| - oo ensuring a finite integral. Therefore, the squared integral is just of finite constants and hence finite.
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A.3 SAMPLING ALGORITHM
Algorithm Sampling
1: Xr ~ X
2: for = .......1. do
3:
Xthin go Xxhin|Xee
4:
No So
5:
Xa = TXoo {X,) Xphin
6:
X,-1 q(x211 Xo, Xx)
7: end for
8: return X,-1
A.4 MODEL SETUP
The classifier to predict Xo n X, is MLP with 3 layers and ReLU as activation function. The mixture of multivariate Gaussian distribution that approximates Xn X, contains 16 components, and the parameters are learned with an MLP of layers and ReLU as an activation function.
Architecture:
Training: All models have been trained NVIDIA A100-PCIE-40GBB We Adam the optimizer and fixed weight decay of 0.0001 to avoid overfitting. To avoid exploding gradients, we clip the gradients have norm lower than 2.
Hyperparameters: We the same hyperparameters for all datasets and types of point processes. In hyperparameter study A.8, we have found 7' = 100 for our cosine noise schedule (Nichol et al., 2021) to give good trade off between sampling time and quality. Further, leverage a hidden dimension and embedding size of 32. For training, we use a batch size of 128 and a learning rate of 0.001.
Early stopping: We train the models up 5000 epochs with early stopping, sampling 100 se- quences from the model and comparing them to the validation split, with WD-SL metric for SPP and the CD-MMD metrie for STPPs.
A.5 EXPERIMENTAL RESULTS WITH STANDARD DEVIATIONS
Table 5: Density estimation results the hold-out test set for SPPs averaged over three random seeds.
Earthquakes
Ciybike
SL43)
MMD(
MMDD
Slr))
MMD(
S14D)
MMD(J)
Lug- Coinsstan Cox
D.RTI 0.014
0.141 0.004
0.00E 0.011
O.AGCE 0UD0
0.104 0.017
0.161 0014
0017
0004
0.233 0005
Regularized Method POINT SET DIEFUSiON
2.361 0.064
0.3913
0.004
0.2551
0.011
0.111 0.003
0.991
0.008
0.342 0.008
0.0991
0.033
0.411 0004
0.038.L 0.003
0.173L 0.004
0.1991 0.002
0.268L 0.016
0.056.L 0.020
0.0921 0.020
0.171 0.003
0.099 0.006
Table Conditional generation results on the hold-out test set for SPP averaged over three random seeds.
Earthquakes
Covid NJ
Citybüke
Pizwheel
MAE()
WD())
MAB(])
WD())
MAE())
WD())
MAB(])
WDX
SET
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Table 7: Density estimation results on the hold-out test set for STPP averaged over three random seeds.
Farthquakes
Covid NJ
Citbike
Pinwhocl
SL(()
MMD(J)
SL(J)
MMD(
SL(()
MMD(()
SL(()
MMD[]]
DeepS
1
DIISTPP
0.088.1 0.009
0.064 0.024
0.3321 0.012
0.146. 0.026
0 0.455
0.6111.011 0.113
0.19611 0.098
00851: 0.005
AutoSTPP
0.033- 0.07
0.0624
0.004
0.3644
0.040
0.280- 0.302
596-
0.047
0.351 0.099
0.127+ 0.004
0. 0.147-0000
POINT Stcr DIITUSION
0.042 0.033
0.025+ 0.003
0,189* 0.006
0.043 0.033
0.02+ 0.044
0,020 0.001
0.023+ 0.003
0.020= 0.001
Table 8: Forecasting results on the hold-out test set for STPP averaged over three random seeds.
Eanhggaakes
Cord
Cityhike
Pinwheel
MADIES
CDV2
MAD
CDC
MADED
CDC
CDL
DPepS
11.21 080.28
6.364 0.313
3,492
0.199
127968
21,29
125.707 1 25,75
15,01 E.EE
15,792 5.32
DRISTPP 1P9
16,027 28.803
17,466
83.788
18,02
63388
14.302
20.216
7.16 21 $1,953
8,00 $1.775
14461 14.800
13,002 83.90
POINT SKT DIFFUIION
7,407 +-0265
10,488 10.118
7,201 -0.02
10,865 10.130
5928 +2.581
7:325 +2.902
6,341 8:0. (08
6,437 #0.124
A.6 PERFORMANCE COMPARISON TO ADD- THIN ON THEIR TPP EXPERIMENTS
We compare POINT SET DIFFUSION to ADD-THIN (Lüdke et al., 2023) their TPP experi- ments. We use the same training and hyper-parameter parameter setup for our model as in the SPP and STPP experiments. For details on the experimental setup, please refer to section 5 of Lüdke et al. (2023).
A.6.1 DENSITY ESTIMATION
Table 9: MMD (4) between the TPP distribution of sampled sequences and hold-out test set (bold best).
Twitter
Yelp!
Yeln2
ADD-THIN
0.02
0.02
0.19
0303
0.02
0.10
0.03
0.01
0.02
0.04
0.04
0.08
0.04
POINT SET DIFFUSION
0.03
0.03
0,19
0.02
0.04
0.07
0.05
0.01
0.02
0.11
0.09
0.06
0.06
Wasserstein distance (L) between the distribution of the number of events of sampled sequences and hold-out test (bold best).
Table 10:
Hawkest 0.04
Hawkes2
PUBG
Reddit-C
Reddie-S
Twitter
Yelp!
Yelp2
THINTHIN
0.02
0.08
0.01
0.02
0.04
0.12
0.03
0.04
0.83
0.01
0.04
0.02
PoINT SET DIFFUSION
0.03
0.03
0.03
0.01
0.03
0.02
0.01
0,03
0.03
0.10
0.01
0.03
0,03
A.6.2 CONDITIONAL GENERATION FORECASTING
Wasserstein distance (1) between forecasted event sequence and ground truth reported for 50 random forecast windows on the test set (lower is better).
Table 11:
PUBG
Reddit-C
Reddit-S
Taxi
Twitter
Yelp1
Yelp2
Average Seq. Length
76.5
295.7
1129.0
98.4
14.9
30.5
55.2
ADD-THIN
17.18
21.32
2.42
1.48
1.00
1.54
POINT SET DIFFUSION
1.98
16.90
16.23
2.52
1.51
0.96
1.50
Table 12: Count MAPE 100% (J) between forecasted event sequences and ground truth reported for 50 random forecast windows on the test set (lower better).
PUBG
Reddit-C
Reddit-S
Taxi
Twitter
Yelp1
Yelp2
Average Seq. Length
76.5
295.7
1129.0
98.4
14.9
30.5
55.2
ADD-THIN
0.45
1.07
0.38
0.37
0.69
0.45
0.50
POINT SET DIFFUSION
0.44
1.13
0.26
0.41
0.60
0.46
0.47
18

---

Published as a conference paper at ICLR 2025
A.7
ADDITIONAL MATERIAL FOR COMPUTATIONAL COMPLEXITY OF STPP MODELS
Table 13: Number of learnable parameters per model.
DEEPSTPP
DIFFSTPP
AUTOSTPP
POINT SET DIFFUSION
450, 000
1, 600, 000
1, 000, 000
25, 000
Table 14: Training runtime in minutes averaged over three random seeds (all models have been trained on A100).
Earthquakes
Covid NJ
Citybike
Pinwheel
DEEPSTPP
32
28
71
17
DIFFSTPP
469
492
832
392
AUTOSTPP
99
156
523
36
POINT SET DIFFUSION
52
42
68
50
A.8 HYPERPARAMETER STUDY T
To provide insight into how the number of steps affects sample quality, we have run hyperparam- eter study for the unconditional STPP experiment the validation of the Earthquake dataset, evaluating T E {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, al- ready matching and even surpassing the quality observed at T" = 200. Although this result may seem counterintuitive those familiar with standard Gaussian diffusion models, it highlights key distinction of our approach: unlike Gaussian diffusion processes, our model employs inherently dis- crete Markov steps-specificallyy the superposition and thinning of point sets with fixed cardinality. As a result, only limited number of points can be added removed over 7 steps, imposing natural ceiling on how much additional steps can improve sample quality.
Table 15: STPP density estimation results on the Earthquake validation set for (20, 50, 100, 200) reported as the average and standard error over three random seeds.
20
50
100
200
SL SL (J)
0.018 + 0.002
0.017 + 0.002
0.014 = 0.002
0.015 + 0.001
MMD (L)
0.020 # 0.0015
0.020 + 0.0002
0.018 0.0012
0.018 = 0.0005
A.9
STPP FORECASTING DENSITY EVOLUTION
Figure 8: Evolution of two STPP forecasts of POINT SET DIFFUSION across time (0 + tmoz); ):
Density plot of forecast for a sliding window of f of the maximum time, black crosses represent history (conditioning), blue ground-truth future.
19


Reviews:
[{"review_id": "HLqXVnjxZa", "reviewer": "Reviewer_XRB4", "summary": "This paper proposes a novel modeling approach to point processes via diffusion on point sets (discrete events), addressing the reliance on the intensity function when establishing or learning the model. It can capture the distribution of point processes and generate a series of events based on noise point sets. Meanwhile, the sampling efficiency of point set diffusion is superior. The overall presentations of both the methodology and experiments are excellent.", "questions": "1. In Figure 5, can the authors show the predicted density of different trajectories in the same masked area?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The idea of using the diffusion-style model to characterize point processes is super interesting. The content is clear and well-written, making the methodology and the results accessible to the reader. The paper also covers unconditional and conditional sampling methods, which have the potential to correspond to two important questions in the point process modeling (first-order and second-order modeling). The authors also provide thorough experimentation to validate the effectiveness of the proposed model.", "weaknesses": "In my opinion, the main weakness, or the most improvement-needed part of the paper, lies in the modeling and experiments of ordered point processes:\n\n1. An important characteristic of the ordered point processes (TPPs or STPPs) is the dependence between future events and past events, which is not considered in the model. The proposed method seems to only consider the first-order statistics of the data (the event intensity/density), and treat these statistics at certain times or locations as fixed values to be learned by the model. For example, if the training data set contains multiple event trajectories sampled over the horizon of $[0, T]$, then the model will assume $p(T/2)$ (the event density at $T/2$) is fixed and to be learned. However, in TPPs, the $p(T/2)$ depends on the history (observation before $T/2$), and is different in each realization of the event trajectory, which violates the assumption of the diffusion model.\n\n2. Although a conditional sampling method is proposed in the paper (Algorithm 3.1), I am wondering about its effectiveness in practice. First, what the $q(X_{t-1}|X_{0}^{c})$ is (line 287) remains unknown. Meanwhile, the (technical/practical) reason for using this conditioning is not shown. The results in Figure 7 are not convincing enough. To me, even though the authors claim that they are solving conditioning tasks and are visualizing the predicted densities for events from different trajectories (panels at the bottom), these density plots would look similar if we overlap them with each other. In other words, I think the model only predicts an averaged event intensity over space, and it has little connection with the conditioned samples.\n\n3. An alternative to prove the effectiveness of conditional generation is to show the predicted intensity/density of events at different times, given a trajectory from the pinwheel dataset. This is the same idea as Figure 5 in [1]. The difference between density functions at different times is more significant and would help validate the conditional sampling method.\n\n4. The conditional sampling task is only experimented with in the spatial domain. An example of showing the evolution of the predicted conditional density of a pinwheel trajectory can support the claim of effective conditional sampling in an ordered (temporal) domain.\n\n5. I am also concerned that there is no log-likelihood metric reported in the paper. The metrics used in the paper are about the first-order characteristics of the data, on which I believe the proposed Point Set Diffusion can perform well. However, they cannot fully reflect the model's fit to the data when second-order data dependencies are involved (e.g., in ordered point processes). On the other hand, the log-likliehood is still the golden standard to suggest the model's goodness-of-fit to the data when it comes to conditional models or tasks [2][3]. Other point process studies that use the diffusion model will also report the data log-likelihood when evaluating the model [4][5]. I am curious about the proposed model's performance on the log-likelihood metric.\n\nAgain, I acknowledge and respect the authors' contribution to the proposed method, and I hope the above questions can be properly answered or addressed.\n\n---\n[1] Chen, Ricky TQ, Brandon Amos, and Maximilian Nickel. \"Neural Spatio-Temporal Point Processes.\" International Conference on Learning Representations.\n\n[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\n\n[3] Reinhart, Alex. \"A review of self-exciting spatio-temporal point processes and their applications.\" Statistical Science 33.3 (2018): 299-318.\n\n[4] Dong, Zheng, Zekai Fan, and Shixiang Zhu. \"Conditional Generative Modeling for High-dimensional Marked Temporal Point Processes.\" arXiv preprint arXiv:2305.12569 (2023).\n\n[5] Yuan, Yuan, et al. \"Spatio-temporal diffusion point processes.\" Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mUg2AqCMUL", "reviewer": "Reviewer_r6U8", "summary": "The paper proposes a Point Set Diffusion model for conditioned and unconditioned generations of point processes (spatial, temporal, and spacial-temporal) without intensity functions. The model treats the latent space of the point process as a whole and applies diffusion to learn how to generate point processes from noise (unconditioned) and conditioning masks (conditioned). At the training phase, the point process is passed through a forward process that gradually thins the original points and adds points from a noise point process. Then, a parameterized model is trained for the backward process that gradually predicts the points in the last timestep conditioned on the current timestep and thins the noise points in the current point process. After training, both conditioned and unconditioned sampling procedures are provided. Numerical experiments illustrate that the proposed Point Set Diffusion model achieves much faster sampling speed than intensity-based autoregressive models. Moreover, it outperforms several baseline autoregressive models on various SPP and STPP tasks, especially density estimation tasks.", "questions": "1. The sampling time and quality of the diffusion model are directly related to the number of forward/backward steps, which I could not find in the paper. Could the authors provide some ablation study on the number of steps, e.g., how the sampling time and quality grow with it?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is overall well-written and easy to follow. The basic concepts are introduced clearly with consistent notations. The forward process, backward process, and final sampling algorithms are well explained. Illustrations (Figure 1-3) are very clear for readers to follow the workflow of the proposed Point Set Diffusion model. The datasets and metrics are also clear in the experiment section. \n\n2. The idea of leveraging diffusion models to generate the whole point process is intriguing, and it is quite different from the common approaches that use autoregressive models with parameterized intensity functions that suffer from sampling speed and are restricted to forecasting tasks. Numerical results are very promising to support the efficacy of the proposed model.", "weaknesses": "1. Currently there are very few baseline algorithms, e.g., for SPP conditional generation there is only one baseline, and for STPP forecasting there are only two. It would be more convincing to compare with more baseline models, or to provide more evidence that the current baselines are already SOTA (which I believe they are).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WzjT2rmhGk", "reviewer": "Reviewer_mmd5", "summary": "This paper proposes a diffusion-based latent variable model for general point processes on metric spaces. By learning stochastic interpolations between data and noise point sets, the model enables efficient, parallel sampling and flexible generation for complex tasks on the metric space. Experiments on synthetic and real-world datasets show that the proposed model achieves state-of-the-art results in unconditional and conditional tasks for spatial and spatio-temporal point processes.", "questions": "1. In the experiments, how are $\\alpha_t$, $\\beta_t$, and $T$ set?\n\n2. The proposed model generalizes the Add-Thin model to general metric spaces. When modeling univariate TPPs, how does the performance of the proposed model compare to Add-Thin in both unconditional and conditional sampling scenarios?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper generalizes the Add-Thin model to define a model for point processes on general metric spaces, enhancing the model's applicability and promising future prospects.\n2. The idea is sound and well-founded. The paper is overall well-written and easy to follow.\n3. Experiments show that the proposed model achieves state-of-the-art results on both conditional and unconditional tasks while enabling faster sampling.", "weaknesses": "1. It would be helpful to discuss the connections between the proposed model and the Add-Thin model when modeling univariate temporal point processes.\n2. In the conditional sampling, the definition of $q(X_{t-1} | X_{0}^c)$ in line 287 was not provided.\n\nTypo: $X_{t+1}^{\\text{thin}}$ and $X_{t}^{\\text{thin}}$ in Eq.(9) should be $X_{t+1}^{\\varepsilon}$ and $X_{t}^{\\varepsilon}$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1K4ItEZHNl", "reviewer": "Reviewer_8FLx", "summary": "This paper proposes a novel diffusion-based approach to model point processes without relying on traditional intensity functions. This model is characterized by its ability to efficiently and flexibly generate point sets through stochastic interpolation between data and noise sets. Experiments on synthetic and real-world datasets demonstrate that the model achieves state-of-the-art performance in generating spatial and spatiotemporal point processes, significantly outperforming existing methods in terms of speed and accuracy of sample generation.", "questions": "- Can you add a discussion on learning time? How does the computational complexity increase, especially with more data points?\n- How can hyperparameters (e.g. number of diffusion steps T or noise scheduling) be determined?\n- Please tell me more about the limitation of the proposed method. For example, how robust is the proposed method in situations where there is little data? Also, will the interpretability of the proposed method be lower than parametric methods (e.g., DNN-based Hawkes processes), or will the number of sensitive hyperparameters increase by using diffusion models as a base?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The approach of modeling point processes using the diffusion model is interesting.\n- Efficient sampling is achieved by making effective use of thinning.\n- The effectiveness of the proposed method is evaluated on artificial and real data.\n- The manuscript is well-written.", "weaknesses": "- There is not enough discussion about computational complexity.\n- Not very clear on how to set hyperparameters.\n- No mention of the effectiveness of the method with respect to the amount of data.\n- No discussion of limitation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)