Point Process Flows
Authors: Nazanin Mehrasa
Ruizhi Deng
Mohamed Osama Ahmed
Bo Chang
Jiawei He
Thibaut Durand
Marcus Brubaker
Greg Mori
Venue:  (2020)
Keywords: Temporal Point Process
Intensity-free Point Process

Abstract: Event sequences can be modeled by temporal point processes (TPPs) to capture their asynchronous and probabilistic nature. We propose an intensity-free framework that directly models the point process as a non-parametric distribution by utilizing normalizing flows. This approach is capable of capturing highly complex temporal distributions and does not rely on restrictive parametric forms. Comparisons with state-of-the-art baseline models on both synthetic and challenging real-life datasets show that the proposed framework is effective at modeling the stochasticity of discrete event sequences. 

Under review as a conference paper at ICLR 2020
POINT PROCESS FLOws
Anonymous authors Paper under double-blind review
ABSTRACT
Event sequences can be modeled by temporal point processes (TPPs) to capture their asynchronous and probabilistic nature. We propose an intensity-free frame- work that directly models the point process as muo-parametric distribution by uti- lizing normalizing flows. This approach is capable of capturing highly complex temporal distributions and does not rely on restrictive parametric forms. Com- parisons with state-of-the-art baseline models on both synthetic and challenging real-life datasets show that the proposed framework is effective at modeling the stochasticity of discrete event sequences.
INTRODUCTION
Data in real-life takes various forms. Event sequences, as a special form of data, are discrete events in continuous time. This type of data is prevalent in broad spectrum of areas, for example, pa- tient visits to hospitals, user behavior on social media, credit card transactions, etc. In this setting, each event is discrete, and the temporal dynamics of the events are complex and asynchronous. It is crucial to understand the characteristics and dynamics of such data, so that plausible future pre- diction, well other downstream applications, such intervention recommendation, can be performed. Despite recent success in modeling images, videos and texts with the power of deep neural networks (DNNs), the asynchronous and probabilistic nature of event sequence data makes challenging to utilize the power of off-the-shelf DNN-based models.
Temporal point processes (TPPs; Daley & Vere-Jones 2007) 2007) provide us with an elegant and effective mathematical framework for modeling event sequences data. A temporal point process is defined as a stochastic process whose realizations consist of a list of events with their corresponding occurring times. These occurring times can either be real numbers from an index set (defined from prior knowledge) or sampled from an intensity function. While other time-series models learn temporal patterns synchronously (with each time-step being treated as an input to the model). Tap-based frameworks directly model the time intervals between events as random variables. With such a setup, it allows for modeling long sequences without vanishing gradients costly memory issues.
Although the temporal point process has shown to be useful in modeling events sequences, it is usually not trivial to come up with simple yet flexible intensity function. An intensity function encodes the rate an event occurs at a specified time-step. Poisson process (Kingman 1992) has been popular hand-crafted design for the intensity which assumes that events independent of each other. More sophisticated design choices are investigated in the self-exciting (Hawkes) 1971) and self-correcting process (Isham & Westcott 1979). The key contribution of these models is to find a functional form of intensity that fits data distribution well by making various parametric assump- tions on the underlying generative process of the data. Although shown effective in modeling simple synthetic datasets, parametric assumptions make such frameworks lack the flexibility to model the generative process for real-life and complex data, hindering wider adoption of TPP-based frame- works.
Recently, learning the intensity function using recurrent neural networks (RNNs) to encode the his- tory has received an increasing amount of attention (Du et al.) 2016 Zhong et al., 2018)) Mei & Eisner 2017 Jing & Smola 2017). In this line of work, history information is encoded and ex- ploited learning the intensity of the point process distribution. In this case, the explicit parametric assumption on the forms of the intensity functions is relaxed. However, the maximum likelihood training criteria on these models still requires the intensity function to be simple for the likelihood to be tractable. Recent work by MMehasa et al. (201)) proposes a probabilistic framework based on variational autoencoders for modeling point process, further facilitating the stochastic generative

---

Under review as a conference paper at ICLR 2020
process. All the literature above is built upon explicit modeling of a temporal point process using the intensity function.
It is not necessary to explicitly model the intensity: few works have tried to formulate TPP in an intensity-free manner. WGANTPP (Xiao et al., 2017 2018) introduce an intensity-free framework for modeling the point process distribution using Wasserstein distance. The model is built upon generative adversarial network (GAN). RLPP (Li et al. 2018) formulates this problem in rein- forcement learning framework and treats future event predictions as actions taken by an agent. Both of these models are optimized by trying to generate sequences of samples that are indistinguishable from the ground-trute sequences (by discriminator Although these models are capable of generating realistic sequences, such training criteria fail to model the data distribution, resulting in intractable likelihood.
WGANTPP and policy learning RLPP).
In this work, We propose novel intensity-free point process model based continuous normal- izing flow and variational autoencoders. The proposed point process flow (PPF) model utilizes a recurrent variational autoencoder to encode the history of a given event sequence and makes proba- bilistic predictions on the next event. It preserves the non-parametric characteristics point process distributions with normalizing flow. The predicted non-parametric point process distribution is ca- pable of capturing complex time distributions of arbitrary shape. leading to more accurate modeling of event sequences. Extensive experiments conducted synthetic and real datasets to evaluate the performance of the proposed model. Experimental results show that our model is capable of cap- turing complex point process distributions well as performing accurate stochastic forecast. The contributions are summarized as follows: (1) A novel intensity-free point process model built upon continuous normalizing flow. The proposed PPF is capable of capturing highly complex temporal distributions and does not rely on restrictive parametric forms; (2) PPF can be optimized by max- imizing the exact likelihood using change of variable formula, relaxing the constraint in previous works that likelihood has to be tractable; (3) Evaluation on both synthetic and challenging real-life point process models.
datasets shows improvement over the
PRELIMINARIES
2.1
TEMPORAL POINT PROCESS
A temporal point process (TPP: Daley Vere-Jones 2007) is mathematical framework for model- ing asynchronous sequences of actions. It is stochastic process whose realization is sequence of discrete events in time t1.m = {t1, t2, ..., tn where t,, is the time when the n event occurred.
A temporal point process distribution is modeled by specifying the probability density function of the time of the next event:
S(())))) x((1H)) exp {-f"" Mul(Hu du duy.
(1)
where the intensity function X(t)H)) is the conditional intensity function. It encodes the expected rate of event happening in small area around and H4 = (41, (2...,, 1 < 1) is the sequence of event times up to time t. Many works explored different design choices of intensity function to capture the phenomena of interest. Here we review two popular hand caadter design choices:
Poisson Process. Poisson process (Kingman 1992) is based the assumption that events happen independent of each other where the intensity is a fixed positive constant X(t) = X and X > 0. In more general case, could be function of time A(t(7t)) A(t) but still independent of other events, which is called inhomogeneous Poisson process.
Self-exciting Process (Hawkes process). Self-exciting process (Hawkes) 1971 assumes that oc- currence of an event increases the probability of other events happening in near future. Its intensity function has the functional form of X(t[HL) = μ + a Ei exp(-(t tt),, where μ and a are positive constants and t < f are all the events happening before time t.
2.2 NORMALIZING FLOW
Normalizing flows are generative models that allow both density estimation and sampling. They map simple distributions to complex ones using bijective functions. Specifically, if our interest is
2

---

Under review as a conference paper at ICLR 2020
to estimate the density function px of random vector X R then normalizing flows assume X = -Z), where ge I2 + R2 a bijective function, and Z E IR is a random vector with a tractable density function pz. We further denote the inverse of go by fo- On one hand, the probability density function can be evaluated using the change of variables formula:
[ 0x
(x(x)= pz(fo(x)) det
where f denotes the Jacobian matrix of fa. On the other hand, sampling from px can be done by first drawing sample from the simple distribution pz, and then apply the bijection x [z).
Given the expressive power of deep neural networks, is natural to construct go neural network. However, it requires the bijection ge to be invertible, and the determinant of the Jacobian matrix should be efficient compute. Several methods have been proposed along this research direction (Rezende & Mohamed] 2013 Dinh et al., 2014) 2017: Kingma et al., 2016; Kingma & Dhariwal, 2018 Papamakarios et al. 2017). An extensive overview of normalizing flow models is given by Kobyzev et al. (2019).
2.3 CONTINUOUS NORMALIZING FLOW
From a dynamical systems perspective, the residual network can be regarded as the discretization of an ordinary differential equation (ODE; Haber & Ruthotto 2017; Chang et al. 2018 Lu et al. 2018).
Inspired by that. Chen et al. (2018) propose neural ODE, where the continuous dynamics hidden units is parameterized using ordinary differentiaal equation specified by neural network:
(z) dt
h(2 tt),
(3)
The neural ODE can be used to construct continuous normalizing flow. The invertibility is naturally guaranteed by the theorem of the existence and uniqueness of the solution of the ODE. Furthermore, using the instantaneous change of variables formula, similar to [Equation 2 the log-density can be evaluated by solving the following ODE:
i Ologp(=(()) -T
(4)
Ot
(2(1)
Grathwohl al. (2019) propose an improved version of neural ODE, named FFJORD, which has lower computational cost by using an unbiased stochastic estimation of the trace of a matrix.
3 PROPOSED FRAMEWORK
We propose intensity-free flow framework to model the timing of events in point process quences. More specifically, we learn a non- non-parametric distribution over the timing of asynchronous event sequences by transforming simple base probability density through continuous normalizing flow, i..,, series of invertible transformations. With our proposed framework, we are able to model complex point process distributions without making any assumption on the functional form of the distribution while being able to evaluate the likelihood of sequences under our model.
3.1 POA-PRRAC POINT PROCESS FLOWS
Let the input be a sequence of asynchronous events {t,, 12, t..... where t; E R_ represents the starting time of the i-th event. We define the inter-arrival time Tn the time difference between the starting time of events f,-1 and f.. Our goal to model the distribution over inter-arrival time Tn given the past history of events inter-arrival times 7im-11 i.e., learning to model the conditional distribution p( -1)
Our approach is to construct the distribution over inter-arrival time 7. by transforming a simple base distribution through normalizing flow transformations. At time-step n. we assume that inter-arrival time Tn was generated by first sampling from simple distribution p(zn) and then transforming the drawn sample ", through an invertible transformation go IR + R+ parametrized by 0:
2n p(2n),
Ta = go (z.))
(5)
3

---

Under review as a conference paper at ICLR 2020
Pp (2,)
Pp (2n-11Frm-2)
Pp (ZulFia-1))
F Flow 80)
Flow ge
Plow go
Po)(11)
Por (Tn-1lRim-2)
Poo (FulFen-1)
Figure 1: This figure shows the overall structure of our non-parametric point process modeling via normalizing flow. We learn a non-parametric distribution over the inter-arrival time of asynchronous event sequences by transforming a base probability density which is conditioned on the past history through normalizing flow transformations.
With this assumption and the change of variable formula discussed in subsection 2.2 the distribution over inter-arrival time Tn as:
write
0
(T(T = p(zn)
(6)
θTn
where fO(TT) = 90 (Tx) Zn. and the scalar Jacobin value dfo/dT,, shows the changes in the density when moving from Tn to Zn. We dropped the determinant term in the change of variable formula, because in our case, the inter-arrival time Tn is a me-diimenioonal variable. With this formulation, we are able model the inter-arrin arrival time distribution, without any specific assumption on the functional form of the distribution. Exact samples of the inter-arrival time distribution can be obtained by sampling from the base distribution スn ~ p(zn) and transforming through flow transformation Tn ge(2z)). We are also able to compute the exact likelihood of Tn by computing ffa
the likelihood of fT( and multiplying it with the associated Jacobian term
8
Current formulation models the inter-arrive distribution of each time-step independent of past his- tory. The future event timing might depend previous events in very complex way, so its im- portant to take the history information into consideration while modeling future events. To capture this dependency, we adapt formulation to construct the point process distribution by learning normalizing flow parameters conditioned on the history. More specifically, we learn the parameters of flow base distribution by time-dependent model parametrized by p that encodes history and provides the conditional base distribution (2 T1:n-1) at each time-step. The overal procedure can be seen in Figure In framework, the base distribution is assumed follow Gaussian distribution:
Pp(Zn ((m=))Tmm-m))>>> -N (p.)
where (Mpp, d² are the parameters of the base distribution. These parameters can be obtained by encoding the history Ti:--1 using various approaches. In this work, we propose two approaches to construct the base distribution of the flow: 1) Base distribution with deterministic parameters. 2) Base distribution with stochastic parameters. In the following sections, we describe each approach in more details.
3.2 BASE DISTRIBUTION WITH DETERMINISTIC PARAMETERS
As discussed in subsection 3.1 we aim to model the conditional distribution p(T, (7) using history information learning the base distribution parameters. Recurrent neural networks (RNNs) have shown to be powerful deterministic models capturing temporal dependencies. Recent works adapt RNNs non-linear mapping of the history to the intensity function to define temporal point process distributions (Du et al. 2016 Zhong et al. 2018: Mei & Eisner (017). As first attempt, we use RNNs to learn the parameters of the base distribution of the flow using the history information.
Figure 2) part (a) illustrates the overall structure of our model. To construct the conditional distribu- tion p Tn T1:n-1). 1),, the RNN takes the history of the past inter-arrival times 7inn-1 and produces
4

---

Under review as a conference paper at ICLR 2020
(a) Deterministic Approach
Flow Module
Tim-1
R
Pp (Z 111m1)
9e
Po.p (1n(
(b) Probabilistic Approach
Flow Module
11m-1-
Encp
N (Hu Gyn)
zn =
Decp Ppp (Z
Po.p (Inl2
Figure 2: Part (a) shows the deterministic approach of utilizing RNNs for predicting conditional distribution (T.p(Ta(Tiss_1) RNN encodes history into the base distribution, then it gets transformed the target distribution by flow transformation go. Part (b) shows the generation phase of incorpo- rating the flow module in a probabilistic framework. During generation, the prior network gets the history and output the latent space distribution for the next time-step. Then a sample of this distri- bution is passed to the decoder which generates the non-parametric distribution over the inter-arrival time of next time-step by incorporating the flow module.
the conditional base distribution Pp(zn. (r(=)))Tmm11)= = = 0 o2.) for the next time-step. Then, the
base distribution is transformed into the conditional inter-arrival time distribution Tn through normalizing flow transformations ga- The RNN is jointly optimized with the flow module by maxi- mizing the logllikeld of observed sequence TI.N under the predicted distribution:
N
N
f 0r,
Co,p(TL.N)= log (T) =
log log
log
(8)
3.3 BASE DISTRIBUTION WITH PROBABILISTIC PARAMETERS
is known that there is a trade-off between the complexity of the bijective transformation and the form of base distribution (Jaini et al. 2019). With the complexity of the bijective transforma- tion fixed, a more flexible base distribution will lead to a more expressive model. In our proposed framework, the fact that flow transformations distribution across time-steps might vary a lot, makes our model more sensitive to the choice of base distribution family. We believe that, if choose model base distributions as Gaussian distributions with deterministic parameters, the bijective transformation might not be able to esti- mate underlying distributions well. We further support our claim by proving Proposition T which, intuitively speaking, says more flexible mase-distributio yields more expressive model.
shared across time-steps and the true underlying
Motivated by this, our second move have a more flexible distriiitribution where the parameters are probabilistic. In order to achieve this, we utilize the variational auto-encoder (VAE; Kingma & Welling 2014) paradigm in modeling the conditional base-distributions. To better illustrate the im- portance of having more flexible base-distee diss-distibutton, we provide motivating example in Appendix A
To avoid confusion, time-step n, we use the notation 2 for the random variable of the normalizing flow base distribution and zare to refer to the VAE latent space. We start by explaining the generation phase, i.e,, how the distributions over inter-arrival time T. are generated by stacking the normalizing flow module on top of the VAE backbone and then describing the training process.
Figure part (b) shows an overview of the generation process. Here, we adapt a recur- rent VAE framework consisting of time-varian prior network parametrized by ib which takes the history of past actions Tim--1 and provides the latent distribution Pap (2 [T Then, sample of this distribution is passed to the VAE's decoder which produces non-parametric distribution over the inter-arrival time Tn by first generating the normalizing flow base distribution Pp(2 | (ee)) and then transforming it through flow transformation ge. By applying the change of variable formula discussed in Equation 6) we can write the distribution over inter-arrival time as:
Generation,
Ofo
po | ..ae)
voe
(9)
irn
5

---

Under review as a conference paper at ICLR 2020
Training. At time-step n of training, the VAE module takes the sequence of inter-arrival times 7im
to approximate the true distribution over the latent space 2n via the help of the recurrent infer- ence network (>< (T:.)) which is parametrized with o. A time-dependent prior network is also adapted to help the model to take use of history information in generation phase pe (= [ [T:--1) Both prior and posterior distributions assumed to follow conditional multivariate Gaussian dis- tributions with diagonal covariance:
(= 11:--1) (o(=n"(/mmm1) = N(uu Dvn.;
(10)
(= ha [imn -N =
(11)
At each time-step during training, a latent code 2,1n is taken from the posterior and is passed to the decoder which aims to generate the conditional distribution (Tml(T The VAE backbone
is jointly trained with the flow module by optimizing the variational lower bound using the parameterization trick (Kingma & Welling 2014):
C (o.0w.
Σ n=1
(= [log p [m]]
(12)
Dri (2 (T (Tmm-1)))
where we compute the log-likelihooe term log pe,p (Tul22 by applying the change of variable formula of Equation 9
EVALUATION
To show the effectiveness of our non-parametric approach, we evaluate the performance of our model on synthetic and real-world datasets and compare it with the state-of-the-art point process models. Please refer to Appendix A for architecture and implementation details.
4.1 DATASETS AND BASELINES
Synthetic Datasets. We create three types of synthetic datasets as follow: (I) Inhomogeneous
Poisson Process (IP) defines the intensity as a function of time but independent of the history.
We simulate sequences of IP process with A(t) Σx i= a (2TTo)) - exp( ca)3 where
k = 6, a = (14, 18, 13, 17, 10, 13), c = (3, 6, 9. 12, 15, 18) and a = (5, 5, 5, 5).. (II) Self- exciting Process (SE) assumes that the occurrence of an event increases the probability of other events happening in the near future.
characterized by X(t) = 1A + 3 Ee g(t t,), where in our case g() = exp(-(), = 1.0, and 3 = 0.8. (II) IP + SE is created by combining the simulated data from the self-exciting process and the inhomogeneous process. For each of IP and SE, we generate 20000 sequences, where the length of each is 60. and split the sequences into train, validation and test sets with proportions of 0.7, 0.1, 0.2, respectively.
Real-world Datasets. We also evaluate our models on real datasets that cover the areas of social media, healthcare, and human activity as follow: (I) Linkedln data is collected from over 3000 Linkedln accounts and contains their job-hopping records with information including the time and company. Our model predicts the time-interval before user's next job-hopping. After pruning users with only one job-hopping record, we collect 2439 sequences. (II) MIMIC (Medical Information Mart for Intensive Care III: Johnson et al. 2016 Pollard 2016) is a publicly available, large-scale dataset which contains the medical records of more than 40000 anonymous patients. Our method models the inter-arrival time of patients' admissions to hospital We keep the record of patients who have at least three visits to hospitals and collected 2377 sequences. (III) Breakfast dataset (Kuehne et al. 2014) contains 1712 videos with 48 action classes related to breakfast preparation. On this dataset, model the inter-arrival times of the actions as well actions categories. We explain this extension in more detail in (subsection 4.2) For the LinkedIn and MIMIC dataset, we also split the dataset into train, validation and test sets with proportions of 0.7, 0.1. 0.2, respectively. For the Breakfast dataset, we use the standard train and test split proposed by Kuehne et al. (2014).
Baselines.
We compare our proposed flow- approach with the state-of-the-art point process models: (I) APP-LSTM is an LSTM that takes the history of past events and predicts the inter- arrival time distribution for the next time-step by mapping the history into the intensity of a point
TThis baseline has comparable performance to Mei Fisner 2017
Du et al. (2016).

---

Under review as a conference paper at ICLR 2020
Model
Dataset
APP-LSTM
PPF-D
APP-VAB
PPF-P
IP
-2942
1.857
> 0.408
0.499
SE
-2.990
-1.615
0.562
0.636
IP+SE
-2.978
-1.07
0.476
0.566
LinkedIn
-0.795
0.973
-1.7733
2.678
MIMIC
-1.962
-0.498
-1.200
1.696
Table 1: Log-likelihood Comparison. LL reported for synthetic and real datasets.
process distribution. We train it by maximizing the likelihood of observed sequences under the predicted distribution. This deterministic baseline has comparable performance to Du et al. (2016). only differs in the way that intensity is defined; unlike Du et al. (2016), its intensity doesn't I explicitly depend on time. Zhong et al. (2018) compare these two design choices, and implicit dependence was shown be more effective in modeling point process distribution.
(II) APP-VAE (Mchrasa et al. 2019) is latent variable framework for modeling marked temporal point process. The model makes conditional predictions by learning conditional latent space. Given a history of past actions, APP- VAE generates two distributions for the next action: one over its timing (by predicting the conditional intensity and using to define point process distributions) and one over its category. On breakfast dataset, we use their original setup with the use of both time and mark data to have a fair comparison with APP-VAEE using mark data could help better capturing the temporal dependencies. For the rest of the datasets, modify their approach to predict the time distribution only
4.2 RESULTS
Log-likelihood Comparison. We report log-likelihood (LL) of test sequences across all models. For our PPF model with the probabilistic approach in learning base distribution parameters (PPF-P; introduced in Isussection 3.3) and APP-VAE baseline, we report the importance weighted autoen- coder (IWAE) bound, which lower bound of the real lig-likelihood. To compute IWAE, each time-step, we draw 1500 samples from the VAE's posterior distribution and follow the standard procedure for computing IWAE. We report the average of log-likelihood along all the time-steps of all sequences in the test dataset. The experimental results are shown in Table The results indicate the better capability of our normalizing flow-based approaches at modeling point process sequence data, especially the real-world data with complicated underlying distributions. Our prob- abilistic PPF-P approach robustly outperforms state-of-the-ar intensity-base baselines across all the datasets. Without any assumption the functional form of intensity, are able model the point process distribution effectively. Our probabilistic approach also has better performance in comparison our deterministic approach introduced in |subsection 3.2 (PPF-D). This demonstrates the advantages of using a more flexible base distribution in the flow in the probabilistic approach.
Point Estimate Comparison. We also report the absolute error (MAE) evaluate the perfor- mance of our model in estimating future events timing. The MAE between the samples of predicted time distribution and the ground-truth is reported. To compute MAE for PPF-P and APP-VAE, at time-step i, we have two stages of sampling: (1) First, we draw samples from the prior distribu- tion 2 (=f pae m)) Then, we pass each to the decoder and (2) draw samples from each predicted distribution Ti ~ (TTl2))) The MAE computation at time-step is as follow:
Eager (x"""" ( (erpp(( (I7 -r-1)]
(13)
where 7." is the ground-truth inter-arrival at iime-step i. We follow a similar procedure for computing the MAE for the deterministic approaches:
Ere-pta ( -
(14)
For PPF-P and APP-VAE, to estimate the expected error, we draw 100 samples from prior dis- tribution P& vae TT: and 15 samples from each predicted base distribution pe 22 For
We drop the use of mark data input and accordingly omit the likelihood calculation of action category distribution from the optimization
7

---

Under review as a conference paper at ICLR 2020
Model
Dataset
APP-LSTM
PPF-D
APP-VAE
PPF-P
IP
6.765
4.7759
0.279
0.278
SE
7.360
4.4205
0.290
0.297
IP+SE
7.163
4.0525
0.288
0.299
Linkedln MIMIC
2.522
2.048
2.495
1.799
23.531
17.709
27.479
26.047
Table 2: Mean Absolute Error Comparison. MAE is reported for synthetic and real datasets.
Model
LL
MAE
Accuracy
APP-LSTM
-8.099
239.624
59.594
PPF-D
-7.637
251.337
61.174
APP-VAE
-6,463
244.019
62.190
PPF-P
-6342
204.913
62.528
Table 3: Comparison log-likelihood, MAE, and accuracy on Breakfast dataset.
PPF-D and APP-LSTM, we sample 1500 predictions from the output distributions p(T. (T+-1) at each step. For our PPF approaches (both deterministic and probabilistie), the corresponding samples of predicted inter-arrival time distribution are obtained using Equation 5) We report the average of MAE along all the time-steps of all sequences in the test dataset. [Talle 2] shows the experimental results for MAE metric. We can see that our PPF-P approach is comparable to the APP-VAE base- line on the synthetic datasets. On the more challenging real datasets, our PPF-based frameworks consistenlty outperforms baseline models (PPF-D on MIMIC, and PPF-P on LinkedIn). The better log likelihood estimations also conformed by lower/competitive MAE which reflects the better quality of generated samples from our PPF approaches.
Extension the Marked Temporal Point Process. On Breakfast dataset, for our model to learn a more powerful encoding of history information, we extend our approach to marked point process which models both the anter-arrival time distribution of future event and also the distribution over its category. At time-step i, given the history of past events including both time and mark infor- mation, in addition to modeling the time distribution of event at time-step + 1, we also model its category distribution. Here, we assume that action category follows multinomial distribution and accordingly, the log-likelinee likelihoood tive and evaluation criterion of our deterministic and probabilistic PPF models APP-LSTM is also extended to the marked case similar to Du et al. (2016). For the experiments on Breakfast dataset, in addition MAE, also report the accuracy of predicting the next action category. To compute accuracy at time-step i, for probabilistic approaches, we draw 100 samples from the prior distri- bution (2 yae [T: (1 and for each predicted category distribution, we select the action category with maximum probability as the predicted class. For each time-step, the most frequently predicted the model's prediction. Table shows the experimental results of comparing
action category distribution modeling is added to training objec-
type is reported log-like log-likelihood, MAE, and accuracy Breakfast dataset. This dataset is much more challenging in comparison to Linkedln and Mimic datasets, because of contains various types of actions. We see that probabilistic approach has better performance in all the metries which shows the effectiveness of our proposed model in capturing the underlying point process distribution.
CONCLUSION
In this paper, we propose Point Process Flows (PPF), an intensity-free framework that directly mod- els the point process as a non- non-parametric distribution by utilizing normalizing flows. The proposed model capable of capturing arbitrary complex time distributions as well as performing stochastic future prediction. The proposed PPF can be optimized by maximizing the likelihood using change of variable formula, relaxing the strict tractable likelihood constraint previous works. Extensive
For our determinisiic/proabiiii approach, assume that at each time-step given the history/latent- code, time and category independent.
8

---

Under review as a conference paper at ICLR 2020
evaluation both synthetic and challenging real-like datasets shows significant improvement over baseline models.
REFERENCES
Bo Chang, Lili Meng, Eldad Haber, Lars Ruthotto, David Begert, and Elliot Holtham. Reversible architectures for arbitrarily deep residual neural networks. In AAAI Conference Artificial In- telligence, 2018.
Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differ-
ential equations. In Advances in neural information processing systems, pp. 657 1-658, 2018.
Daryl Daley and David Vere-Jones. An introduction to the theory point processes: volume II: general theory and structure. Springer Science & Business Media, 2007.
Laurent Dinh, David Krueger, and Yoshua Bengio. NICE: Non-linear independent components
estimation. arXiv preprint 141.v: 16, 2014.
Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. In International Conference Learning Representations (ICLR), 2017.
Nan Du, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez-Rodriguez, and Le Song. Recurrent marked temporal point processes: Embedding event history to vector. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1555-1564. ACM, 2016.
Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, and David Duvenaud. Scalable reversible generative models with free-form continuous dynamics. In International Conference on Learning Representations, 2019. URL /ttp:/sss///opeeeeeeee..... net forum/ormmpp (it.rruunnn--rrxxxKCCCCCC.
Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks. Inverse Problems, 34(1):014004, 2017.
Alan G Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 1971.
Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural compulation, 9(8): 1735-1780, 1997.
Valerie Isham and Mark Westcott. self-correcting point process. Stochastic Processes and their Applications, 1979.
Priyank Jaini, Ivan Kobyzev, Marcus Brubaker, and Yaoliang Yu. Tails of triangular flows. arXiv preprint ar:iv: 19070048,,, 04481, 2019.
How Jing and Alexander J Smola. Neural survival recommender. In Proceedings of the Tenth ACM International Conference Web Search and Data Mining, pp. 515-524, ACM, 2017.
Alistair EW Johnson, Tom J Pollard, Lu Shen, H Lehman Li-wei, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G Mark. Mimic-iii, freely accessible critical care database. Scientific data, 3:160035, 2016.
Diederik Kingma and Jimmy Ba. Adam: method for stochastic optimization. In International
Conference on Machine Learning, 2015.
Diederik P Kingma and Max Welling. Auto-Encoding Variational Bayes. In International Confer- ence on Learning Representations (ICLR), 2014.
Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible Ix1 convolutions, In Advances in Neural Information Processing Systems, pp. 10215-10224, 2018.
Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Im- proved variational inference with inverse autoregressive flow. In Advances in neural information processing systems, pp. 4743-4751 2016.

---

Under review as a conference paper at ICLR 2020
J.F.C. Kingman, Poisson Processes. Oxford Studies in Probability. Clarendon Press, 1992. ISBN 9780191591242.
Ivan Kobyzev, Simon Prince, and Marcus A Brubaker. Normalizing flows: Introduction and ideas. arXiv preprint arXiv: 1908::900.09900 09257,, 2019.
Hilde Kuehne, Ali Arslan, and Thomas Serre. The Language of Actions: Recovering the Syntax and Semantics of Goal-Directed Human Activities. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2014.
Shuang Li. Shuai Xiao, Shixiang Zhu, Nan Du, Yao Xie, and Le Song. Learning temporal point processes via reinforcement learning. In Advances in Neural Information Processing Systems (NeurIPS). 2018.
Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In International Conference on Machine Learning. pp. 3282-3291, 2018.
Nazanin Mehrasa, Akash Abdu Jyothi, Thibaut Durand, Jiawei He, Leonid Sigal, and Greg Mori. A Variational Auto- -ncoder Model for Stochastic Point Processes. In IEEE Conference Com- puter Vision and Pattern Recognition (CVPR), 2019.
Hongyuan Mei and Jason Eisner. The Neural Hawkes Process: A Neurally Self-Modulating Multi- variate Point Process. In Advances in Neural Information Processing Systems (NeurIPS), 2017.
George Papamakarios, Theo Pavlakou, and lain Murray. Masked autoregressive flow for density estimation. In Advances in Neural Information Processing Systems, pp. 2338-2347, 2017.
Alistair EW Pollard, Tom J abd Johnson. The mimic-iii clinical database. http://dx.do doi
0rq/10.1300 13de100013022/2xxx223 2016
Danilo Rezende and Shakir Mohamed, Variational inference with normalizing flows. lional Conference on Machine Learning, pp. 1530-1538, 2015.
Interna-
Shuai Xiao, Mehrdad Farajtabar, Xiaojing Ye, Junchi Yan, Le Song, and Hongyuan Zha. Wasserstein learning of deep generative point process models. In Advances in Neural Information Processing Systems (NeurIPS), 2017.
Shuai Xiao, Hongteng Xu, Junchi Yan, Mehrdad Farajtabar, Xiaokang Yang. Le Song, and Hongyuan Zha. Learning conditional generative models for temporal point processes. In Thirty- Second AAAI Conference on Artificial Intelligence, 2018.
Y. Zhong, B. Xu. G.-T. Zhou, Bornn, and G. Mori. Time Perception Machine: Temporal Point Processes for the When, Where and What of Activity Prediction. In arXiv 1808.04063, 2018.
A APPENDIX
A.1 ARCHITECTURE
The overall architectures are illustrated in Figure 2 For the deterministic approach introduced in subsection 3.2] a long short-term memory (LSTM: Hochreiter & Schmidhuber 1997) network is used model the conditional distribution Pz(z.. Tim-1) (Mpp,, o/,,) ). The flow module go is where the derivative function h(.)) is modeled by
neural ODE model described in subsection 2.3
multilayer perceptron (MIP).
both the prior distribution Pa (2p (-im-1) 1) and the are modeled by LSTMs. The log-likelihood term log (o.p(Tn is computed in two steps. First, decoder network maps the latent variable 27 zae to a base distribution Pp (p(>l12200) pac), which also a normal distribution. The decoder network is a MLP that outputs the parameters of the base distribution. After that, the flow module ge generates the distribution the anter-arrival time po,p (T. deterministic approach.
For the probabilistic approach in subsection 3.3 approximate posterior distribution (..""
(T.| z00). The architecture of go is the same as in the
10

---

Under review as a conference paper at ICLR 2020
197-
010
01-
0.05
80
0
30
32
24
0
20
22
14
Probahii by
Probubility
pdt
6
19F.D
IFF
005
0.
360
30
12
14
22 14
(a)
(b)
Figure 3: Part (a) shows the results of modeling conditional distribution P(r, 11s-1 (4,
Part (b) shows the results modeling the conditional distribution P(r++1 (1 5 (4, 1) + (10, 1). The top/bottom figure of each sub-figure shows the generated samples by PPF-P/PPF-D. As we can see, PPF-D is not able handle this case very well becausethere could be no one-to-one transformation that can map two Gaussian base distributions exactly into a uni-modal and a multi- modal distribution respectively at the same time. However, PPF-P perform much better modelling the true underlying distributions.
A.2 IMPLEMENTATION DETAILS
For the LSTM cells, choose hidden size to be 128 for the synthetic data and Breakfast dataset, 64 for LinkedIn and MIMIC datasets. The dimension of the latent space of VAE models set to be 256 for Breakfast and synthetic datasets and 64 for LinkedIn and MIMIC For PPF-P model, the latent code was decoded into the mean and variance of the base distribution by two separate decoders, each with two hidden layers of size 256. For all continuous normalizing modules, one block of network with 3 hidden layers of 64 dimension. We use Adam optimizer (Kingma & Ba 2015) for all models with a learning rate of 0.001.
A.3 A MOTIVATING EXAMPLE FOR PPF-P
When using flow techniques for density estimation, the expressiveness of the model is not only limited by the complexity of normalizing flow transformations, but also by class of base- distributions. In our proposed framework, the fact that the flow transformations are shared across time-steps and the underlying distribution across time-steps might vary a lot makes our model more sensitive to the choice of the base-distribution family. By introducing latent variable z0ee such that the 2,, follows different Gaussian distributions conditioned on different samples of saae the distributions of 2 after marginalizing 200 becomes highly flexible.
To motivate this argument, we make the following proposition, show its proof and substantiate with experiment results on PPF-P and PPF-D models.
Proposition Let f IR + R be bijective singular mapping that satisfies the following: N Npo. and x = (z) follows mixture of Gaussian distribution of two components, N N(H1, of) and N (uz, oz), with weights and 1- a for some a E (0, 1). There exists E (1,2), such that if is sampled from component i of the Gaussian mixture distribution ~ N (11,,0)) (x) does not follow a Gaussian distribution.
In summary, Proposition says if normalizing flow can maps a Gaussian distribution to a mixture of Gaussian distributions, which is multi-modal, we can not obtain one of the mixture distribution's 's components by applying the same normalizing flow to another Gaussian distribution. Continuous normalizing flow defines continuous bijective mapping from R to IR and therefore it must be sin- gular increasing or decreasing. The normalizing flow we used is shared for all time steps and its derivative is independent of the parameters of the base distribution. It is also worth noting that this
11

---

Under review as a conference paper at ICLR 2020
proposition can be extended to scenarios of any Gaussian mixture distributions with finite compo- nents.
Proof. Without loss of generality, consider the following two cases:
Case 1 of > oz. We show that the inverse mapping of (Nn,, of) is not Gaussian by contradiction. Suppose f (x) follows a Gaussian Wistribution (us, oz). By our assumptions and the change of varaible theorems we have the following
Po (x)
(0(z) (x(
Ox dz
det
(The Jacobian is independent of the distribution.)
where
1
(z Ho)
pz(z)
exp
2TTO
20
(x - 11)2)
(x μ22
(x(
exp
+ (1 0)
exp
V
/2m
20
22
20
1
(z 22)
(0(z)
exp
V2T03
20
(x - 11 )
(x())
exp
20
Rewriting the equality ahove, we get
(x()
(z)
21 (x)
(z(z)
0 V2n07
exp
(x-H))² 20f
+ (1
a)
V2x03
exp
(x-N2))
201
LHS
(x-my)²
V2noz
exp
20
01
+ (1
a)
exp
x² +
cx +d
02
20
20
(2- )2
2
exp
20
RHS
(a (a=
exp
v 2
205
03 exp
(z 113)2
(z - -aa
203
20
=expee2 + fz + g)
for some constants c, d, c, f and g where one of and is non-zero. Since of o2. know 1 1 x² + cx d + -c and LHS- a > 0 as x - or -00. When x - 00 or -00, 207 20
we have z - Do or oo as well since f is bijective on R and singular. However, the RHS can only converge to 0 or diverge
when taking the limit of We get contradiction.
Case 2 of = =03. We also show that the inverse mapping of X ~ N of) is not a Gaussian by contradiction. Following similar steps in Case 1, we get
LHS a (1 a) a1 exp (cx + d)
T2
RHS exp(c2² fz 9
12

---

Under review as a conference paper at ICLR 2020
Model
1 LL
LL score
PPF-D
-2.022
0.331
PPF-P
-1.785
0.044
Table 4: Log-likelihood comparison of PPF-D and PPF-P. LL score represents the difference of log- likelihood under the true underlying distribution and log-likelihood under the learned model. Arrow (1)(L) shows higher/lower values are better.
for some constant c, d, f and where must be non-zero and at least one of and f is non-zero. Taking the limit of such that ex + oc, we have LHS a. By the bijectivity and singularity of f. we know z goes to either oo or -o. However, in either case, the RHS can only diverge to co or converge O. We get contradiction.
To complement Proposition| 1 generate sequential data with the following property train PPF- D and PPF-P models to fit the data: the underlying distribution values at each time-step switches between mixture of Gaussians distribution and one component of the mixture distribution. More specifically, the underlying distribution of observations at even time steps follows A 1) and the underlying distribution for odd time steps follow Gaussian mixture distribution of two components (44, 1) and N (110, 1) with equal weights. We created dataset of 1000 sequences where each sequence has the length of 15, and trained both PPF-D and PPF-P on this dataset.
shows the experimental results for this experiment. We can see that PPF-D is not able to handle data generated by this distribution very well. The output distribution of PPF-D tries to cover both components of the Gaussian mixture distribution, but most of the samples are concentrated in an area of low probability. In contrast, we can see that PPF-P, with more flexible base distribution, much better at modeling sequences sampled from our synthetic switching distribution. Most of the data sampled from PPF-P model lie the high-probability region: odd time step, the sampled data can hit both components of the Gaussian mixture model and at even time step, the sampled data can also recover the ground truth distribution. Table 4 shows the log-likelihove under the distribution learned from model the liglliiel under the true distribution. The better estimation of PPF-P is conformed by a higher log- ligeliilli We also reported the difference of log- lieeeood under the distribution and log-likelihood under the learned model. PPF-P has a lower score which shows it performs better in estimating the true underlying distribution.
Figure 3
test sequences
13


Reviews:
[{"review_id": "rJlg7oH2Kr", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a model for point processes using normalizing flows. The conditional distributions of next inter-arrival times are modeled as normalizing flows with base distributions conditioned on the history. Continuous-time flow based on neural ODE was employed. \n\nOverall I find this paper incremental. There have been several works using deep generative models to temporal data, and the proposed method is a simple combination of well-established existing works without problem-specific adaptation. \n\nI don’t get the point of using VAE in the likelihood estimation. The model (PPF-D) already defines a flexible conditional distribution. The only reason I can imagine to introduce VAE type model is when the dimension of the latent variables (which should be strictly the same as the observed variable) is too large to model directly. In such case one may choose to optimize a lower bound computed from the variational distribution defined over the lower-dimensional latent spaces. Hence in case of temporal point processes where the dimension is one, I see no point of doing VAE. The authors stated that VAE based model (PPF-P) performs better than ordinary flow model (PPF-D) because PPF-P has more flexible base distribution. My guess is that PPF-P has one more stochastic layer so the architecture itself is more expressive than PPF-D. PPF-D with more complex structure (e.g., more layers in base distribution parameter network) may result in a similar performance.\n\nThe authors stated in coclusion section that “The proposed PPF can be optimized by maximizing the exact likelihood”, which is not true for PPF-P optimizing the lower bound. ", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "SyeFu0fRqr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a new intensity-free model for temporal point processes based on continuous normalizing flows and VAEs. Intensity-free methods are an interesting alternative to standard approaches for TPPs and fit well into ICLR.\n\nThe paper is written well and is mostly good to follow (although it would be good to integrate Appendix A.1 into the main text). The paper proposes interesting ideas to learn non-parametric distributions over event sequences using CNFs and the initial experimental results are indeed promising. However, I found the presentation of the new framework and the associated contributions somewhat insufficient.\n\nThe proposed approach seems to consist mostly of applications of existing techniques and of only few technical contributions. There is also no real theoretical analysis of the advantages of the new approach beyond general statements. In addition, the experimental analysis is missing comparisons to\n- other intensity-free methods (e.g., [1, 2]) \n- other NeuralODE based methods (e.g, [3, 4]) \nand would also benefit from a closer analysis of the models advantages and/or additional tasks. While each of these points on its own would not be very severe, I found that the combination of all of them is problematic in the current version of the paper. I hope that the authors can address this in their response or future revision.\n\nFurther comments:\nThe results on Breakfast of the competing methods seem quite lower than the results published in (Mehrasa 2019). What is the cause for the differences here? For instance, APP-VAE in (Mehsara 2019) would outperform the results of PPF-P both in terms of LL and MAE (142.7 vs 204.9)?\n\n[1] Xiao et al: Wasserstein Learning of deep generative point process models, 2017. \n[2] Xiao et al: Learning conditional generative models of temporal points processes, 2018. \n[3] Chen et al: Neural Ordinary Differential Equations. \n[4] Jia et al: Neural Jump Stochastic Differential Equations", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "H1gYWzbatr", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a method for learning models for discrete events happening in continuous time by modelling the process as a temporal point process. Instead of learning the conditional intensity for the point process, as is usually the case, the authors instead propose an elegant method based on Normalizing Flows to directly learn the probability distribution of the next time step. To further increase the expressive power of the normalizing flow, they propose using a VAE to learn the underlying input to the \"Flow Module\". They show by means of extensive experiments on real as well as synthetic data that their approach is able to attain and often surpass state of the art predictive models which rely on parametric modelling of the intensity function. The writers have put their contributions in context well and the presentation of the paper itself is very clear.\n\nThough the final proof is in the pudding, and the addition of the VAE to model the base distribution yields promising results, the only justification for it in the paper is to create a more \"expressive\" model. There are multiple ways of increasing the expressiveness of the underlying distribution: moving from RNNs to GRU or LSTMs, increasing the hierarchical depth of the recurrence by stacking the layers, increasing the size of the hidden state, more layers before the output layer, etc. A convincing justification behind using a VAE for the task seems to be missing. Also, using the VAE for a predictive task is a little unusual.\n  \nAnother, relatively small point which the authors glance over is the matter of efficient training. The Neural Hawkes model suffers from slow training because of the inclusion of a sampling step in the likelihood calculation. I believe that since the model proposed by the authors allows easy back-propagation, their model ought to be easy and fast to train as well. Including the training time for the baselines, as well as the method proposed by the authors, will help settle the point. \n\nMinor point:\n\n - The extension of the method to Marked Temporal Point Processes in the Evaluation section seems out of place, esp. after setting up the expectation that the marks will not be modelled initially, up till footnote 2 on page 7.", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposed to use  normalizing flow to model point processes. However, the reviews find that the paper is incremental. There have been several works using deep generative models to temporal data, and the proposed method is a simple combination of well-established existing works without problem-specific adaptation. 
Decision: Reject