Variational Autoencoders for Highly Multivariate Spatial Point Processes Intensities
Authors: Baichuan Yuan
Xiaowei Wang
Jianxin Ma
Chang Zhou
Andrea L. Bertozzi
Hongxia Yang
Venue:  (2020)
Keywords: VAE
collaborative filtering
recommender systems
spatial point process

Abstract: Multivariate spatial point process models can describe heterotopic data over space. However, highly multivariate intensities are computationally challenging due to the curse of dimensionality. To bridge this gap, we introduce a declustering based hidden variable model that leads to an efficient inference procedure via a variational autoencoder (VAE). We also prove that this model is a generalization of the VAE-based model for collaborative filtering. This leads to an interesting application of spatial point process models to recommender systems. Experimental results show the method's utility on both synthetic data and real-world data sets.


Published as a conference paper at ICLR 2020
VARIATIONAL AUTOENCODERS FOR HIGHLY MULTI- VARIATE SPATIAL POINT PROCESSES INTENSITIES
Baichuan Yuan', Xiaowei Jiangin Ma?, Chang Zhou²,
Andrea L. Bertozzi', Hongxia Yang?
'Department of Mathematics, University of California, Los Angeles DAMO Academy, Alibaba Group
ybcmath@@maiin com, daemon.. ma fx13ffomthu@@eeeeoooooo gma com, ericzhou, wc@.zibabaabbbbbaaaaoaaaaaabbbaa com, bertozzi i @math. ucla edu, yang. whw.ayxxaallla
com,
yang.yhx0a
com
ABSTRACT
Multivariate spatial point process models can describe heterotopic data over space. However, highly multivariate intensities are computationally challenging due the curse of dimensionality, To bridge this gap, we introduce declustering based hidden variable model that leads to an efficient inference procedure via varia- tional autoencoder (VAE). We also prove that this model is generalization of the VAE-based model for collaborative filtering. This leads to an interesting ap- plication of spatial point process models recommender systems. Experimental results show the method's utility on both synthetic data and roal-world data sets.
INTRODUCTION
Multivariate point processes widely used model events of multiple types occurring in dimensional continuum. This paper focuses on multivariate spatial point processes (SPP), which can uncover hidden connections between subprocesses based the correlations of their spatial point patterns. Often we encounter missing data problems, where some subprocesses are not fully observable. The underlying connections could further contribute the prediction of these subpro- cesses over the unobserved areas. Mooreno-Müño et al. (2018) has shown the effectiveness of this joint model for Gaussian processes with heterotopic data. Multi-output models in Lian et al. (2015) such coregionalization and cokriging can outperform independent predictions. However, there is limited literature on the statistical methodology of the highly multivariate spatial point processes, according to the very recent paper (Choiruddin et al. 2019).
Inference for multivariate spatial point processes intensities still a challenging problem (Taylor et al. 2015), especially with large number of subprocesses. For popular Gaussian processes-basse approaches (Williams & Rasmussen, 2006). the multivariate intensity often consists independent and multi-output Gaussian processes. The complexity of the models and the curse of dimensional- ity hinder this approach for highly multivariate data, such as friendship networks and recommender systems with millions of users. In these problems, we only partially observe the events (e.g. users teract with items, locations) for each subprocess (user). It is necessary to jointly infer the preference of each user based on their hidden correlations. For example, a common approach recommender systems, collaborative filtering (He et al. 2017). predicts the item interests of each user with the help of the collection of item preferences for large number of users.
To address these problems, we propose multivariate spatial point process model with a nonparamet- ric intensity. We extend the well-known kernel estimator in Diggle (1885) to the multivariate case. This generalization is achieved through the introduction of hidden variables inspired by stochastic declustering (Zhuang et al.) 2002). The latent variables naturally lead to a variational Bayesian in- ference approach, which is different from the frequentist point estimation in the kernel estimator. To reduce the complexity in the highly multivariate case, we consider an alternative set of hidden vari- ables that are designed to work well as latent variables for variational autoencoder (VAE) (Kingmal & Welling 2014). This amortized inference (Gershman & Goodman 2014) approach leads to fast inference once the model is fully trained. Further, we show the equivalence for these two different

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settings of hidden variables using the properties of the spatial point process. This efficient approach makes it possible to apply multivariate spatial point processes in many areas, including location- based social networks and recommender systems with many users. Moreover, the nonparametric method for analyzing spatial point data patterns is not related to specific parametric families of models, which only requires the intensity be well-defined.
Our approach is not a direct replacement for current inference methods on few-variate spatial point processes (Jalilian et al., 2015). In contrast the classical methodology, VAE requires a large number of training data. The highly multivariate data that are widely available in social networks and recommender systems can be ideal applications for our approach. In fact, it can be shown that our model is a generalization of a tfte-of-theear the-hhr-attt art VAE-based collaborative filtering model (Liang jet al. 2018). Our model nonparametrically fits the underlying intensity function. Compared with the multinomial distribution used in Liang et al.] (2018), 8), this leads to not only a smoother intensity over space but also better predictions terms of ranking- -asedd losses. Compared to a univariate model, such the trans-Gaussian Cox processes (Williams & Rasmussen 2006), our multivariate model enhances the predictive ability on missing or unobserved areas, which is consistent with the results of heterogeneous multi-output Gaussian processes (Moreno- Muñoz al. 2018).
The contributions of this paper are three-folds We first build a novel multivariate spatial point pro- cess model and find direct connection with the VAE-base collaborative filtering through detailed theoretical analysis. Secondly, this connection introduces amortized inference for an efficient mul- tivariate point process estimation. Finally, point processes generalize the discrete distribution used in (Liang et al. 2018) and lead to a better modeling of spatial heterogeneity. We validate these benefits through experiments for multiple multivariate data sets, showing improvement over classic SPP methods and potentials on collaborative filtering applications.
PRELIMINARIES
Spatial point process A point process (PP) is a random counting measure Nzx) on a com- plete separable metric space R (here always assume that R C R") that takes values on {0, 2, Ufoo). While the major theory of point processes centers around the temporal dynam- ics, spatial point process models (Diggle al. 1983) are established in forestry and seismology, focusing on the stationary and isotropic case. We focus on the (first-order) intensity function which is the expected rate of the accumulation of points around a particular spatial location r. We
functtoon A(x).
write
E [[N(A |
Xxx)
lim
Ar
where Ar is small ball in the metric space, e.g. the Euclidean space R", with the centre and measure |Az The second-order intensity function is naturally defined as
E [N(A2)
A (I, 9) lim |
A |Ayl
measuring the chance of points co-occurring in both Ax and Ay. Normalizing this leads to the pair-correlation funcciiooonnnno g(,,, v) A12) /// (x)))()) g(x, y) > 1 indicates that points are more likely to form clusters than the simple Poisson process where g( v) = 1.
Common models in SPPs include the Poisson process with a non-stationary rate X(x), and the Cox process with nonnegative-value intensity process A(x), which is also a stochastic process. Cox processes conditional on a realization of the intensity process A(x) >(z) are Poisson processes with intensity A(x). To model the aggregated points patterns, Poisson cluster (Neyman- -ccmat)--oot) pro- cesses generate parent events from a Poisson process. Then each parent independently generates a random number of offsprings. The relative positions of these offsprings to the parent are distributed according to some f.d.f K2 (x)) in space (Diggle et al. 1983). Many point process models, including most Cox processes, are in fact Poisson cluster processes. The duality between Cox processes and cluster processes is widely used to construct Cox process models. For example, the kernel-basee intensity process A(z) Σe= Kx 24) with I; from Poisson process. is essentially a Poisson cluster process. The number of offsprings is from a Poisson distribution with À = 1 and the relative position distribution is K2 (2:). Repulsive SPPs, on the other hand, model that nearby points of the
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process tend to repel each other. Higher order intensities often considered in this case, such as determinantal PPs.
Alternatively, if terpretation, the Trans-Gaussian Cox process provides a tractable way to construct the Cox process using a nonlinear transformation a Gaussian process (1:). Popular choices for A(x) include the log- ogg-Gussiian Cox process (LGCP) with cxp(S(z:) and the permanental process with S(x)². Re- cent works on Cox processes have been extensively focused on the cases that modulated via the Gaussian random field, due to its capability in modeling the intensity and pair-correlations between subprocesses. In the next section, we develop a more explicit approach to model interactions for fast inference and the generalization ability for new subprocesses.
more interested in the realization intensity X(z) than the mechanical in-
Inference for Point Processes Inference methods for point processes are mainly based on the or- der statistics or likelihood function. The order statistics are often estimated nonparametricall,, such as the kernel estimator (Diggle, 1985) of the intensity function. For the likelihood based inference, we assume that one observes events X { of the underlying spatial point process over the area R. The log-likelihood for the inhomogeneous Poisson process over space is
N
logpl pl ( = Σ log(A(a;)) - A(x)dx.
The integration term is the log void probability and can be viewed as a normalization term for the likelihood. For Cox processes, the likelihood is the expectation over the Poisson likelihood. It is difficult to directly integrate over the distribution of A. Monte Carlo methods (Adams et al. 2009) commonly used to approximate the expectation. To improve the scalability of the expensive sampling. many methods such as variational inference (Lloyd et al., 2015). Laplace approximation
(Williams Rasmussen 2006) and reproducing kernel Hilbert spaces (Flaxman al. 2017) are proposed.
Variational Autoencoder As stochastic variational inference algorithm, VAE Kingma & Welling 2014) is maximizing the evidence lower bound (ELBO) of the log- likeliieldoood function
logp(X(O) 2
K (= p()))
The hidden variables have simple multivariate Gaussian prior p(z) (2; 0). The true pos- terior, which is often intractable as in the Cox process, is approximated via a multivariate Gaussian (x)xXX) (2; n (X), a (X)) The KI divergence term in the ELBO can be calculated analyti- cally. VAE uses multilayer perceptron (MLP) to learn the mean and variance of the approximated posterior directly from the data. The most related work here is recent VAE-based model for col- laborative filtering (VAE-CF) (Liang et al. 2018). They assume that each user is a multinomial distribution over items with the log-likelihood log Po (Xxl==)) > Σi=
Xiv log T((=u) ) for each user u. Here X,, is the observed data of user clicking items, 7, (zu) is the probability that user u clicks the item and Xau. is indicator function whether the user clicked the item i.
3 MULTIVARIATE SPATIAL POINT PROCESSES
Here we consider a multivariate case of the SPP, with U interdependent univariate point processes on the sample space R. The intensity function is measured in similar way as the univariate via dr (x) = lim Jo (E [N, (Ar]] Axl), where (A())) is the number of events within a set Ax for the subprocess u.
3.1 A NONPARAMETRIC MODEL
The observed data of multivariate SPP include the location of N., events X, = [., associated with each subprocess u. For each u, the observed event locations follow a Poisson process with spa- tial intensity A₂ (x), which is a realization of the random intensity A, (z). Using the nonparameteric kernel estimator, the intensity of the subprocess is estimated by
No.
>m(x) =
K2 - 2x
(5)
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Here (x) is kemel function and we usually adopt the radial basis function kernel (RBF) where We ignore the end-correction (Diggle 1985) for now.
Ko (x)
In real-world applications. however, one often encounters the missing data problem, where not directly observe points in certain areas for some subprocesses. Instead, we seek to infer the hidden data from other fully observed subprocesses. In our model, assume that each subpro- cess reflects the stochastic and heterogeneous patterns. For example, users an e-commerce plat- form usually prefer different categories. As Poisson cluster processes, this naturally leads to events clustering in specific areas. Another real-world example is the aggregation of check-in ac- tivities around the home and workplaces for social network users (Cho et al. 2011). Note that
N = E"=,NN, is the total number of events. We introduce hidden variables Y, for each event T; I, N and subprocess
I,..,, U, where Y;" 1 if the subprocess includes event and y = otherwise. EY,;" = P3 is the probability that event v; is from the subprocess u. Then the intensity process for multivariate SPP model is
N Σ Ko --:),
(x(x)
(6)
for each subprocess This model generalizes the kernel density-based intensity to the missing data case. Similarly to the original method, can applied to estimate the intensity for both cluster processes such Cox processes and repulsive ones like determinantal PPs. In order to incorporate prior information and model the data uncertainty, we adopt a variational inference approach for the hidden variables.
3.2 VARIATIONAL INFERENCE
major drawback of current inference methods for SPP is the introduction of large number of parameters in the highly multivariate case. For our model, we use amortized inference approach
VAE (Kingma & Welling 2014) avoid the computational complexity of directly estimating the posterior for each subprocess
The generative process of our model can be described as follows: For each subprocess u, it has K K-dimmnnionnl hidden variable 2u with multivariate normal prior zu (0, Lg). Here use a low-dimensional representation and then nonlinear mapping fo fz(z)) = {pp Jiml transforms 2u so that it has the same dimension the number of events N. Finally the spatial points of the subprocess u are sampled according the intensity (² = Σ Ko (x-x,), We approximate the intractable posterior distribution of (x([X) with a multivariate Gaussian N (M (X) (x)). As in Liang et al. (2018), we use MIPs learn the nonlinear function fo(z) with parameters 0 and the mean and variance with parameters Φb The variational bound of our multivariate Cox process model is then
1N
logp(X.,10) 2 (x. (po Xx|zz)) (L (x))p((z)) L.
The first term in L is essentially a complete likelihood function. For each subprocess u, it has the following (expected) intensity function
N p K x
Ix.(e.lx.)AA10) = Σ
(8)
and Poisson process log-likelihood function from and
No
No p (x
No
(eo(ss)X..) X.)) Log X1, (ulzu) =
log
Σ 1=1
rp K K2 (x -xx") )d..
(9)
For applications without explicit spatial information, we embed each event into latent space vector. First, we obtain a similarity graph for all events. Then the embedding 2; of it., event in this graph is obtained via graph neural networks (GNNs) such GraphSAGE (Hamilton et al. 2017). See Figure for an illustration of our framework. Both ø; and 2, are learned jointly combining the information of item embedding and user hidden variable.
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Events Embedding
Koft,
(Optional)
Σ(X)
Z
f(z)
A(-1z)
SPP
μ(X)
X
Poisson Likelihood
KL
Figure I: Visual illustration of our spatial point process model via VAE during the training.
3.3 ALTERNATIVE MODEL
Recall that the hidden variables Y describe whether the event z, is from the subprocess u. By definition, we have Eum Y.u - and E 1 =1 for any i. During the training process, is dif- ficult to normalize the probability p over all subprocesses (have use the full data). Moreover, this constraint leads to Due < for и * o, implying mutual-inhibitioon behaviors between subprocesses. Instead, we consider an alternative model where p the probability that the subprocess generates an event 2; i.e. N. p. = 1 for each u. During the training, the total number of events N, is not viewed as hidden variable for each subprocess. Thus the alternative model essentially normalizes Au by a constant. With the reparameterization trick in Kingma & Welling (2014), we sample the log-likelihood function using all events within mini user batch and compute the gradient. This approach incorporates all information about the user so that negative sampling is not needed. See Algorithm | for our training procedure. For the model prediction, the normalized intensity of new subprocess can be efficiently calculated in O(N) using the approximated posterior Ro(2|Xneuu and nonlinear function fo(2) with parameters 0, inferred from data. We can further reduce the computational challenge (Liang et al. 2018) for large N due to fz(2 by discretizing the space.
Now we show equivalence of our multivariate model and the alternative one. There are two probabilities to consider. The first one is the conditional probability of observed events X,, in the subprocess u with an intensity function (z). given that there are N₂ events within the metric space R. The second one is the probability of sample X,, of size N, from the normalized density (x) >u(x)/ JR (a(s)ds. For general SPPs data, we have
Theorem 1. A spatial point process on a measurable set R C.C. B with an intensity function (u(x) equivalent to Na iidd samples within with p.d.f hi hu(()) (x) = >u(x)/ JR (u(s)ds, given we know N₂ = (s)dd) which the number of points within R for the point process model.
Proof. See Section B in Appendix.
According to this theorem (see supplementary material), we can replace the log- likelhhodd function 9 in the ELBO with
No E (x., log po (oo = Σ log hn (x + C.
(10)
Here C' is related to the log-likelihood on the number of events N.. which is constant because J Au(s)ds = N₂, is observed. One drawback of this approach is that, for the prediction actual missing data, cannot infer the number of missing points. Instead, our VAE-based model gener- ates the normalized intensity predicting the possible locations for the missing events. We use the alternative definition of pf from now on.
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This result shows that VAE-CF is special of this multivariate SPP model over discrete space X of events. In fact, VAE-CF is the altermative model with a delta function as the kernel (2,, (zz)/ Jx (s)ds p)), which is equivalent to the SPP model according to the theorem. To better model the spatial heterogeneity of events, one can replace the delta function with other kernels
more advanced SPP intensities. We simply RBF kernel here, resulting in exp(1 /
(x) 1i=1 P
Algorithm 1: Training VAE SPP with stochastic gradient descent.
Input: Traning subprocesses
Ur with their point locations X,
Result: Parameters 0 and ø Initialize and randomly; while not converged do Sample a subprocesses batch Ub from Ur and their points X, = UnEU, Xu:
forall u E Ub do
Sample zn (N(u (x..)) with reparameterization trick; Compute fa(za) = (p" Jæ,Ex;
forall X, do
Compute sampled normalized intensity ha(x)) ~ Ex, Ko (x- xz);;
end
Compute noisy gradients of the ELBO L w.r.t 0 and
end
Average noisy gradients over batch; Update 9 and with the Adam optimizer Kingma & Ba 2015 end
One benefit of this alternative model is its resulted consistency. The nonparametric kernel estimation for the point process intensity is unbiased. see this, for any measurable set R, we take the expectation of the estimated intensity A(x) over the Poisson point process distribution
f A(x)dox /
- r;)dæ -Ka) JaJe Ko y)ply)dyds = - ply)dy,
E
(11)
where p(y) is the true intensity function. Then EA(x) = p(x) under mild conditions, egg., a spa- tially continuous assumption on p. But it inconsistent due the non-vanishing variance without normalization. For our alternative model, the normalized intensity function (x) is still unbiased. And according the standard theory of the multivariate kernel density estimation (KDE), the con- sistency of h., (I) is also guaranteed. Another benefit of using this alternative form can be seen from the cross pair-correlation function. For the alternative model, we remove the undesirable restriction of negative correlations between all users (guv < for u * v) and can incorporate more diverse relationships between users. To see this, we first consider the auto and crass pair-correlation func- tion Juv AA, LEA,,,, For our original model, it is straightforward to prove that Juu > and Juv < 1, u * v (see supplementary material). The auto pair-correlation functions show that our model is more aggregate than the simple Poisson process.
4 EXPERIMENTS
We compare our model (with the RBF kernel, VAE-SPP) with both VAE-CF LLiagg et al. 2018) and univariate spatial point process models using standard KDE (Diggle 1985 or TGCP (Williams & Rasmussen 2006) as intensity functions. We adopt the experiment setting in VAE-CF. We split the data into training, validation and testing sets. For the multivariate model, the training data used to learn the parameters f,, é. For KDE and TGCP models, we omit the training data because different subprocesses are assumed to be independent and also because of the computational complexity of fitting highly multivariate TGCP. We assume that only 80% of the events in the validation and test sets are observed. The remaining 20% are viewed as missing data to be inferred by different models. Hyperparameters selected on the validation data as in Liang et al. (2018). Finally, compare the prediction performance of different models on the missing data given the partially-observe events. We standard ranking losses such as NDCG@ K and Recall @ K defined in Appendix D.

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4.1 MULTIVARIATE SPP ON SPATIAL DATA
Synthetic data sets We simulate two different data sets using multiexponential and multisine models. For the multiexponential data set, we simulate 5,000 Poisson processes with Ax(2) age=b22 k = 1, .... 5000, E [0,30] training data. Here Oę and by are uniformly sampled be- tween [5, 10] and [0.1, 0.2) separately. 500 validation and 500 test subprocesses are generated the same way with parameters sampled from dg and bg. The multisine data set is generated via replac- ing the intensity function with d (4)) = max(aş (in(b)xz), 5) and sampling ag and by uniformly between [5, 10] and [1, 2] separately. Each realization of the spatial point process is discretized using a uniform grid over x with grid spacing 0.01.
Table I: Testing results on the simulation data sets. Both the mean and variance are percentages
(same below).
Multicxp
Multisine
Name
NDCG@ 100
Recall 50
Recall 100
NDCG@ 100
Recall @50
Recal 100
VAE-CF
6.78(0.28)
7.25(0.40)
14.5(0.52)
3.30(0.15)
2.49(0.13)
4.64(0.18)
VAE-SPP
7.11(0.31)
7.34(0.40)
14.9(0.54)
3,53(0.15)
2.58(0.13)
4.90(0.18)
KDE
5.27(0.15)
5,85(0.12)
11.8(0.1)
3.23(0. 15)
2.29(0.12)
4.55(0.27)
TCGP
3.11(0.14)
3.32(0.11)
6.44(0.11)
3.77(0. 14)
1.88(0.11)
3,92(0.17)
Location-based based Social Network. We consider the Gowalla data set (Cho et al. 2011 in New York City (NYC) and California (CA). We use a bounding box of 1-2244099, 32.5343, 1184,1308, 42.0095 for CA and -74.0479, 40.6829, -73.9067, 40.8820 for NYC (both from flick() Each user with at least 20 events (check-ins) is viewed as subprocess. There are 673,183 events and 6,728 users for Gowalla-CA. We randomly select 500 users the validation set and 500 users the testing set. We use the remaining users for training. For Gowalla NYC, there are 86,703 703 events from 1.171 users. We the size of both validation and testing sets to 100. For the spatial tessellation, we use uniform grids (32 x 32 for NYC and 64 x 64 for CA). Both our model and VAE-CF can work without grids. We further compare the performance of our model with VAE-CF by viewing each location as an item.
Table Testing results on the Gowalla data sets with uniform grids.
CA
NYC
Name
NDCG@I0K
Recall@50
Recall @ 100
NDCG@ 100
RecalI@50
Recall 100
VAE-CF
41.8(1.5)
64.8(2.0)
70.0(2.0)
43.6(2.3)
73.9(2.9)
86.2(2.2)
VAE-SPP
42.3(1.5)
65.2(2.0)
70.2(1.9)
44.8(2.4)
74.5(2.9)
86,2(2.2)
KDE
34.5(1.5)
59.2(2.0)
64.0(2.0)
41.2(1.5)
69.9(2.0)
83.6(2.0)
TCGP
31.8(1.3)
56.5(2.0)
60.9(2.0)
37.3(2.3)
59.9(3.3)
75.9(2.8)
Table 3: Testing results on the Gowalla data sets without discretization.
CA
NYC
Name NDCG@ 100
Recall @
Recall 100
NDCG@ 100
Recall @00
Recall 100
VAE-CF 21.3(..77)
16.6(0.74) 74)
32.8(0.97)
16.0C1.7)
13.2(1.7)
26.3(2.4)
VAE-SPP 21.6(0.77)
17.0(0.80)
33.5(0.76)
16.1(1.7)
13.7(1.8)
27.1(2.5)
In Table ! we summarize the performance of both multivariate and univariate models on the sim- ulation data sets. It is clear that the multivariate models outperform the univariate ones. Moreover, testing on multivariate models takes less time because it only evaluates the posterior probability and intensity function. This illustrates the power of multivariate models using amortized inference. Within the multivariate models, our continuous model further improves upon the discrete VAE-CF. This is due to the fact that these simulation intensities are continuous over R. For real-world appli- cations, the results on the location-based social network prediction and recommendation with and without grids are presented in Table and We observe the same pattern in both NYC and CA.
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Estimated density functions for Gowalla user in NYC (log scale). The first from left to right: observed check-in locations (in red), held-out check-in locations (in blue, as missing data) and the estimated intensity from VAE-SPP. The second from left right: the estimated intensity (or density) from VAE-CE, KDE and TGCP.
Figure 2:
We stop using univariate models from now on due to their inferior performances, especially for col- laborative filtering applications. Moreover, our model improves discrete VAE-CF regardless of the choice of spatial grids. For visualization purposes, in Figure we plot a user's check-in locations in Gowalla-NYC and intensities estimated via different methods. Comparing with VAE-CF, model generates a continuous intensity. The univariate models overfit the training data and lead to inferior predictions of the missing data.
4.2 MULTIVARIATE SPP WITH A LATENT SPACE
MovieLens data (ML-100K and ML-IM) include the movie (item) rating by users and binarize the rating with threshold of 4. In the spatial point process setting, we view each user as a subprocess over the latent space of item embeddings. Here the item embedding is generated via a GNN. This framework is natural generalization of the multimodal distribution over items. The item-item graph constructed based on item-item similarities. We use the Jaccard distance to measure the similarities between items, which are further viewed as the sampling probabilities for GNN. Currently, we only consider I-hop connections. Both GNN and VAE are trained jointly, which is more expensive than VAE-CF but leads to better performance compared to separate training (see Appendix D). For movie recommendation tasks, we compare the discrete VAE- EF to our joint model with GNN. The results in Table4shov again the improvement of our model over the baseline.
Table 4: Testing results on the MovieLens data sets.
MLI00K
MLIM
Name
NDCG 100
Recall@220
Recal 100
NDCG@ 100
2ccall@220
Recal @ T00
VAE-CF
40.8(2.8)
32.3(2.8)
57.6(3.3)
41.6(0.76)
33.1(0.81)
56.8(0.88)
VAE-SPP
41.5(2.9)
31.3(2.7)
59.0(3.5)
42.3(0.77)
33.9(0.82)
57.6(0.88)
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5 CONCLUSION
In this paper, we introduce novel spatial point process model for efficient inference on the highly multivariate case. Through amortized inference, our model makes possible investigate correla- tions between myriad of point patterns based on a large number of training data, and the theoretical analysis on the density and intensity for SPPs builds the connection between our model and VAE- CF. There are many promising directions of future works including the extension for multivariate spatiotemporal PPs (Mohler al. 2011] Yuan al. 2019) and using features as covariances. There multiple ways to estimate the mcan rate (h x)) of spatial point process overall events, includ- ing Gaussian mixture models, Gaussian processes and flow-based models. For future work, we can investigate the connections between model and other density-based estimations for point pro- cesses. Another interesting application is to handle real-world recommender systems via improving the joint training efficiency and comparing thoroughly with simpler algorithms as in Dacrema et al. 2019).
ACKNOWLEDGMENTS
We would like to thank the comments from Frederic P. Schoenberg and George Mohler. Andrea L. Bertozzi and Baichuan Yuan want to thank the support of NIJ fellowship 2018-R2-CX-0013 and NSF DMS-1737770.
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Peter Diggle. A kernel method for smoothing point process data. Journal of the Royal Statistical Society: Series C (Applied Statistics), 34(2):138 147, 1985.
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Xiangnan He, Lizi Liao, Hanwang Zhang, Liqiang Nic, Xia Hu, and Tat-Seng Chua. Neural col- laborative filtering. Proceedings of the 26th International Conference on World Wide Web, pp- 173-182. International World Wide Web Conferences Steering Committee, 2017.
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Diederik P Kingma and Max Welling. Auto-encoding variational bayes. ICLR, 2014.
Wenzhao Lian, Ricardo Henao, Vinayak Rao, Joseph Lucas, and Lawrence Carin. A multitask point process predictive model. In International Conference on Machine Learning, pp. 2030-2038, 2015.
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Published as a conference paper at ICLR 2020
A
TABLE OF NOTATIONS
Table 5: Notations.
Notation
Definition or Descriptions
(x)
counting measure on metric space R the number of events of subprocess intensity function of subprocess u intensity process of subprocess
Nu
(x)
Aa(2)
(x(r)
kemel function
U
# of subprocesses on the space subprocesses in batch
U
X
events set
X,
observed events of subprocess
X6
all the observed events in a batch of sub- processes
4;
embedding/locatiinn of the ies event
Y!"
hidden variables indicate whether the subprocess u includes the 4e event
2u
K-dimensional hidden variable repre- sents subprocess u
pM
probability of the ish event occurs in subprocess u
Φ,
parameters of encoder(ju,, aT) and decoder(fo)
Juv EA,
auto and cross pair-correlation function normalized density
Nu (x)
B PROOF OF THEOREM
Proof. We define our model as a point process on R with the intensity function X²
The alternative model is Ny i.dd samples within R with p.d.f (z(x), given that we know No J (a(sdds is the number of points within the point process model.
1) Our model has the following probability generating functional
/
G(o) = exp(-
v(x)] A(dx))
(12)
Rdd
2) Given Nu.
MI,, (N..)
II hns)))
(13)
3) Our alternative model (a counting r.v. (zx) with locations according to h,,())) has the following characteristic functional
Go(v) = Σ (RN = n)B(exp( / (oes) = n)
(14)
0=0
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Published as a conference paper at ICLR 2020
Using 2), can evaluate this conditional probability
(og()(s)) = =n|= = [ (u())u(s)ds )
Elexp(
(15)
Xu (s)
Using 2) again and because the point process observation probability is
- p(w) (R) N.)p((1,,..,,.... IN. nn
12
Au (x.)) Saappppj J )(x)dx),
(16)
we have
/ (1))do((1 + Σ ///l/
1 (())(()dd))))=aa = =xp( ((s)( 1)ds). (17)
==1
The theorem follows from Ge(u) = G(u) as the probability generating functional completely deter- mines the probability structure of the point process.
We show that (10) holds in the main paper. Corollary 1.1.
E go x.) (ogl((X))zu)
Σ
log(h, (x()) +
(18)
Proof. Define \x(x)= Aul))
(x.( log Xul Kulza) = log (p(N(R)
(19)
N₂
= log(h.((z?)) + log(p(N( = N.))
(20)
log(p(N(R)) = N.)) = n nlog( f {(x)de)) - log(n!) - Max)dr
(21)
only function of N..
C AUTO AND CROSS PAIR-CORRELATTION FUNCTIONS
We show that (u, y) for u = u and ( y) < for u = v for our orginal model.
(x, v) = EA, (x)
(22)
(A,,
We have
N
N (I xx:)) Σ 3=1
(A,(A)A. = E
Σ
Y Kn - =z3))
(23)
i=1
EY,YY" Kn Kn) xxz).
(24)
Similarly,
N (x) (E
Σ
YI Kn ::: Σ KnyKn xz))
(25)
N N
x:) - - xz).
(26)
Note that EV_ Y.x 1 and E PM = When + j. we have EY.Yy pppp for any u, e.
When = j, EY" = pp¹ > ) for u = u and EY."Y"" = 0 p.P." Then it is easy to see Bu. (x y) for and gu, (r, y) for + v.
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Published as a conference paper at ICLR 2020
D MORE ON EXPERIMENTS
D.1 METRICS DEFINITION
The ranking performance evaluated through recall at K (Recall@ K) and normalized discounted cumulative gain at K (NDCG@K), In VAE-SPP model, the predicted rank of the held-out items I, for each user u are obtained from sorting the intensity function (z)).
Here keep the definition in Liang al., (2018). Recall @K K is defined
Recall @K Σ I(ra E Iu)
(27)
NDCG@ @ K calculated by normalizing discounted cumulative gain (DCG@K) with ideal DCG@K (IDCG@ K). The definition are as follows
K 2((re
DCG@K
DCG@K
NDCG@K = IDCG@K
(28)
log2 (i +1)
where I is the indicator function and r; is the 7th item among held-out items; IDCG@K is the ideal DCG @ K when the ranked list is perfectly ranked.
D.2 HYPERPARAMETERS
We implement models in Tensorflow based on VAE-CI We keep the same MLP architectures and hyperparameters for both of them. We use 3-E with as suggested. The only additional hyperparameters for model is the o² in the kernel function, which is determined using grid search on the validation set. We conducted the experiments on a single GTX 1080 TI 1 11GB GPU.
For simulation data, train both models for 200 epochs using Adam optimizer with 8 0.2, Ir = 5 x 10-5 We use mini-batches of size 20. Our architectures consist a one layer MLP with K 50. For VAE-SPP, o² 0.001. For Gowalla-NYC data, we train both models for 200 epochs using Adam optimizer with 8 = 0.2, Ir = 5 x 10-4, We use mini-batches of size 20. Our architectures consist of a one layer MLP with K 50. For VAE-SPP, o² 1 10-5 For Gowalla-La LA data, we train both models for 200 epochs using Adam optimizer with / = 0.2, Ir x 10 3 We use mini-batches of size 20. Our architectures consist of a one layer MI.P with K = 50. For VAE-SPP, x² = 0.001. For MI-1M data, we train both models for 100 epochs using Adam optimizer with B 0.2, Ir
10-3 We use mini-batches of size 5. Our architectures consist of one layer MLP with K 200. For VAE-SPP, o² = 10 5 For ML-100K data, we train both models for 100 epochs using Adam optimizer with 3 0.2, IT = x 10 We use mini-batches of size 5. Our architectures consist of
layer MLP with K 200. For VAE- SPP, 0² = 1 x 10 The one-layer GNN in ML data is trained using GraphSAGE, for which the embedding dimension 32 and the number of neighborhood is 10 for item and for user. The graph is consist of the edges between users and items as well as the edges between items based on their Jaccard similarity.
We use python statsmodel for the KDE and GPy for TGCP Bandwidth for KDE is selected auto- matically. The hyperparameters for TGCP are determined with grid search on the validation set. For simulation data sets, we set an RBF kernel with variance=l, lengthscale=0.1 for TGCP For Gowalla- CA data sets, we set an Matern32 kernel with variance=1e-3, lengthscale=0.1 I for TGCP. For Gowalla-NYC data sets, we set an Matern32 kernel with variance=le-4 lengthscale=0001 for TGCP.
D.3 ADDITIONAL EXPERIMENTS
On the training of VAE and GNN, we tried different training settings (separately or jointly) and choose train them jointly. We also tested the point estimate version of the VAE-CF called DAE- CF (Mult-DAE in Liang et al., (2018), with the same setting), which can improve the result under
https::/giite hhtp://tttttuttcce www.daww//wwwooooo.ooooo...... /
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Published as a conference paper at ICLR 2020
certain metries. One can easily extend our work to DAE-SPP obtain point estimation for the SPP intensity.
Table 6: Testing results on Moviel_ens-100K. These methods share the same network and trained
with 100 epochs. The test data is evaluating the model with the best performance during the valida- tion. Separate means that GNN is trained separately with VAE-SPP.
NDCG@ 100
@ecall@20
Recall@ 100
VAE-CF
40.88
32.32
57.63
DAE-CF
40.98
29.29
58.80
VAE-SPP
41.50
31.34
58.99
-AE-Sat-Sepparae
41.43
31.15
58.82
We also did experiments the MLPs for VAE. For Movie Lens 1M, the larger network in VAE-CF leads to a 40.3 NDCG@ 100 for VAE-CF and 41.9 for VAE-SPP. As a result, we use the smaller one instead.
14


Reviews:
[{"review_id": "rylSHQ6aYH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 work describes an application of a spatial point process for solving problems with missing data. The authors introduce a novel method based on a non-parametric definition of point intensities for the multivariate case. The method incorporates VAE framework to effectively handle missing points via smooth intensity estimation and enjoys amortized inference for efficient computations and quick prediction generation. Using a sequence of mild assumptions, the authors show connection to a popular VAE-based collaborative filtering model, which turns out to be a special case of their approach.\n \n\nThis is a rigorous study providing theoretically justified evidence on the effectiveness of the proposed approach. Apart from the issues in the last part of experiments with classical collaborative filtering task (which will be detailed below), the work presents a solid research. I would therefore vote for accepting it.\n\n\nThe text is well structured, and all key points are clearly explained. The problem solved by the authors is well described, and the motivation for this work is convincing. The way point process theory is applied constitutes a rigorous probabilistic approach. The authors convincingly justify the need for all approximations and simplifications made in the model. One of key results making the entire model feasible is supported by the corresponding theorem proved by the authors. I haven’t carefully verified all the derivations, though.\n\n \nMy major concern is related to the last part with experiments on the Movielens data. As the authors state, “applications without explicit spatial information, we embed each event into a latent space as a vector.” “No spatial information” is exactly the case with the standard collaborative filtering task, which the authors attempt to solve. This leads to an introduction of an additional model like GNN, which is unrelated to the main approach. As GNN is involved it’s not immediately obvious that the improvement over standard VAE architecture, observed in the experiments on ML-100K and ML-1M, is due to a better point process modelling.  No evidence is provided to argue that this is not simply due to a good compression or a good data preprocessing achieved by a GNN architecture itself. Therefore, the results on a pure recommender systems part are not convincing. What would happen if GNN was trained and fed into another (simpler) algorithm? Maybe a simple KNN based algorithm would produce comparable or even better results? As indicated by the work of [Dacrema, Cremonesi, Jannach 2019] on “A Worrying Analysis of Recent Neural Recommendation Approaches”, VAE-CF (along with several other recently proposed neural network-based methods) is inferior to even properly tuned kNN-based models. I would not be surprised, if a kNN model trained on GNN output would produce even better results than the proposed VAE-SPP.\n\nAnother related question is how incorporating GNN affects the training time? Is it comparable to that of VAE-CF or is it much worse? Computational performance is an important part in making practical decisions and should be also considered.\n\nFurthermore, both ML datasets used for tests are too small and not very representative to make any generalized conclusions. Even on a larger ML-20M dataset an optimal SVD-based model can be trained within several minutes on a standard CPU on a laptop (according to my experiments, VAE-CF would take at least twice longer on Tesla K80). Therefore, it can hardly be considered a realistic example. In practice, there could be millions and hundreds of millions of items. The authors even mention it in the in the introduction, using it as a vehicle to motivate their approach. However, computing similarities between that many entities can be a laborious task on its own, which adds an extra layer of complexity and again is not directly related to the main approach. It can easily become a bottleneck or make further computations inefficient. More efficient similarity computations may in contrast reduce the resulting accuracy.\nThe issue can get even worse, because, unlike classical MF methods, there’s still no proper support for sparse operations in NN frameworks. In the VAE framework it means that, during the training, user batches will be converted into dense arrays and may become inefficient to work with in terms of memory and CPU utilization (a few non-zero entries vs. hundreds of millions of explicitly stored zeroes).\nIn spite of all this, I’d also suggest rephrasing “We validate these beneﬁts through extensive experiments” as it sounds a bit exaggerated (if we are considering real recommender systems applications). I agree that the proposed approach is potentially applicable in real cases for recommender systems, however, there’s still not enough evidence for this. In fact, I don’t even think that completely removing the part with ML-100K and ML-1M datasets would make the whole work any worse. Clearly stating the region of applicability of the proposed approach would be enough. Right now some statements in this section in contrast are raising concerns rather than convincing the reader. The wording should be at least changed, so that readers do not get an impression that the case with classical CF task is solved purely by the proposed VAE-SPP approach.\n\nOther remarks to help improve the text:\n1) “… points are more likely to … form clusters than the simple Poisson process …” the sentence seems to be inconsistent.\n2) “The generative process of our model can be described as follow:” -> … as follows:\n3) Page 4, last paragraph, line 6 – shouldn’t the upper bound for summation be N_u instead of just N?\n\nReferences:\nDacrema, Maurizio Ferrari, Paolo Cremonesi, and Dietmar Jannach. \"Are we really making much progress? A worrying analysis of recent neural recommendation approaches.\" In Proceedings of the 13th ACM Conference on Recommender Systems, pp. 101-109. ACM, 2019.", "extra_scores": {"experience_assessment": "I do not know much about 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 carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "rygQxkld9B", "reviewer": "AnonReviewer4", "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": "ICLR review\n\nIn this paper, the authors propose to tackle the multivariate spatial point process model with a variational inference approach. The variational inference is implemented with MLP (amortized inference). In experiments, the results show that the proposed approach outperforms VAE-based collaborative filtering on Gowalla datasets and MovieLens datasets.\n\nMy overall judgement of this paper leans to acceptation. However, by doing a quick comparison with Liang's VAE-CF paper. The proposed algorithm seems to be an extension or modification of Liang's framework. Therefore I guess the main contribution of this paper is Eq. (9). If so, I'm expecting more detailed comparison with VAE-CF in the paper (not only quantitive evaluation).\n\nQuestions:\n\n- In Liang's paper, I see they report the evaluation results on both ML-1M and ML-20M dataset. However, in this paper, the results are reported on ML-100K and ML-1M. Why not evaluate the model on the bigger ML-20M dataset?\n\n- In Liang's paper and also He's paper (Neural Collaborative Filtering), they report NDCG@10 on ML-1M dataset. The NCF approach reaches a NDCG@10 of 0.426, Multi-DAE (denoising version of Liang's model) reaches 0.446. You are reporting NDCG@100, which is a different measure. However, the numbers in Table 4 seem to be lower than what I expected. Can you compare with them in the same condition?", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "SkxY1KDaKH", "reviewer": "AnonReviewer2", "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": "In this paper, the authors propose a VAE model for spatial point processes. The model generalizes the kernel density-based intensity and applies variational inference. The model is applied to synthetic datasets, a location-based social network dataset, and a recommender system dataset. \n\nThe paper is well motivated and clearly written. I found the probabilistic modeling interesting. My major concern is that, with added complexity, the experimental results suggest that VAE-SPP is not significantly better than the existing VAE-CF on most tasks.\n\nFor the MovieLens task, it seems that the GNN is an important component in the pipeline, but no further detail is provided about it, including the network architecture. The authors might also want to provide the definitions of NDCG@k and Recall@k, at least in the appendix.\n\n---\n\nI appreciate the author's detailed response and updated paper. I have changed my rating.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents a novel VAE-based model for multivariate spatial point process which can realize efficient inference by amortization and handle missing points via smooth intensity estimation. Authors also provide interesting theoretical analysis to connect their method to a popular VAE-based collaborative filtering method.
Overall, all reviewers appreciate the methodological and theoretical contributions of the paper. During the reviewer discussion, one reviewer decided to update to the score to Weak Acceptance. There are still some concerns regarding experimental validation, I think the paper provides enough theoretical contribution to the community and would like to recommend acceptance. 
Decision: Accept (Poster)