TEN-DM: Topology-Enhanced Diffusion Model for Spatio-Temporal Event Prediction
Authors: Yuxin Liu
Kaiming Wang
Chenguang Yang
Yulia Gel
Yuzhou Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Spatio-temporal point process
Diffusion model
Topological data analysis
Primary area: learning on time series and dynamical systems

Abstract: Spatio-temporal point process (STPP) data appear in many domains. A natural way to model them is to describe how the instantaneous event rate varies over space and time given the observed history which enables interpretation, interaction detection, and forecasting. Traditional parametric kernel-based models, while historically dominant, struggle to capture complex nonlinear patterns. In contrast, deep learning methods leverage the representational power of neural networks to aggregate historical events and integrate spatio-temporal point processes. However, existing deep learning methods often process space and time independently, overlooking the spatio-temporal dependencies. To address this limitation, we propose a novel method called Topology-ENhanced Diffusion Model (TEN-DM), including two key components namely spatio-temporal graph construction and multimodal topological feature representation learning. Further, we use temporal query technique to effectively capture periodic temporal patterns for learning effective temporal representations. Extensive experiments show the effectiveness of TEN-DM on multiple STPP datasets compared to state-of-the-art methods.

Published as a conference paper at ICLR 2026
TEN-DM: TOPOLOGY-ENHANCED DIFFUSION MODEL FOR SPATIO-TEMPORAL
TEMPORAL EVENT PREDICTION
Yuxin Liu' Kaiming Wang Chenguang Yang'
Yulia R. Gel Yuzhou Chen'
DDpartmeet of Statistics, University of California, Riverside
Department of Statisties, Virginia Tech [yliu1020, kwang280, cyang314, yuzhou. wuzhou.chen)@ucr.cee ygl@vt.en
ABSTRACT
Spatio-temporal point process (STPP) data appear in many domains. A natural way to model them is to describe how the instantaneous event rate varies over space and time given the observed history which enables interpretation, interac- tion detection, and forecasting. Traditional parametric kernel-based models, while historically dominant, struggle capture complex nonlinear patterns. In contrast, deep learning methods leverage the representational power of neural networks to aggregate historical events and integrate spatio-tempord. point processes. How- ever, existing deep learning methods often process space and time independently, overlooking the spatio-temporal dependencies. To address this limitation, we pro- pose a novel method called including two key components namely spatio-temporal graph construction and multimodal topological feature representation learning. Further, use temporal query technique to effectively capture periodic temporal patterns for learning ef- fective temporal representations. Extensive experiments show the effectiveness of TEN-DM on multiple STPP datasets compared to state-of-the-ar methods.
Diffusion Model (TEN-DM),
INTRODUCTION
Stochastic point processes are loosely speaking random sets of points (marks) scattered over some domain. Such processes appear wide range of natural and manmade phenomena and can be also used to characterize various human activities, with applications ranging from earthquake occur- rence emergency calls heart beat. Some point processes can be put in correspondence with an index. If such index represents time, call this stochastic process a temporal point process (TPP) like, for instance, high-frequency trading order book events in finance or patient's 's decline towards septic shock in medicine which often modeled by Hawkes processes (Limal 2023 Kuang al.) 2024 Laub et al., 2025). If the points live in some d-dimensional domain (c.g., Euclidean space manifold), for example, patterns of Ice Age archaeological sites Jayalath al. (2015), spatial arrangements of trees and animals in ecology (Samarasekara et al.) (025) or distributions of young stellar objects in astronomy (Retter et al.) 2019), call it spatial point process (SPP). In turn, spatio-temporal point processes (STPP) extend these concepts by considering stochastic processes that integrate both spatial and temporal dimensions. STPP allow to consider phenomena in which events occur within a spatial domain, with the time of occurrence serving as a distinguishing feature (mark) associated with each event, with applications ranging from earthquake tracking to crime de- tection to monitoring infectious diseases (Zhu & Xie 2022 Bernabeu et al. (0255. just to name few. SPP, TPP and STPP have been a longstanding area research within statistical sciences (Cox) & Isham 1980 Daley & Vere-Jones 2003 Diggle 2001). However, such more traditional statistical approaches predominantly either impose some parametric model restrictions or tend to rely heav- ily unrealistic assumptions about the event sequences' generative processes, exhibiting limited abilities to scale for long historical records and massive event sets.
In turn, numerous recent efforts have been dedicated to developing DL for modeling TPPs and STPPs (see the most recent comprehensive overview by Cheng et al. (2025)). Some of the earlier thrusts in this direction include combination of event encoder, aggregation encoder, and decoder

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parametrization for event sequence prediction (Du et al. 2016 Shchur al. 2021 The more cent techniques advance the concept of deep STPPs which encompass such methods as deep kernels, neural latent processes with transformers, and deep generative models (Zhu et al. 2020 Chen al. 2020 /Zhou et al. 2022 Cheng et al.) 2025). Despite these advances, due to inherent non-stationarity and complex interplay within and between time and space dimensions, existing DL tools often ex- hibit limited capabilities to capture intrinsic structural organization of the underlying STTPs and to distill intricate latent spatio-temporal interdependencies. especially under sparse regimes, which leads to deficiencies in predictive performance (Cheng et al. (025). Our paper is, hence, motivated by the following tightly interwoven questions: How can we deseribe the complex
STTP interdependenccis, especially under spurse and noisy scenarios? ? and How can we distill latent structural STTP characteristics that play a particularly important role for predictive tasks?
We argue that these fundamental questions in STTP modeling can be approached by fusing the rais- ing paradigm of diffusion models with the emerging tools from computational topology, particularly, the concepts of zigzag persistence. Why graphs? As shown by a number of recent studies (Jin) ct al. 2024), graph abstraction offers a flexible and versatile framework to describe higher-order interdependencies in multivariate spatio-temporal processes which otherwise cannot be systemati- cally assessed by more traditional methods. Despite this success and despite the existence of STTPs on graphs and manifolds, graph abstractions have never been used to model STTPs. We fill this gap by designing a STPP graph construction strategy with different views that convert STPP into a graph abstraction and learning node (i.e.. event) embeddings. By leveraging temporal query and self-attention on data in temporal dimension, we then capture period patterns and temporal-wise dependencies. This allows us to enhance prediction in spatial and temporal domains. Why topol- ogy and why zigzag persistence? In a nutshell, computational topology extracts shape properties of the data that are intact under continuous transformations. Integrating such extracted topological descriptors to DL has shown to result in enhanced model performance and robustness gains. Zigzag persistence (ZP) advances these ideas toward distilling the most essential shape signatures of the data that manifest over time. While ZP has been studied in conjunction with graph diffusion, the utility of ZP for modeling STTPs has never been explored. To leverage such important aware shape information, we transform the observed STPP time series of images and then, armed with ZP, learn the most essential topological characteristics that reveal over time. Finally, Why dif- fusion? Thanks to their advanced capabilities capture complex relational structures within the observed data, diffusion models have recently emerged as new powerful machinery for a variety of downstream tasks, from anomaly detection prediction. While there are number of studies on diffusion models for TTPs (Lüdke et al., 2023 Zhang et al. (024) to the best of our knowledge, neither of them yet consider diffusion for STTP. As such, this paper advances STTP modeling along multiple directions: by leveraging zigzag- enhrnneed diffusion and STTP graph representation, we propose a novel versatile Topology-ENhanl Ehhancee Diffusion Model (TEN-DM) for capturing complex spatio-temporal dynamics of STTP under sparse and noisy regimes.
In summary, the paper makes the following key contributions:
We design (i) a STPP graph construction and learning (GCL) module to preprocess STPP into the graph format, enabling the GNNs learn complex spatio-temporal interactions; (ii) a novel temporal topological learning (TTL) framework coupled with cubical zigzag persistence, which captures topology-aware spatio-temporal information over STPP; and (iii) temporal query-guided self-attention mechanism (TQ-SA) to capture temporal de- pendencies.
Based on the GCL, TTL, and TQ-SA, we introduce TEN-DM, a novel diffusion model to address the "dynamic spatio-temporal dependencies learning dilemma" " in STPP.
Extensive experiments on real-worke STPP datasets show the proposed TEN-DM achieves sfte--ofthhe art prediction performance in both spatial and temporal dimensions.
2 TECHNICAL BACKGROUND
Notations & Problem Formulation. We are given a sequence of spatio-temport events X =
{æ, = 1,2,....N ... whose number of events N. Each event is described as z; = g:), where t; denotes i-th occurrence time, gi denotes i-th geospatial information (c.g.. latitude, longitude, or zipcode), and 0 < 4 < <.... lT < T (i.e., sequence of strictly increasing arrival times). The goal
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of forecasting model Fg with weights 0 to predict future spatial and temporal information based on the history until time t denoted as H. = (011.... In <t. i.c., B(+1 = Fo(()))
Diffusion Models. Diffusion models probabilistic generators that learn data distribution by corrupting samples with Gaussian noise in a forward Markov chain and then training a neural net- work to iteratively denoise in reverse. This framework has exerted significant influence on state- of-the-art results in computer vision (Rombach et al.] [022 Saharia et al. 2022 Ho et al. (022) and
natural language processing (Gong et al. 2022; He al. 2022 Li et 2022). Given their abil- ity to capture data distributions, diffusion models are increasingly studied for spatio-temporal data forecasting and generation in variety of scenarios. DiffSTG (Wen et al. (023) adapts diffusion to ppatio-temporal graphs and introduces a UNet style temporal module with graph convolutions. KSTDiff (Zhou et al. 2023) tackles urban flow generation by combining a region-customized dif- fusion process guided by learned volume estimator with knowledge graph-enhanced denoising network. Dyffusion (Rühling Cachay al. 2023) introduces a dynamics informed diffusion model by embedding temporal dynamics into the diffusion steps and training a stochastic time conditioned interpolator with predictor. Diff-RNTraj (Wei et al. 2024) focus on vehicle trajectory data, which pretrains continuous embeddings of road information as denoising condition and decodes back with a spatial-validity loss. ControlTraa (Zhu et al.] (024) develops autoencoder that learns road seg- ment embeddings and combines road-network topology constraints, merging them into geographic UNet to guide the denoising process.
Point Processes. Point processes widely used model sequences of discrete events across di- verse domains (Daley & Vere-Jones] 2008 Reiss 2012 Karr 2017). Classical TPP models focuses
on conditional intensity function (Rasmussen 2018), including Poisson process (Kingman 1992),
Hawkes process (Hawkes) [971], and self-correcting process (Isham & Westcon 1979). The simple patterns of occurrence can be captured by the classical TPP models, while the neural TPP models can perform better in capturing complex dependencies (Shchur et al. (021). Decoupled Marked Temporal Point Process (MTPP) (Song et al., 2024) uses Neural ODEs to decouple the influence of each past event into its own latent continuous trajectory. Neural Jump-Diffusion TPP (NJDTPP)
(Zhang et al. 2024) proves the equivalence of stochastic differential equations (SDEs) for classical TPPs, and uses neural sump-diffusion SDE (NJDSDE) which provides a unified SDE view with the- oretical footing. For the SPP, it is well introduced in Moller & Waagepetersen (2003:
: Illian et al.) 20088. Continuous normalizing flows (CNF) and Time- Varying CNF (TVCNF) can be used for modeling spatial distribution where the latter considers the dependence on the timestamps (Chen) al. 2018 Yuan et al., (023a). Beyond TPP and SPP, STPP takes spatial and temporal information into consideration. Classical STPPs are extended from the point process including (in (in)ooooeenoous Poisson process (Daley & Vere-Jones 2003). Neyman-
process (Gabriel & Diggle 2009), in-
hibition process (Gabriel et al. 2013), strauss process (Cronic & Van Lieshout 2015 and Cox process Cox 1955 Diggle 2013) Diggle et al. 2013). Recent neural approaches extend STPPs along two main directions, i.e., Influence-kernel based models and Intensity- models (Cheng jet al. 2025). Deep Non-stationary Kernel (DNSK) (Dong et al., 2022) develops a novel and general low-rank decomposition to approximate the influence kernel and representation through deep neu- ral networks. However, real-world STPP often exhibit complex and nonstationary spatio-temporg dependencies, which leads to significant challenges in accurately predicting spatio-temporal events. In contrast, our TEN-DM introduces spatio-temporal geometric and topological learning paradigm, which can effectively introduce the graph structural and dynamic topological information into the diffusion model, therehy being able to capture complex spatio-temporal dependence between dis- crete events.
3 METHODOLOGY
To address the limitations of existing STPP approaches and leverage the strengths of geometric and topological representation learning, we propose TEN-DM, unified diffusion model framework that integrates structural, temporal, and topological information for enhanced spatial and temporal forecasting. As illustrated in Figure T the framework comprises three core components.
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Temporal Input
Hemporas
Spatial Input
Diffusion Model
TST-M9A
Time-series Image Input
TLL
Graph Input
Figure I: Overview of the TEN-DM model: (1) sequence of events; (2) extracted temporal, spatial, time-series image, and graph information; (3) key components including temporal encoding, spatial encoding, temporal topological learning module (TLL), graph representation learning (GNN), and topology-guided spatio-temporal multi-head attention (TST-MHA).
3.1 SPATIO-TEMPORAL POINT PROCESS GRAPHI LEARNING
Compared with existing approaches which neglect spatio-temports point process inherent interac- tions between different event properties, to address this limitation, we propose new strategy to capture the nuanced spatial and temporal relationships by generating a graph structure and learn the corresponding high-level graph representation. There exist multiple types of edges between nodes (i.e., events), and each type of edge has different role and impact node representation learning. For example, for the graph representation learning in crime incidents, different types of relationships between crime incidents exhibit various dependency semantics such as thefts in specific census tracts can be temporally linked to seasonal trends (e.g., spikes during holidays), robberies tied to highways transit stations, and burglaries often higher in neighborhoods with high poverty rates and hous- ing instability. Therefore, in this section, we aim to answer the question: How to build a STPP graph by fully utilizing the rich tiating the types of edge connections between nodes in the heterogeneous graph. Then we aggregate graph structural information with different importance weights. We denote our
information?. We first generate multiple graphs by differen-
the relation-guide generated graph as G, (A,, X,)) (where r {1.... R); note that, in our study, R = 3 includes time, latitude, and longitude information) with r-th node feature matrix X, and the adjacency ma- trix of r-th graph A,. To achieve this, we first construct a e-graph G, (Calder & Trillos 2022). Specifically, we quantify similarity 8ne between events u and v as follows:
" = (x", 3 x/)((((()) [11a<a11))))1))) eny {(u,)) v) S > L"))
where O denotes dot product. In this work, we use cosine similarity to calculate event similarity. By using a cell list find event pairs that are within given cut-off distance, we can efficiently solve the problem with time complexity O(kN) where & is the maximal number of neighbors within the radius. Hence, node interactions with various relation semantics will have different structural char- acteristics. To capture such multi-typed node dependencies, we assign different weights to different adjacency matrices A. >R A,
adjacency matrices and aggregate R edge-type-spent Note that the set of importance weights {a, are updated adaptively during training.
To deliver high-level graph- structured information into diffusion model framework. we pretrain and @ denotes the concatenation)
GNN with the node features (where X @(X, X2........ Xx))
and joint adjacency matrix A, and adopt a pooling layer (i.e., (ool(.) ()) to get the graph representation
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which can be formalized as:
og Pool(GNN((X (X, A)).
3.2 TIME-SERIES IMAGE TOPOLOGICAL REPRESENTATION LEARNING
To enhance spatio-temporal prediction by incorporating dynamic topological information, we create time-series images for spatio-temporal point process data and propose an effective method, i.e., dynamic image topology learning module that captures dynamic visual scenes.
Formulating time-series image representation by the sequence of events. Given a sequence of events X = {m) we first divide X into patches which can be either overlapped or non- overlapped. Here we set the patch length (i.e., scale) to be P and the stride to be S (i.e., the non- overlapping region between two neighboring patches). Then we can obtain a sequence of patches which is denoted by X = {x(1), xx(2) "..")......N........... .N', E PxN where x(() E R² and N = =I(T P)/S| + 2. After that, for each patch i, create image m, E RR"* which is 2D grid and clipped to the target data region. For instance, given the Philadelphia crime incident data, the 2D grid is bounded by the set of latitude (min: 39.86; max: 40.14) and longitude (min: -75.28; max: -74.55). Within the patch i, we rasterize the events' geo-coordinates onto the 2D image by recording as each pixel's value the associated temporal attribute. Specifically, we convert the spatial locations of events into grayscale image, where each pixel corresponds to a discretized spatial cell. Pixels containing least one event are assigned a value of (white), while pixels without events are assigned 0 (black). Thus, we can create a binary image that encodes the spatial distribution of events within the patch. That is, m = [i] which represents j-th timestamp in i-th patch (+). Thus, we can generate the time-series image data M as series of images {1)(1, m)(2) ...., (N)).
Topology learning on time-series image data. Persistent homology (PH) is branch in topolog- ical data analysis which tracks the evolution of the various data shape patterns along various user- selected geometric dimensions (Zoorodiian & Carlsson 2004 Edelsbrunner & Morozov 2012). Despite the generality of simplicial complexes in PH, cubical complex is more natural representa- tion for 2D images or 3D volumes. See Appendix A for formal definition of the cubical complex.
Definition 3.1 Let H be a cubical complex, and suppose f Qx - TR where Qx denotes the set of elementary cubes in X, satisfies (i) f(Q') S f(Q) whenever Q' is face of Q. Let H (a) = (i--i(((oa ((-00, a)), and notice that (i) implies H (a) is a subcomplex of H for every a E TR. on to be the values of on the cubes of A and denoting H (ax) Hay we say that the following increasing sequence of subcomplexes is a filtration associated with f
Taking 01 a2
Ø = -oo C Ha, C...C =.
In our setup, we use the lower-star filtration to extract the topological information encoded in 2D images. However, given time-series image data, standard persistence algorithms capture only in- dependent spatial topological information, and unaware of the temporal topological information and complex spatio-temporal dependencies. Furthermore, beyond capturing dynamic structures and temporal dependencies, it is equally important to learn multi-scale temporal information, i.e., under- standing temporal information by leveraging multi-scall time-related data such as time units (e.g., minutes, hours, and days). To address the above two challenges, we propose a more flexible temporal topology learning (TTL) framework coupled with zigzag persistence which is capable of capturing vital time-aware topological information on time-series images across different time scales.
Zigzag persistence. Zigzag persistence (ZP) is a special type of quiver representations and gener- alizes conventional PH by enabling the analysis of topological spaces connected through inclusions in both forward and backward directions (Carlsson & De Silva 2010)). Unlike the above standard PH which requires nested sequence of spaces, ZP accommodate more flexible filtrations which makes it particularly well suited for capturing the evolving topological structure of image time-series data. This capability has led to growing interest in ZP across range of data analysis tasks involv- ing time- vimeinarying signals. Specifically, for time sequence of images (m( m²)).... zigzag filtration over image snapshots is constructed by the bidirectional arrows as follows:
..... m (N)), }. the
m m U (2) (2) - _.... m(-N1 - m (N-1) U (M) (N)
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To compute the homology groups and the corresponding topologiaal feature of the zigzag filtration, for each timestamp i, we first apply the cubical complex (i.e., with lower star filtration) to the image m(() time step within the i-th patch and construct simplicial complex H'. Then we compute the union of two adjacent simplicial complexes (i) and ((+1) denoted by X x(+++1) x ((+1), i.e., we include simplex o ((..+1 if and only if p E X(() p E x which preserves features that appear (born) or disappear (die) across time steps. Leveraging the zigzag sequence of images, based on fixed scale parameter a, we can compute the zigzag persistence of the sequence of vector spaces as follows (where Hp(xX) denotes p-th homology group of .x) )
()(x - (p(( U x()) - H2((x(9) _ Hp(xX(M-1) U U.X(M)
- ()))((N)
The extracted topological information can be summarized in the form of a multiset in R²² i.c., so-
called persistence diagram (ZPD) Dgz {p, (bp, dp) IR2 dp > 6n} U A (here A {(T,T)[T E R} is the diagonal set containing points counted with infinite multiplicity; including A allows us to compare different ZPDs based on the cost of the optimal matching between their points). For each persistence diagram Dgz, we compute its zigzag persistence image (ZPI) denoted as PIz via the vectorization (denoted as Vee(.)) scheme (Adams et al., 2017). We provide an exmaple of ZPI generation pipeline in Figure _ Appendix A In our experiments, we use both O- (i.e., connected components) and 1-dimensional (i.c., holes) topological features.
Theorem 3.2 (Zigzag stability for time-series images) Let Q C 2² be finite and m0 : - R I..... .N) be grayscale frames. For a E R define the lower-star cubical sublevel complexes Aa = (a C Ω cubical: maxxuvert(o) m ml()(v) S a) and bridges (1t.511) = Fix p 2 0 and a field k. For each o, let
=Hll) U ((+1)
Xa X(L) - -x(2) - xx
...... X((N Va = =Xpiiiaakk),
and have V : Va- For another sequence m with [lm()) 100 I E for all define V analogously. Let sha be the shift (shaV),. Va+5- Then there exist natural transformations + d + sh. V and V : V + + sh. V making an s-interleaving: (sh, (b) a nj and (she V) o n, where T), ij are the canonical inclusions to the 28-shift. Hence dr(V, V) S e, and (since these modules are pointwise (mite-dimensional on a finite grid) their zigzag persistence diagrams satisfy:
ds ds (V),,(V), (?), Se.
The proof of Theorem 3.2 is provided in Appendix B The theorem suggests that time-zigzag per- sistence on image sequences is robust to small grayscale fluctuations, so differences observed in the resulting zigzag diagrams reflect genuine structural change rather than noise, which supports reliable comparison across time windows and trustworthy use in downstream analysis.
Temporal topological learning framework. To address the challenge of processing multi-scale temporal information, we aggregate multiple time-scale topological features into on one uni- fied representation. More specifically, according to different temporal scales, we use Q dif- ferent patch lengths P [Pi, P2..... ........) P&) and obtain the corresponding Q time-series data MI = (Mp M Pp..... Mrp....,,p.... Mr by using the proposed patching strategy. Given multi-scale time- series image M, we first employ the 7P and vectorization method to generate zigzag persistence images with different scales, and then integrate them into a mixup zigzag persistence image with different coefficients. That is:
PIz = E,PTT PT Ver(ZP(Mr,)),
(3)
q=1
where Bg represent the importance coefficient for q-th temporal scale (in this paper, we consider different time scales and hyperparameters / = { are equal to 0.25 (i.e., 31 = B2 = B3 =
B4 0.25)). Next, apply two-layer CNN over the mixup zigzag persistence image PIz, yielding latent dynamic topological representation:
where LayerNorm denotes the layer normalization to maintain the feature scale, and FC denotes a fully connected layer which flattens convolution results.

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3.3 MODELING SPATIAL AND TEMPORAL INFORMATION
Temporal encoding. Positional encoding is crucial design in the Transformer architecture for making use of the order of the sequence. To effectively utilize temporal information of STPP, for each event time 4, we map it into the temporal embedding & by using a positional encoding (where D denotes the embedding dimension). In summary, we have:
cos sin
100000)1770) when j is odd, bi
[t
(5)
100000-1770
when j is even.
For accurate temporal prediction, is vital to model temporal dependencies. as well as trend shift. Temporal query (TQ) techniques have been successfully applied to learn robust multivariate corre- lations from multivariate time-series data (Kulkarni et al. [01 Lin et al., 2025). Inspired by this, in this work, we develop TQ-aware self-attention module to effectively and adaptively identify temporal patterns inside the sequence of events. Specifically, given the skeleton the self-attention
module, i.e., self-Attention Softmax QK \V (where Q, K, and V are queries, keys, and val- ues respectively), we integrate TQ learnable matrix into the query matrix, and integrate temporal encoding output into both key and value matrices, i.e.,
vd
@rok: VVde
Self- Self-Attentionro (t) Softmax
V
(6)
where QTo = WTO WO, K; iW' V = tWV. WTo is a learnable TQ matrix, Wº, WK and WV are projection matrices, and d; is the dimensionality of the queries and the keys. Then the output of the positional encoding, i.c., anism, and the self-attention output is added back to its input (i.e., é) via a residual connection and then normalized with the layer normalization, which stabilizes optimization and gradient flow while preserving the initial embedding as new contextual information is integrated:
(t1.t2.... tx) fed into the self-attention mech-
8² LayerNorm(f + ceff-Attentionq(()).
Spatial encoding. Given the spatial information 9i of event i, we apply a lightweight MLP to learn the spatial embedding and present the output gt- In our study, the MLP consists of two connected layers with ReLU activation, which are defined as = Linear(Relaa (eeeaareLLLLinear((gg
For spatial encoding, apply the regular self-attention mechanism over the initial spatial embed- ding ğ = (g1.92.... gk), yielding the final spatial representation:
LayerNorm(@ + Self-Attentionng)).
Following Eqs. and we can obtain latent embeddings in temporal (ō;) and spatial (ō.) domains separately. However, to jointly learn spatio-temporal representations, the summarization operation is not enough to seamlessly link two domains without an adapter. To integrate spatial and temporal embeddings, next we introduce an unified topology-awan fusion framework.
3.4 SPATIO-TEMPORAL FUSION WITH TEMPORAL TOPOLOGY LEARNING FRAMEWORK
The topology-awer aware fusion framework integrates spatial, temporal, and dynamic topological em- beddings, leveraging their complementary strengths to (i) capture spatio-temporal dependencies and (ii) narrow the gap between spatial and temporal domains. The dynamic topology embedding from TTI. encodes spatio-temporal topological patterns and serve as queries in the topology-guine spatio-temporal multi-head attention (TST-MHA) mechanism, while the concatenation of spatial, temporal, and graph embeddings denoted as F @(t. g, og) serve as keys and values. The TST- MHA is defined as:
(QkK), Vdly
TST-MHA(Q, K, V) = + (head),........ head,) W head, = Softmax
h
Vn.
(9)
where Q₁ 2W K, FWK V, FWV. and 8. W W. and W learnable projection matrices. dy denotes the head dimension, and H is the number of attention heads. The proposed fusion framework integrates spatial, temporal, and topology- dynamic information,
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capturing both local and global dependencies. We derive the Lipschitz bound for TST-MHA, Theorem B.I in Appendix B
Finally, the combined embedding (i.c., from spatial domain, temporal domain, and TST-MHA) will be transformed by the feedforward layer, which formally computed as:
o TST
--eeeeeFFFoe Forward(LayrrNomm + TST-MHA K, V))).
(10)
3.5 SPATIO-TEMPORAL FORWARD AND REVERSE DIFFUSION
For each event x; = (Ti, 9:) in the sequence (where т; is the time interval since the last event), we perform the forward diffusion process as Markov process over the spatial and temporal dimensions as (a), ... xxx²
where K is the number of diffusion steps. That is, we add small amount of Gaussian noise step by step the space and time values until they are close pure Gaussian noises. The forward process of our diffusion model on spatial and temporal dimensions can be written as:
(ar la. -) = (r 9..001995 1),
(11)
NN(o"; vI Bexx Ba1), denotes the Gaus- sian distribution used to generate the noise, is the identity matrix, and o = II==, f1.K E (0, 1)). The purpose of the reverse process is enable our diffusion model to learn the denois- ing ability of noisy spatial and temporal information. Specifically, we aim to reconstruct the point x; = (Tr.g) 9:) with the learned model over K steps x - x
where the recursive formula is (10="11*
(1= (1- B,) (where
x.. Further, we also in- corporate latent spatio-temporal embedding denoted as ō,-1 (where ,;-1 = @ o into the backward diffusion process which helps to guide the denoising process towards the clean sample. The denoising transition step is outlined as follows:
p | o,- = Po(TT-11/1 1
of 0.-1)Po(ot- "> ō,-1).
(12)
In our experiments, we employ the cross- conditional denoising decoder (Wang et al. 2024) which incorporates predicted values -k++1 g. k+1 in temporal and spatial dimensions respectively, graph learning output og (see Eq. 2) and denoising step i with positional encoding and leverages the latent spatio-temporal embedding o for the guidance in the conditional denoising process. To
predict future event, we utilize the inference framework proposed by Yuan et (2023a).
EXPERIMENTS
4.1 EXPERIMENT SETTINGS
Datasets. In our experiments, we use 5 real-world datasets. i.e., JPN Earthquake: Earthquake with a magnitude at least 2.5 in Japan from 1990 to 2020; COVID-19: COVID-19 dataset is collected from publicly released COVID19 cases in New Jersey state from March 2020 to July 2020; US Earthquake: The US earthquake dataset contains earthquake occurrences from December 2023 to January 2024 in US: Theft: The theft data is collected by the Philadelphia police department from January 2025 to April 2025 in Philadelphia: 311 Service: Similar to theft data, we collect Philadel- phia 311 service dataset from OpenDataPhilly from January 2025 to June 2025. More details of data resources and train/validation/test split ratio are in Appendix A
Baselines and Evaluation Protocol. We compare TEN-DM with 17 baselines, including 3 SPP
baselines, 10 TPP baselines, and 4 STPP baselines. SPP baselines: (i) Conditional Kernel Density
Estimator (KDE) (Chen al., 2018;; (ii) Continuous Normalizing Flow (CNF) (Chen al. 2018
and (iii) Vaainn Continuous Normalizing Flow (TVCNF) (Chen et al. 2018). TPP Base-
lines: (i) homogeneous Poisson process (Kingman 1992); (ii) Hawkes Process (Hawkes) 1971
(iii) Self-coreecing process (Isham & Westcott 1979): (iv) Recurrent Marked Temporal Point Pro-
cess (RMTPP) (Du et al. 20166;; (v) Neural Hawkes Process (NHP) (Mei & Eisner 2017): (vi)
Transformer Hawkes Process (THP) (Zuo et al. (020); (vii) Self-Attentive Hawkes Process (SAHP)
Zhang et al.) 2020); (vii) Log Normal Mixture model (LogNormMix) (Shchur et al.) 2019): (ix)
Wasserstein GAN (WGAN) (Xiao et al.
2017); and (x) Neural Jump- Diffuuiifssion Temporal Point
Process (NJDTPP) (Zhang et al. (024). STPP Baselines: (i) Neural Jump Stochastic Differen- 2019) (ii) Neural Spatio-temporal Point Process (NSTPP)
tial Equations (NJSDE) (Jia & Benson
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(Chen et al., 2020); (iii) Deep Spatio-temporal Point Process (DeepSTPP) Zhou al. 2022); and (iv) Spatio-temporal Diffusion Point Processes (DSTPP) (Yuan et al. 2023b). We have included a detailed introduction about baselines in Appendix [ We evaluate prediction the next event in both space and time. The spatial error is measured by the Fuclidean distance between the predicted and ground truth location, and the temporal error by root-mean-square error (RMSE) between the predicted and ground truth time interval.
Implementation Details. We run our experiments on 4 NVIDIA RTX A5000 GPU cards with 24GB memory. Optimization uses Ada = 0.9, 32 = 0.99) and learning rate warm-up from 0 a peak selected from {le -3 3e 3) followed by linear decay to 5e - for 1000 epochs. More details can be found in the Appendix A Evaluation metrics report spatial Euclidean distance and temporal RMSE on 3 runs with different random seeds. Code is available at htttps: / thub. ngc TEN-DM All datasets used are publicly available; we include raw data resources, and the train/val/test splits used in our experiments. Hyperparameter settings are reported in "Implementation Details".
4.2 RESULTS
Table D compares the forecasting errors of TEN-DM with 17 baseline models across 5 real-world datasets in spatial and temporal domains. Lower Euclidean distance and RMSE indicate higher forecasting accuracy. As shown in Table TEN-DM significantly outperforms all baselines across all 5 datasets in both Euclidean distance and RMSE except for JPN carthquake data in temporal dimension. Notably, on COVID-19, US earthquake, and 311 service, our TEN-DM is statistically significantly (with p-value < 0.05) better than runner-ups in both spatial and temporal domains. Su- perior performance against SPP baselines. TEN-DM outperforms all 3 baselines; compared with SPP runner-ups, TEN-DM achieves 21.97% 42.97%, 9.37%, 0.29%, and 0.55% relative improve- ment on JPN earthquake, COVID-19. 19, US earthquake, theft, and 311 service datasets respectively. Superior performance against TPP baselines. TEN-DM statistically significantly outperforms all TPP baselines with 6.74%, 42.53%. 57.14%. 38.02% and 13.89% relative improvements on JPN earthquake, COVID-19 19. US earthquake, theft, and 311 service datasets respectively (compared with runner-ups). Better performance against STPP baselines. Compared with the runner-up (i.e., DSTPP), TEN-DM achieves 7.16% and 6.90% relative improvement on COVID-19 dataset in spa- tial and temporal dimensions respectively; a 17.08% and 3.66% relative improvement on theft and 311 service datasets respectively in temporal dimension. Computational complexity and running time are provided in Appendix A
Table 1: Performance evaluation for predicting both time and space of the next event. We use Eu- clidean distance and RMSE to predict errors of the spatial domain and temporal domain respectively. Here denotes valuee < 0.05 (i.e., statistically significant results).
JPS Eaniagaake
COVIS-T
US Earbbqukk
Tieh
311 Service
Moskd
Scalell
Tenporal
Speril
Tomp-ni
Speril 4130000000
Eempini
Syoria
Irupural
Spario
Condeora XDB
010021111
140010054
0.072100000
01006:00 01000
TVENP
K110580001
0.500=0000
0072400000
0.000:11000
0631:00017
0.66310.021
1.001.0.000
0.626-000055
1.259
Hawless
0541-000
060
122 -1
0.625 1107
1.880 10
Selfcourestigg
11:200:04406
2.83050.14 141
5.300 346
0.509=0.024 02024
2.526 123
RMTEP
0424200005
1320100024
1.00 130
0.583=00025
1,142-0.00
NHPP
1800-11171
10110:00000
0.517-0000 0107
2:400=0
SAMI
LegNormiix
0,501-00000 105
0.1066-0.011
D. 11474:20116
0.501=01000
2.075
WGAN
1177
0112440.000
11 (1766221100
9:099::10019
SIDTTP
0.306-11001
0,70040.006
0.53--001110
0.506-00064
20002-
SUSDE
0.1. 130001
IL 10000301000
CORO
NSTEP
110:00000
0,547=00000
0.560=0000
0.145100.000
1002::0.000
0.0972000000
0.554:0.045
01072
0.570=00007
6770200000
0.375:0.000
2,119-0.000
0.09360.000
38.89200.10
0.77-00000
1.700
TEN-IAI
$290 6341
TLWT
10101
T0A1
TC300
0.017
"0.36T
820012
4.3 ABLATION STUDIES
To examine the effectiveness of the proposed components, we conduct experiments without graph learning (i.e., W/o Graph), TQ-SA (i.e., W/o TQ-SA), and TTL (i.e., W/o TTL) on JPN earthquake, COVID-19, 311 service datasets. From Table 2 in Appendix A the results indicate that employing graph learning, TQ-SA, and TTL significantly improves model performance which demonstrate the effectiveness of the TEN-DM model architecture. For instance, (i) removing graph learning leads
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to Euclidean distance 0.24% and 0.26% increases JPN earthquake and COVID-19 datasets spectively; (ii) removing TQ-SA leads to Euclidean distance 1.28% increases on COVID-19; (iii) removing TTL severely limits TEN-DM's ability to capture spatio-temporal topologiaal informa- tion, i.e., leading to RMSE 4.60% and 3.03% increases on COVID-19 and 311 service datasets respectively.
CONCLUSION
We present TEN-DM, novel diffusion model that leverages geometric and topological learning frameworks to capture dynamic local and global spatio-temporal dependencies for STPP forecast- ing. Integration of the GCL, TQ-SA, and TTL modules enhances the diffusion model's ability to learn complex spatio-temporal interactions, periodic patterns, and local and global dynamic topo- logical information. Extensive experiments on 5 real-world datasets demonstrated the effectiveness of TEN-DM, achieving state-of-the-arr performance across diverse datasets and dimensions. Our work establishes new direction in STPP forecasting by highlighting the potential of geometric and topological DI. in capturing intricate temporal and spatial dependencies.
ACKNOWLEDGMENTS
The authors would like to express their gratitude to the AC and Reviewers for providing construc- tive and thought provoking feedback. This work was supported in part by NSF Awards TIP/ITE- 2333703, BAC-2530471/2530442 and DMS 2533984/2533985, of Energy (DOE), Office of Science, Advanced Scientific Computing Research (ASCR) program un- der the Scientifie Discovery through Advanced Computing (SciDAC) Institute "LEADS: LEarning- Accelerated Domain Science".
well by the U.S. Department
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THE USE OF LARGE LANGUAGE MODELS (LLMS)
We only use large language models (LLMs) for rephrasing sentences and correcting grammar. important note that the LLMs not involved in the data analysis, research methodology. or experimental design.
A
MORE DETAILS
Cubical complex. An elementary interval a closed subset of R of the form [z, 2 1] with & E Z, together with the degenerate interval [z, 2| (i.e., point). An elementary cube is any Cartesian product
las where d is the dimension of the space; for each we have I [zj, 2j + 1] with 2g E Z. The dimension of Q is:
Q J
dim Q = # (1...., I is nondegenerate ([zj, 2j + 1])}
A cubical complex X is finite collection of such axis- elementary cubes in some R that satisfies: if Q E X and F obtained from Q by replaeing one or more nondegenerate factors [zj, 2j + 1] with endpoint {=y} or {=j + 1), then F E Xx; for any Q, Q E H, the intersection Q'Q' is either empty or a (possibly degenerate) common face of both.
Persistent homology. By utilizing a multi-scale approach to shape description, PH addresses the intrinsic limitations of classical homology and allows for the retrieval shape patterns that tend to persist over multiple scales and, hence, are likelier to play an important role for given downstream task. The main idea is to select some suitable scale parameters a and then to assess changes in shape (or more formally homology) that occur Specifically, let f be a filtration function that maps every simplex to the maximum function value of its vertices (in our case the grayscale value) and let m. f--((--o,, a)), a E RR. Setting increasing sequence of similariy thresholds a, i.c., ag < a1 < ... < ah sub-images are generated in a nested sequence of cubical complexes (and their connected components, loops, and voids are recorded), i.e., mao C Mas C .... Mai and we can construct the corresponding sequence of complexes, i.e., Kap C Kaa, C C... C Has which are referred as the lower-star filtered cubical complexes. Based on the evolution of these simplicial complexes through a sequence of thresholds, the homology groups are induced as (Hp(Hoo)) H (,,,,),. ...,, H (HH..)) } where H,,)),) represents the p-th homology group of May.
image m, which evolves with respect to
Zigzag persistence image generation pipeline. Figure illustrate zigzag persistence image gener- ation pipeline.
00:00-23:59
00:00-05:59
06:00-11:59
12:00-17:59
18:00-23:59
Pr
Po
Pl
00:00-03:39
04:00-07:59
08:0011:599
12:00-1:99
16:0018:59
20:00-2369
Figure 2: Pipeline for generating ZPI with different time scales.
Datasets. In our experiments, we use five real-world STPP datasets, i.e., Japan Earthquake: Earth- quake with a magnitude of least 2.5 in Japan from 1990 to 2020 were sourced from the U.S. Geo- logical Survey ! It contains 91,897 events in total. Sequences are formed using sliding windows of 30 days. The dataset is partitioned into nonoverlapping splits containing 950 training sequences, 50 validation sequences, and 50 test sequences. Sequence lengths range from 22 to 554. COVID-19:
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COVID-19 dataset is collected from publicly released COVIDI9 cases in New Jersey state from March 2020 to July 2020 by The New York Time:" This dataset includes 161,307 recorded cases, aggregated at the county level with a 7-day window size. It is split without overlap into 1450 train- ing sequences, 100 validation sequences, and 100 test sequences. Sequence lengths range from 5 to 287. US Earthquake: The US earthquake datasef contains 9,451 earthquake occurrences from 12/27/2023 to 01/26/2024 in US, and is divided into 21 training sequences, 5 validation sequences, and 5 testing sequences. Sequences lengths range from 5 to 512. Theft: The theft data is collected by the Philadelphia police department (PPD/jand we select 11 11,405 cases from the most recent theft incidents from 01/01/2025 04/30/2025 in Philadelphia. We use 24-hour sliding windows to form the dataset that contains 100 training sequences, 9 validation sequences, and 9 testing sequences. Sequence lengths range from 6 to 138. 311 Service: Similar to theft data, we collect Philadelphia 311 service dataset from OpenDataPhilly In this work, we focus on 9,597 illegal dumping reports from 01/01/2025- 06/30/2025, and split the dataset into disjoint training, validation, and test sets with 151, 15, and 15 sequences respectively. Sequence lengths range from 9 to 111.
We evaluate methods that model continuous spatial density function. A learned parameterized conditional kernel density estimator (KDE) (Chen et al.) 2018) that models p(z[t) as a Gaussion mixture model conditioned on historical events. Continuous normalizing flow (CNF) (Chen al., 2018) defines the invertible flow neural ODE and learns a continuous prob- ability density over space. Time-varing Continuous normalizing flow (TVCNF) (Chen et al.] 2018) extends CNF by making the flow dynamics dependent on timestamps. TPP Baselines: Homoge- neous Poisson process (Kingman| 1992) models the probability of an event as proportional to the time interval length. Hawkes Process (Hawkes) 197T) a self-exciting process where the historical occurrence of events can positively influences the probability of future event occurrence. Self- correcting process (Isham & Westcott [179) is opposite to self-exciting processes, which means historical occurrence of events can decrease the probability of future event occurrence and the in- tensity is negatively influenced upon a new event happened. The Recurrent Marked Temporal Point Process (RMTPP) (Du et al., 2016) simultancously model the event timings and the markers by viewing the intensity function as a nonlinear one and apply a recurrent neural network to embed the event history. Neural Hawkes Process (NHP) (Mei & Eisner 2017) uses neurally self-modulating multivariate point process where the intensities of each event type change by a novel LSTM. Trans- former Hawkes Process (THP) (Zuo et al. 2020) replaces RNNs with self-attention mechanism to better capture long-term dependencies and keeps computational efficiency. Self-Attentive Hawkes Process (SAHP) (Zhang et al. 2020) also uses self-attention mechanism and modify positional en- coding so that time intervals become phase shifts in sinusoidal encoding, which improves the usual "order-only" encoding. Log Normal Mixture model (LogNormMix) (Shchur al.] [2019) learns probability density by log-normal mixture model. Wasserstein GAN (WGAN) (Xiao et al. 2017) transforms nuisance processes to a target one, providing an intensity-free approach for point pro- cesses modeling. Neural Jump-Diffusion Temporal Point Process (NJDTPP) (Zhang et al.) 2024) which formulates a neural Sump-diffusion SDE with neural parameterizations of the drift, diffu- sion, and jump coefficient functions. STPP Baselines: Neural Jump Stochastic Differential Equa- tions (NJSDE) (Jia & Benson 2019) learns hybrid continuous discrete dynamics and generates a latent trajectory to model the temporal point processes. The spatial distribu- tion is modeled with a Gaussian mixture model. Neural Spatio-temporal Point Process (NSTPP) (Chen al.) 2020) proposed two novel neural architectures which adds event-time updates and attention for long histories non-trivially. Deep Spatio-tempores Point Process (DeepSTPP) (Zhou) let al.) (022)) proposes a nonparametrie space-time intensity governed by neural latent process. Spatio-temporal Diffusion Point Processes (DSTPP) (Yuan et al., (023b) uses conditional diffusion that learns the joint distribution of next event's time and location.
Baselines. SPP Baselines:
Implementation Details. We run our experiments on 4 NVIDIA RTX A5000 GPU cards with 24GB memory. Optimization uses Adam (dd, = 0.9, 3, 0.99) and a learning rate warm-up from 0 to a peak selected from 3 -4) followed by linear decay to 55 for 1000 epochs. Training and sampling timesteps are selected from (200, 500). We tune the batch size over (32, 64). We consider three losses applied to the diffusion objective: e loss, Eg loss, and Euclidean loss. The length of the learnable TQ vectors is selected from {7, 24, 303. The number of attention heads is selected from
"hatp::/githuu.cmmm// http://www....oo/o////o///////da
https://arrthquakc.uss.....u wttp:///wwww...../////////o/caaa 10/csv.ppp
hhtp://opendauuphily..rrg
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(2,3.4) 4). For ZPI, we set the grid size to be 50 50. For graph pretraining, learn per-sequence graph embeddings with a graph auto-encoder (GAE) (Kipf Welling 2016) with graph attention network (GAT) (Velčkkoićć al. 2017) encoder trained each graph independently. Optimization uses Adam with learning rate of 0.01 over 400 epochs. Evaluation metrics report spatial Euclidean distance and temporal RMSE 3 runs with different random seeds.
Abltioon studies.
Table 2: Ablation studies.
Farthquake
COVID
Model
Spatial
Tempporal
Spatbal
Tempural
Spatial
Tempural
Spatial
Temporal
TEN.DM
0.37T 0.00330.39
0.087 0.000 0.0154
0.792 0.0700 0.0001 0.363 16.077
Wo Graph
000.--00000000332200000000 392 0.000
088=0.001 0.054 0003 O 798+0 0 3.070 +0.000 0.391 +0.015
TQ-SA
6.663=0.051 0.3733+0.00 0.396=0.004
0.0880000000 000000000000000000 0548=0 0001 0.807 000 0.072200 0001 0.374=0.020
W/o TTL
6.6633+0.0270.3222222 000.0000...2200000.00.0...0.. 0.003 0. 0.091=0.000
0.0549+0 0002
6-8
06+-0.004 004 0.0701 +0.0001
0.425+0.001
Computational complexity. The computational complexity of GNN with 1 layers is O(ElE) E di) where & denotes the total number of gradient descent. The computational com- plexity of cubical zigzag persistence for connected components and is O(U log" U log U) and OU log" n + U log U + n log where
Σ.T A,, n denotes pixels per frame, and T the number of frames. As shown in Table 3 we also report the running time (training time per epoch) of our TEN-DM model on all 5 datasets.
Table 3: Running time (in seconds (s)) per epoch.
Model TEN-DM
JPN Earthquake COVID-19
US Earthquake
Theft
311 Service
14.096
22.873
1.846
5.982
4.570
B PROOFS
Proof of Theorem 3.2
Proof: The bound [m( m S implies, for every and a,
x(1) Hater
2111) x(()
(1,++1) and the reverse inclusion with tildes and non-tildes swapped. Objectwise, these inclusions assemble into morphisms of zigzags
and likewise for bridge nodes: X. (1.++1) Xa+s
La X. Eate, which commute with all internal arrows. Moreover, on each node,
Ja X. Xater
Ho Hate - Hotz. Ho - Hoto 4 Hat2er so (sh; T) JJ la and (sh: J) ola lao the canonical 2e-shift inclusions of zigzags. Applying Hp(-;k) yields natural transformations
Φ. HTT(a)) V. Vater V. Hp(Ja): V. Vater
satisfying (sh. 1) Va = na and (sh. v) &a = Tec Hence V and V are s-interleaved, so (V. V) I e. Since Ω is finite, the modules are pointwise fiite-dimenniionl and constructible; algebraic stability for zigzag persistence then gives d (ZPD,, ZPD, S dv V e.
Theorem B.1 (Lipschitz bound for topology-guided multi-head attention) Let z be topology embeddings and ÿ be concatenated spatial, temporal, and graph embeddings. Consider a single multi-head attention block with h heads,
TST-MHA(2, F) = <n (Sottmax Qn(*) (1)
v(()))
WO
Vdk
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where Qh. Kr: Vn, W are linear maps with operator norms bounded by MQ. MK, My. Mo. and logits are uniformly bounded by B (e.g., via clipping or LayerNorm). Then the map (2, r) + TST-MHA(Z, F) is jointly Lipschitz:
[TST-MHA(Z, r) THA f) S L (
with L - =c(h, B) Mo (Mq + MK + Mv) for an explicit c(h, B) depending only on the number of heads and the softmax Lipschitz constant on a B B-bounded domain.
Proof: For head h E {I...., H} define
(2, = Softmax (L), PM(2, (2 (f). LA (2, = (z) KT)
vds
Then TST-MHA(2, F)
WO (# (n W). Let S₁ Sn f'), , Pk Ph(Ee, p). By
addition/subtractionn,
Pr P = (S, 5,) Vir + So (V.(r) - v((r)))
Assume the logits uniformly bounded, i.e., L (z, bo [L F')le I B (e.g., via Layer- Norm/clipping). The row-wise softmax Lipschitz on this B bounded set: for some Loott (B) > 0,
S - S s Loon (B) (2. r) ( + Il.
Moreover each St is row-stochastic with entries controlled by B, hence ||Sall2 S CB( for some (B(
With operator norm bounds |[Quilop S Mo. || S Mk. VVVillop S My and submultiplicativ- ity,
Mr (12-~"1e+|1i=>>1) [V S My My 1
(x -x
(2
Vdx
Also (F' S c(B) My for harmless constant c(B) (absorbing input-norm control due to normalization).
Applying these bounds to Pn - PP = (Sh -% (F) + (V.(F) - VV))),
Pr lrll - ||S l (F))22
||51|12 (f) V.
Ma (12-
S cB Mv Loon (B)
2'1 F + || flle) Cs(B) Mv || -flle.
vdk
Hence, for constant ei (B) depending only on B,
Pr PXIpp s (B) (Mo + Mv) (11= E'(lr + 1|8- f'(l))
Concatenation across heads gives
H Pr PP) F S VH max Pr. Pr.PPille S VHei(B) (MoMx.MMMM)))) =12-=111 -
Finally, multiplying by wo with | - Mo.
TST-MHA(E, TST-MHAA f) S Mo v ci (B) MM (1 [|==2"|=+(liã+111
Using AM-GM to bound Mo My I C (MQ Mr Mv) and absorbing numeric factors into c(H, B) /H e, (B) yields the stated form
(F S c(H, Mo (MQ+ ( [| ||++11 1-111-1'11 F'lle)
(SSTMHAA(E
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C ADDITIONAL EXPERIMENTS AND RUNNING TIME
C.I ADDITIONAL EXPERIMENTS ON NEW DATASETS
We have also run additional experiments on human mobility (during natural disasters) data (Wang & Taylor 2016), wildfire data (Mathur al. 2025), and Twitter data (i.c., geo-tagged Tweets from the United States from January 12 to 18, 2013)). As Table T shows. TEN-DM outperforms DSTPP (the next best competitor) on both spatial and temporal dimensions but the gains vary with respect to the dimension and type of the data. In particular, for the spatial dimension on mobility and wildfire datasets, TEN-DM achieves more substantial gains, including significant results on the mobility data (0-value ~ 0.09; with "), while the difference between TEN-DM and DSTPP on the temporal dimension is negligible.
turn, on the twitter data, TEN-DM outperforms DSTPP with highly statistically significant gain (p-value so 0.001; with ***), and yield similar performance on the spatial dimension. These phenomena suggest that TEN- -DM captures some inherent latent higher-order structural properties of STTPs that its non-topological competitors such as DSTPP cannot.
Table 4: Performance evaluation for predicting both time and space of the next event on human mobility, wildfire, and Twitter datasets.
Human Mobility
Wildiire
Twitter
Model
Spatial 1 Temporal
Spanial
Temporal
Spatial
Temporal
DPTPP 0.3330 0.0149000000 11 4241 + 0.0232 0.3602 FO 1..0000 0.0761 0.0001 0,7102 F0.013 TEN-DM
1 5.5592.1..187' 1.013011.0018 1.408410.00330 0.3600-10.000 0.0760 0.0001 0.6774.10.009
C.2 ADDITIONAL ABLATION STUDIES
We have conducted an additional ablation study on Theft data (Table 5 below). We observe that removing GNN pretraining/temporal query-guided self-attention mechanism (TQ-SA)temporal topological learning (TTL) causes performance degradation in both spatial and temporal dimen- sions. Specifically, on the temporal dimension, compared to without GNN pre-training or TTL, the performance degradation is statistically significant (*).
Table 5: Additional ablation study on Theft data.
Theft
Model
Spatial
Temporal
TEN-DM 0.070010.0001 0.36311.011
W/o Graph 40.000T +0.391+0.005
W/o 0.0002....00 0.37410.020 W/o TTL 0.0000 "0.425:10.00
To evaluate the sensitivity of topological hyperparameters. including filtration resolution, patch size, and zigzag directionality, we conducted additional ablation experiments using the 311 service dataset. As Table shoows, replacing the original zigzag persistence configuration with an alternative filtration resolution (using smaller step increment), patch size (using the doubled the original patch size), zigzag direction (using the reverse time direction). and weighted multi-scale ZIPP fusion yields only marginal performance changes across both temporal and spatial predictions: 0.7920 + 0.026 0.7928 + 0.0091 or 0.0547 + 0.0002 vs. 0.0549 + 0.0002. These results demonstrate that TEN-DM is robust to variations in topological hyperparameters, and the overall improvements from extremely small. This also suggests that our model's performance tends to be not sensitive to the specific choices of topological settings, which makes it particularly attractive for limited data availability sparse data regimes.
these modifications
We have also conducted additional experiments comparing the TEN-DM model using zigzag per- sistence versus using standard persistent homology (PH) on the 311 service dataset. We found that
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Table 6: Additional ablation study topological hyperparameters.
311 Service
Model
Spatial
Temporal
TEN-DM (original)
0.0547= 0.900000000000000000000000000000
TEN-DM with New Filtration Resolution TEN-DM with New Patch Size
0.0548==0.000 0.000000000000027222000000000000
0.0549 0.000000000000000000000000000202
TEN-DM with Another Zigzag Directionality
0.05461:0.000 (. 0.990000000000000000000070
TEN-DM with Weighted ZPI
0.0547++000001 0.8029++0.0014
TEN-DM with 7P (i.e., 0.793+00.02 on the temporal dimension) outperforms TEN-DM with ordi- nary PH (i.e., 0.800++0.000 on the temporal dimension), which can be intuitively expected. Indeed, while both ZP and PH distill inherent higher-order properties of the underlying data generating STPP, ZP captures the most essential topological properties along the temporal dimension.
On the spatial resolution, the optimal spatial resolution can be found by the cross-validation. Tables 7 shows
prediction performance comparison between 256 x 256 tiles (i.e., TEN-DM) and 64 x 64 tiles (i.e., TEN-DM with new resolution). Overall, we have found that the results stable with respect to the considered resolutions, which indicates the overall robustness of TEN-DM.
Table 7: Additional ablation study on the spatial resolution.
311 Service
Model
Spatial 0.054710.0002 0.0200
Temporal |
TEN-DM (original)
TEN-DM with New Resolution 0.0548:10.0002 O. 8082
C.3 SENSITIVE ANALYSIS
In the datasets we consider, we have 0.04% of such events in the 311 Service and such events in JPN Earthquake, US Earthquake, and COVID-19. The model performance for the 311 data with or without the restriction on no duplicate events essentially the same, while the standard errors become smaller (see Table & below).
Table 8: Additional experiments on 311 service data (with and without duplicated events).
311 Service
Model
Spatial
Temporal
TEN-DM w/o Duplicated Events 0.0547+0.0001 0.0020-0.013 TEN-DM
0.054740.0002 0.790
To find the threshold c, we the standard cross-validation argument. In the considered case stud- ies, we have found that our results are overall stable with respect to the selected (see Table 9 with over the set 10.1, 0.2, 0.3, 0.4, 0.5]). However, we agree with the Reviewer that it may be prob- lem in general. Following the line of research on information criteria, we also suggest to consider tradeoff between the model performance and network sparsity during the cross-validation, with a penalty associated for the increased computational costs. Alternatively, optimal e can be selected as Bayesian prior (i.e., "expert knowledge"). Furthermore, the aggregation weights a,, can be con- figured as fixed hyperparameters optimized as trainable parameters during model training. The weights a, will be updated adaptively, when treated as learnable parameters. We have conducted new experiments by adding an adjacency matrix with another threshold and set to be trainable.
The results on the 311 data (Table indicate that adding an additional adjacency matrix and setting a, to be trainable dimension, the updated model's performance a bit worse than the previous version, however, is still better than other baselines.
improve prediction performance in the temporal dimension. In the spatial
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Table 9: Additional experiments on 311 service data with different
311 Service
Model
Spatial 0.0551=0..000 0.598
Temporal
TEN-DM with adaptive a,
TEN-DM with = 0.1
0.0547=0.0002 0.7920::000006
TEN-DM with = 0.2
0.0555=0.0002 0.8206=0.0222
TEN-DM with = 0..
0.0550=+0.0001 0.8176:10.0205
TEN-DM with 0.4
0.0552++00000 0.8307++0.0011
TEN-DM with = 0.5
0.0551 0.00055
To explore this hypothesis our TEN-DM, i.e. useful under the conditions of noisy, we randomly select 5% of the latitude and longitude coordinate pairs in the test set of the 311 service data and perturb them by adding Gaussian noise (with μ = O and a = 0.01). As Table T0 demonstrates, TEN-DM outperforms DSTPP both spatial and temporal dimensions. Specifically, temporal dimension, TEN-DM achieves statistically significant performance (with p-value se 0.02). In fact, this surprising as TEN-DM is equipped with the most essential higher order properties distilled via ZP.
the
Table 10: Robustness analysis 311 service data.
311 Service
Model
Spatial
Temporal
DSTPP 0.8200=0..033 0.055010.0002 DEN-DDM0.78791222 0.0040:10.0001
C.4 RUNNING TIME OF ZPI GENERATION
Remark C.1 (Running Time of ZPI Generation). We generated the ZPI representations offline on a local CPU machine (Apple M4 Pro with 24GB memory). For instance, when working on images with a resolution of 20 x 20 and sequence length of 1, the average processing time of 0.00040 seconds per event. When the sequence length increases to (with the same resolution), the average processing time of 0.00046 seconds per event. For images with resolution of 50 x 50 and sequence length of 1, the average processing time of 0.0013 seconds per event.
21


Reviews:
[{"review_id": "kl72bsgome", "reviewer": "Reviewer_rwpn", "summary": "The paper introduces Topology-ENhanced Diffusion Model (TEN-DM) for predicting future events in spatio-temporal point process (STPP) data.\n\nThe central problem the authors address is that existing methods often fail to capture the complex, higher-order dependencies between the spatial and temporal dimensions. Their solution is a conditional diffusion model where the conditioning signal is a sophisticated, fused representation of the event history. It fuses Spatio-Temporal Graph, Spatial information, temporal information and Topological Learning (converts the STPP data into a time-series of 2D images). This fused embedding guides the diffusion model’s denoising process to accurately predict both the time and location of the next event.\n\nExperiments on five real-world datasets (e.g., earthquakes, crime, COVID-19) show that TEN-DM achieves state-of-the-art performance", "questions": "Could the author please clarify the rasterization process in Section 3.2?\n\nWhy did you choose the image-based cubical complex representation?\n\nIn Section 3.1, how are the adjacency matrix aggregation weights $\\{\\alpha_r\\}$ \"updated adaptively\"? Are these learnable parameters optimized end-to-end, or set via a separate process?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The motivation is straightforward: conditioning a spatio-temporal generative model on more available information generally improves performance. The approach proves highly effective, achieving top results across five diverse real-world datasets and outperforming 17 baselines.", "weaknesses": "The proposed pipeline is exceptionally complex. It involves GNN pre-training, data-to-image conversion, multi-scale cubical zigzag persistence computation (which is notoriously expensive), a CNN on persistence images, and a conditional diffusion model. This complexity may make the model impractical.\n\nThe conversion of event data to a 2D image (Section 3.2) is a critical step, but it is underspecified and potentially lossy. The paper states, \"we rasterize the events geo-coordinates onto the 2D image by recording as each pixel's value the associated temporal attribute.\" But What happens if two or more events fall into the same grid and same time patch? How to choose the grid resolution?\n\nThe GCL module feels less developed than the TTL module. The graph is described as a similarity-based $\\epsilon$-graph (using cosine similarity), which is known to be highly sensitive to the choice of the threshold $\\epsilon$ (or $\\mathbb{R}^r$ in the paper). The mechanism by which the aggregation weights $\\{\\alpha_r\\}$ are \"updated adaptively\" is not explained", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6bPYQbAd05", "reviewer": "Reviewer_Wz4A", "summary": "This paper presents TEN-DM, a Topology-Enhanced Diffusion Model for spatio-temporal point process (STPP) prediction. The authors aim to address the limitations of existing deep learning models in capturing complex, non-stationary spatio-temporal dependencies, especially under sparse and noisy conditions. TEN-DM introduces three core components: (i) a graph construction and learning module to model event interactions, (ii) a temporal topological learning (TTL) framework based on zigzag persistence to extract dynamic topological features from time-series images, and (iii) a temporal query-guided self-attention (TQ-SA) mechanism to capture periodic patterns. The model is evaluated on five real-world datasets and outperforms the baselines in both spatial and temporal prediction tasks.", "questions": "- Scalability: How does TEN-DM scale to larger datasets (e.g., >1M events)? What is the time and memory complexity of ZPI generation and TTL module with respect to image resolution and sequence length?\n\n- Topological Sensitivity: How does the model performance change with different filtration functions, patch sizes, or zigzag directions? Have you tried adaptive weighting for multi-scale ZPI fusion?\n\n- Cross-domain Evaluation: Have you tested TEN-DM on non-urban, non-seismic data, such as animal movement, social media check-ins, or climate anomalies? How general is the topological assumption?\n\n- Comparison with TDA Baselines: Why not compare with other TDA-enhanced models? This would better highlight the unique value of zigzag persistence.\n\n- Real-time Forecasting: Is TEN-DM suitable for real-time or online forecasting? Can the ZPI and TTL modules be incrementally updated as new events arrive?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Originality: This is the first work to integrate zigzag persistence and diffusion models for STPP forecasting. The use of topological data analysis (TDA) in the form of zigzag persistence images (ZPI) to capture time-evolving shape patterns is novel and well-motivated. The idea of converting STPP data into image time-series and analyzing their topological evolution is creative and technically sound.\n\n- Technical Quality: The paper is mathematically rigorous, with formal definitions of cubical complexes, filtrations, and zigzag persistence. The stability theorem (Theorem 3.2) provides theoretical grounding for the robustness of ZPI under noise. The Lipschitz bound for the proposed attention mechanism (TST-MHA) further demonstrates the model’s theoretical controllability.\n\n- Clarity: Despite the complexity of the methodology, the paper is well-organized and clearly written. Each module is introduced with both intuition and formalization. Figures (e.g., Fig. 1, Fig. 2) effectively illustrate the pipeline and help readers understand the workflow.\n\n- Significance: The paper addresses a real-world problem, forecasting discrete events in space and time (e.g., earthquakes, crimes, disease outbreaks), and proposes a unified, interpretable, and robust solution. The integration of geometry, topology, and diffusion opens a new research direction in spatio-temporal modeling, especially for sparse and irregular data.", "weaknesses": "- Scalability and Efficiency Concerns: While the model is effective on small-scale datasets (e.g., ~10K events), its scalability to large-scale urban data (e.g., millions of taxi trips or tweets) is unclear. The zigzag persistence computation and image rasterization steps may become prohibitively expensive for high-resolution or long-duration data. A complexity breakdown or runtime scaling analysis is missing.\n\n- Limited Ablation on Topological Hyperparameters: The paper does not thoroughly explore the sensitivity of ZPI parameters, such as filtration resolution, patch size, or zigzag directionality. The multi-scale mixing uses fixed weights (βq = 0.25), but adaptive or learned weighting could be more effective. An ablation on these choices would strengthen the contribution.\n\n- Generalization Across Domains: Although the model is tested on five datasets, they are all from the US or Japan, and mostly urban or seismic events. There is no evaluation on human mobility, social media, or climate events, which are also common STPP scenarios. A cross-domain generalization test would better support the claim of universality.\n\n- Baseline Diversity: While 17 baselines are included, few recent graph-based or transformer-based STPP models are missing. Also, no comparison with other TDA-based methods is provided. This limits the completeness of the empirical evaluation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z3u9mQoHuk", "reviewer": "Reviewer_UFd9", "summary": "This paper proposes a TEN-DM framework for modeling Spatio-Temporal Point Processes (STPPs). The method integrates three key components: a Graph Construction and Learning (GCL) module to represent STPP data as graphs, a Temporal Topological Learning (TTL) framework using zigzag persistence to capture evolving topological features, and a Temporal Query-guided Self-Attention (TQ-SA) mechanism. The authors combine these into a diffusion model to learn complex spatio-temporal dependencies, particularly under sparse and noisy regimes. Extensive experiments on five real-world datasets are presented with promising results.", "questions": "See the weaknesses above.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Strengths:\n\n- The paper proposes several advanced paradigms, such as the graph neural networks and diffusion models, into a single framework for STPP modeling.\n\n- The paper provides a thorough experimental section, benchmarking TEN-DM against a set of baselines across multiple real-world datasets. \n\n- The proposed model is decomposed into clear, modular components, making the architecture easy to understand.", "weaknesses": "Weaknesses:\n\n- One of the main issues of this paper is the lack of motivation and problem analysis. Although there are some discussions for the choices of the key components, the justifications are somewhat insufficient. For example, the rationale for using graph abstraction is \"graph abstraction offers a flexible... framework\" and \"never been used\". This does not articulate what specific challenges or limitations exist in current STPP methods.\n\n- Similarly, the motivation for using diffusion models is that they are \"a new powerful machinery\" and have not been applied to STPPs before, which is somewhat insufficient. Thus, I suggest that to provide a theoretical analysis to show that under what conditions the proposed method is better than the previous methods.\n\n- Furthermore, it would be better if it could provide some case studies to illustrate the key increment and its intuition compared with other STTP methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)