TRIDENT: Cross-Domain Trajectory Spatio-Temporal  Representation via Distance-Preserving Triplet Learning
Authors: Guan Yi Jhang
Jeng-Chung Lien
Yu Hui-Ching
Hsu-Chao Lai
Jiun-Long Huang
Venue: ICLR 2026 Poster (2026)
Keywords: Spatiotemporal representation learning
Trajectory analysis
Cross-domain generalization
Triplet loss
Distance metric learning
self-supervised representation learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose TRIDENT, a self-supervised trajectory embedding framework that fuses spatio-temporal features with NTAP and a distance-preserving triplet loss, aligning native-space to reduce distortion and improve cross-domain retrieval.

Abstract: We present the TRIplet-based Distance-preserving Embedding Network for Trajectories (TRIDENT), a spatio-temporal representation framework for compressing and retrieving trajectories across scales, from badminton courts to large-scale urban environments. Existing methods often assume smooth, continuous motion, but real trajectories exhibit event-driven annotation, abrupt direction changes, GPS errors, irregular sampling, and domain shifts, exposing the inefficiency, limited generalization, and inability to robustly integrate temporal order with spatial sequence structure of prior models. TRIDENT addresses these challenges by combining Graph Convolutional Network (GCN) spatial embeddings with temporal features in a Dual-Attention Encoder (DAEncoder), along with a Nonlinear Tanh-Projection Attention Pooling (NTAP) module that preserves local order and robustness under noise. For metric learning, we introduce a Distance-preserving Multi-kernel Triplet Loss (DMT) to preserve pairwise spatio-temporal distances in the native feature space and their rank order within the embedding, thereby reducing geometry distortion and improving cross-domain generalization. Experiments on urban mobility and badminton datasets show that TRIDENT outperforms strong baselines in retrieval accuracy, efficiency, and cross-domain generalization. Furthermore, the learned embeddings capture spatio-temporal sequence patterns, facilitating tactical analysis of badminton rallies via silhouette-guided spectral clustering that provides more actionable insights than direct trajectory classification.

Published as a conference paper at ICLR 2026
TRIDENT: CROSS-DOMAIN TRAJECTORY SPATIO- TEMPORAL REPRESENTATION VIA DISTANCE-
PRESERVING TRIPLET LEARNING
Guan-Yi Jhang', Bang-Chng Lien', Hui-Ching Yu', Hsu-Chao L Jiun-Long Huang! National Yang Ming Chiao Tung University "hh University of Queensland *Chang Gung University {jhangguanyi.c lillian. he wh..hh...u. yu. jlhuang]engs edu.tw, jengchung. lien@gmail com, hsuchaolai@cgu.
ABSTRACT
We present the TRIplet-based Distance-preserving Embedding Network for Tra- jectories (TRIDENT), spatio-temporal representation framework for compress- ing and retrieving trajectories across scales, from badminton courts to large-scale urban environments. Existing methods often assume smooth, continuous mo- tion, but real trajectories exhibit event-driven annotation, abrupt direction changes, GPS errors, irregular sampling, and domain shifts, exposing the inefficiency, lim- ited generalization, and inability to robustly integrate temporal order with spa- tial sequence structure of prior models. TRIDENT addresses these challenges by combining Graph Convolutional Network (GCN) spatial embeddings with tem- poral features in Anal-Attention Encoder (DAEncoder), along with a Nonlincar Attention Pooling (NTAP) module that preserves local order and robustness under noise. For metric learning, we introduce a Distance-preservvin Multi-kernel Triplet Loss (DMT) preserve pairwise spatio-temporal distances in the native feature space and their rank order within the embedding, thereby reducing geometry distortion and improving cross-domain generalization. Exper- iments on urban mobility and badminton datasets show that TRIDENT outper- forms strong baselines retrieval accuracy, efficiency, and cross cross-domain gener- alization. Furthermore, the learned embeddings capture spatio-tempel temporal sequence patterns, facilitating tactical analysis of badminton rallies via silhouette- spectral clustering that provides more actionable insights than direct trajectory classification.
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INTRODUCTION
With the development of general imaging facilities and location infrastructure, large volumes of tra- jectory data are collected and used to capture human or vehicle movements (Hsu et al. 2019 Zheng
2015). which results in various research emerging in trajectory analysis (Chang
al. 2023a; Si
et al. 2024). Trajectory clustering a core task in trajectory analysis. Its aim is to discover similar trajectories, which facilitate applications in hotspot detection (Fang al. 2020 movement pattern mining (Chen et al.] (201)). and abnormal activity prediction (Fernando et al. 2017). Despite this progress, challenges remain. Handcrafted distances and raw-data-based encodings the point, seg- ment, or polyline level struggle to generalize across datasets with different lengths, sampling rates, and motion patterns, and they trade off local spatial detail against global trajectory shape (c.g., EDR vs. Hausdorff). In contrast, deep learning approaches learn task-specific representations directly from data, which reduces dependence on predefined metrics and can improve generalization across dataset types. Furthermore, existing similarity measures either focus solely on local spatial depen- dencies (point-based) or emphasize only global relationships between trajectories (shape-based), which makes it challenging to capture both levels simultaneously. Finally, trajectory data often suf- fer from measurement and sampling issues that distort the paths and their derived features, which from GPS
in turn degrades similarity estimation and clustering quality. When trajectories sampled coordinates (Zheng) 2013). urban canyons weaken positioning signals and introduce large emors. If the sampling rate is low irregular, the few available points cannot faithfully represent

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the true movement Li et al. 2018). When trajectories come from marker based video annotation in Wang et al. (2023b). differences in marker placement such as left or right foot and forefoot or heel lead to inter annotator variability and additional noise [Wang et al. 2023a).
In time sensitive applications such as traffic planning TTra et al. 2020p and surveillance (Yu et al,
2019), trajectory analysis often involves continuous GPS traces (see Section 2 In space sensitive applications such as movement pattern mining (Chang et al.] 2023a Lucey et al. 2013). the tra-
jectories of badminton players (Wang al., 2023a) are discrete traces (see Section
2 We discuss these two settings together because they represent complementary challenges. Most prior work fo-
cuses on smooth driving data (Si al. 2024; Fang et al. [022] Chang et al., 2023b Zhang al., 2020 Yao et al. (019), while sports trajectories remain under explored. To provide foundational understanding. we present three figures that highlight key movement differences. Fig. Ta and Tb depict taxi trajectories collected in Beijing and Rome, each vehicle is equipped with GPS device that samples and transmits location data (latitude and longitude) at 7-seconds intervals, yielding low-curvature trajectories. In contrast, Fig. Ic illustrates badminton players' movements at hit mo- ments. Rapid, nonlinear motion patterns and high-frequency shifts pose challenges to modeling. The state-of-the-an (SOTA) trajectory clustering models fail to capture the underlying patterns in the Badminton dataset, revealing a gap that has limited progress on this topic despite the existence of such movement patterns (Appendix A.T). Currently, we encounter several challenges:
(a) T-Drive datasct
(b) Rome dataset
(c) Badminton dataset
Figure 1: Trajectory comparison among different datasets.
(1) Scalable Unified Trajectory Representation across Sampling Regimes and Movement Styles: Self-supervised modeling of badminton trajectories is challenging. Many trajectory models target densely sampled and smooth GPS traces, whereas research on badminton trajectories remains scarce due to their discrete characteristies. We use single unified architecture that can be trained on different data types. This avoids redesigning distances and model components when data collection shifts from continuous event-driven annotation. It also keeps nearest neighbor queries consistent within each dataset when sampling density or annotation style varies. Many emphasize only one aspect raw coordinates representation, which limits tactical insight and reduces robustness.
(2) Complexity of Trajectory Clustering: Badminton trajectories combine discrete shot locations and continuous player coordinates with sequence dependencies. This combination increases the complexity of trajectory clustering, as traditional methods struggle to integrate spatial and temporal sequence attributes (Hamdi al. 2022). Taxi trajectories suffer from GPS errors and interruptions, which induce drift and jumps that persist after map matching. Variable and missing sampling yields trajectories that are sparse overly dense, hindering alignment. Device heterogeneity and cross-city variations in network density, traffic restrictions, and signal control lead to distinct distributions.
To address these challenges, we integrate multiple dataset into the TRIplet-based Distance- preserving Embedding Network for Trajectories (TRIDENT) to evaluate its effectiveness in dis- tinct trajectory analysis. We use a Graph Convolutional NetWork(GCN) to capture spatial and Dat))Vec(D2V) to capture temporal features. Using Dual-Attention Encoder (DAEncoder)), sequentially fuse temporal and spatial embeddings while preserving their cross cross-domain interac- tions and frequency-domain dependencies. In addition, introduce the Nonlinear Tanh- Projettroo Attention Pooling (NTAP). By capturing higher-order nonlincar interactions and dynamically reweighting subsequences through learnable context vector, NTAP resolves the discrete event-
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driven structure commonly found in badminton trajectories, enabling consistent representation. We further propose Distance-present prioritizes trajectories of varying lengths and uses a multi kernel mechanism accommodate di- verse characteristics and scales across discrete and continuous feature spaces. The geometry is induced by the distances among the anchor, positive, and negative samples, directly tying the loss to the triplet objective, which we call TRIDENT. Our contributions are summarized as follows:
Multi-kernel Triplet Loss (DMT), a geometry-awart loss that
We propose a data driven approach for unified trajectory mod- eling and self-supervised clustering. To the best of our knowledge this is the first unified representa- tion that can learn either continuous movement trajectories or discrete action sequence trajectories, and it achieves superior performance. The method reduces the bias toward continuous trajectory models, exemplified by smooth taxi trajectories, and works well on Badminton data, closing a key literature gap. Validation spans selected heterogeneous datasets, covering different task (badminton and taxi) to different domain datasets (Rome and T-Drive), which further tests robustness under data drift.
(1) Generalist Trajectory Learning:
(2) Geometry Preserving Representation Learning: We present a representation learning frame- work that unifies triplet objectives with loss design while preserving spatial and temporal geom- etry. The method anchors the distances between the positive and negative sample, combine with DMT loss to leam key information and structure from the negative similar and the most similar targets. We apply temporal embedding distance to preserve the desired pairwise temporal relations. To our knowledge we the first preserve the geometry of the original data features within the embedding space during training. Our method achieves 34.8% and 72.4% reduction in total train- ing time compared to SOTA on Badminton and T-Drive. The average query latency 8.3% lower than SOTA and remains far below the original TP. Across three datasets our method improves over the SOTA by 127.8%, 11.2%, and 10.9%. All results are obtained on the same device, and details are provided in Appendix D.6 After spatiotemporal convergence the learned representations yield clusters and analyses with meaningful class separation and stronger representativeness (Fig. and Fig. 4T.
(3) Sequence Aware Spatiotemporal Integration: Most existing trajectory models consider spatial features and sometimes temporal attributes, but rarely account for sequence dependencies, limiting their applicability to event-driven sports trajectories. Badminton movement players often return to center position, producing fragmented trajectory sequences (Wang et al. (2023a) and Fig. 77. As a result, models relying solely on average pooling or linear attention tend to miss cross-segment dependencies. To address discrete sports actions, NTAP captures higher-order feature interactions through a nonlinear projection, and employs a learnable context vector to dynamically reweight different segments. Through this sequence-aware integration, event-driven sports trajectories can achieve representation quality comparable to continuous GPS trajectories. The method overcomes the earlier focus on smooth continuous trajectory models, and achieves strong results on Badminton data, thereby filling an important gap in the literature.
2 PRELIMINARIES
Unified Definition. We define a trajectory T= (81, S2....,,. as a sequence 2D locations over time. A sequence s; (loc,, t.) includes the 2D location loc; (I,, (x4,34) the coordinates and the observed time t; of the location. We model the trajectory in trajectory network as a directed graph G (L, F), where is the set of trajectory vertices and E is the set of directed edges corresponding to feasible trajectory segments. Each vertex f E denotes a location in which its coordinate is (z,, v.). An edge eij (4,, E E models a trajectory segment from & to Ey.
Trajectory Characteristics. Different trajectory sources lie along a spectrum of properties and aren't 't limited, as the examples shown below:
Urban Mobility: constrained by spatial topology, continuous, low tortuosity.
Basketball or Soccer: no spatial topology constraints, continuous, moderate tortuosity.
Badminton or Tennis: spatial topology constraints, discrete, moderate-high tortuosity.
We define the characteristics of discrete trajectory are being (1) event-triggered, (2) returning to reset position (Wang al. 2023a), and (3) following an episodic temporal structure. In contrast,
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Spatio
Weights
Eg. Baciminton
Other Analysis
Clustering
GCN
DAEncoder
NTAP
Trajectory
Trajectories
Spatto-lamppral
Represemalion
Anchor
Temporal
Representation
DMT Loss
NTAP
Scallo tamporal Represemtatinn
Embedding
Training
Spato
Temporal
Figure Overview TRIDENT framework.
the characteristics of a continuous trajectory are (1) continuous, (2) no-reset, and (3) constantly adjusted behavior. Furthermore, utilize Average CCR defined in Yu al. (2025) quantify tortuosity. A higher Average CCR indicates greater tortuosity. In our datasets, urban mobility exhibits relatively low tortuosity, whereas badminton exhibits high tortuosity (see Appendix D.2 for details). The distinction between different trajectories is non-binary and arises from their inher- ent behavioral organization and control mechanisms, rather than from differences in sampling rate. Therefore, we choose urban mobility and badminton as two extreme cases.
Urban Mobility. For urban mobility, a taxi with GPS as an example. GPS records latitude and longitude over time, where z; is the latitude, Y: is the longitude, and f; is the timestamp of the GPS reading.
Badminton Movement. The half-court layout is shown in Fig. as is the range of movement of the player instead the full court (6.1 13.4 m). Each half-court measures 6.1 x 6.7 m, and the spatial unit is 0.1 m. The coordinate origin (0. 0) at the lower left corner and the axis extending to (61, 0) and (0, 67). To avoid duplicate patterns in different absolute locations, for any location (a,, y) with y > 67, the mapped location is (2 (2-30.5 as a stroke-by- location of the half court where the player plays the stroke the stroke index t, Additionally, we sequentially index all integer grid coordinates in the standardized half court as vertices (61 67 vertices). Therefore, the vertex is instead n rloo.)). where IR2 I. denote the nearest-vertex projection.
2-67-1 107-20 y). Player movements are captured sequence, ordered by the stroke index within each rally. As (I; ys) is the
Learning Target. Before learning, need choose the similarity measure learning target. There are many measures available, such as Yi et al. (199) for spatial metrics, TP /hang et al. (017). DITA Shang et al. 2018), and (CRS|Yuan & Li (2019) for more advanced trajectory-based metrics. In this study, we chose TP as the main similarity measure. The reasons are: (1) Less Emphasis on Sequence Order, (2) Considers Both Spatial and Temporal Dimensions, and (3)
Nearest Neighbor Matching (see Appendix D.5 for details).
METHOD
Our method consists of (1) a hyperparameter-free training strategy for constructing informative triplets, (2) a Distance-preserving Multi-kernel Triplet Loss (DMT) that aligns embedding spacee distances with native trajectory-space distances across multiple scales, and (3) a Dual-Attention En- coder (DAFnccdder) equipped with Nonlinear Y'anh-Projection Attention Pooling (NTAP) for mod- cling both continuous and discrete trajectories.
Three limitations motivate our framework: (1) classic contrastive and triplet losses enforce relative ranking but not absolute geometric constraints, often causing dimension class collapse (Jing al. 2022 Xuan et al. 2020); (2) classic contrastive and triplet losses rely on sensitive hy hyprraarmeeters like margins, temperatures, positive/negative ratios, and mining strategies (He et al. 2020 Schroffi ct al. 2013), triplet diversity or data augmentation (Chen et al., 2020), making training unstable and distorting the underlying geometry; (3) Previous trajectory representation methods handle continu-

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trajectories well, but are less effective for discrete trajectories. Our design is guided by three principles:
Preserve intrinsic geometry:
Distances in the embedding space must respect distances in the native trajectory space at both local and global scales, preventing collapse.
Eliminate hyperparameter sensitivity: Triplet construction and the loss should operate without margins, temperatures, or mining heuristics.
Maintain structural stability across trajectory types: ous and discrete trajectories using the same architecture.
The system must handle continu-
Below we describe the training strategy, loss formulation, and encoder architecture.
3.1 TRAINING STRATEGY
We leverage similarity measure TP to construct triplets in self-supervised fashion. For each anchor trajectory, we identify its most similar trajectory as the positive example. This captures the strongest local structural relationship while avoiding oversmoothing due to multiple positives. Negatives are drawn randomly from trajectories that are not the anchor nor its top-1 positive. This supplies unbiased global diversity without hard-negative mining.
This top-1 positive + random negative strategy reflects two motivations: (1) local identity is defined by the closest neighbor, and mutual top-1 relationships already imply strong constraint without artificially compressing distances; (2) global geometry is learned through diverse negatives without requiring the user specify ceaddive-to-postiivv ratio.
3.2 DISTANCE--PRESERVII MULTI-KERNEL TRIPLET LOss(DMT)
A limitation of classic triplet loss even with margin is that it only enforces relative ordering (d dan), without constraining the magnitude of distances. This often distorts the embedding geometry. collapsing local neighborhoods, or exaggerating long-range separations, which degrades retrieval accuracy and cross-domain generalization. To mitigate this, we introduce the Distance-preservee Multi-kerner Triplet Loss (DMT), which explicitly aligns embedding space distances with ground truth native trajectory space distances, while weighting errors across multiple scales.
Aligmment. Let A', P',N' Rexdamm denote the predicted representations of anchor, positive, and negative samples of batch size Let d'p: dan E T be the predicted embedding space distances and dap, dan E R& be the ground truth trajectory-space distances. We compute the embedding and trajectory space distances as:
[d d
[dap dlan
dap = || P'112,
=1==|AA -'.=
N' d' =
d
E
2b
(1)
The objective is to align embedding space distances with ground truth native trajectory space dis- tances minimizing the squared error between d' and d as:
E (d' - d)² TRxbbx1
However, by using native squared error, the error would be dominated by large-distance deviations, which fails to control local neighborhoods and does not retain global structure. Therefore, we apply a multi-scale Gaussian reweighting that normalizes error contribution by distance scale.
Multi-kernel Gaussian weighting. To compare distances across heterogeneous scales, normal- ize the error constructing multiple Gaussian kernels. We first compute base bandwidth ohase from the batch median ground-truth distance and multiply with fixed multipliers [0.5, 1.0, 2.0) different scales of o.
(3) Small d% emphasizes accurate matching for small native distances (local neighborhoods), while larger di spreads attention to larger distances (global structure). Each kernel defines weights:
m [m...... mnIT. Tbssc = median d),
Obscc m
W exp (
d2
12 122x
20 +
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Scale-normalized residual. For each scale k, weight the squared errors E² with W
A = W E R
(5)
Then the scale specific loss is normalized as:
1 A
4 1 W +E ERR k= 1. ,......,K
(6)
so that each kernel only penalizes deviations meaningful to its own distance regime. This prevents small-scale kernels from being dominated by large distances and prevents large-scale kernels from overweighting tiny local perturbations to further improve training stability.
DMT loss. Finally, the scale specific loss averages across scales:
Litmt
ER
(7)
3.3 MODEL STRUCTURE
Each trajectory is represented by spatial embedding (Appendix C.1 and a temporal embedding (Appendix C.2) We propose Dual-Attention Encoder, which employs Nonlinear Tanh-Projection Attention Pooling mechanism to integrate these spatial and temporal representations, producing a unified representation architecture for continuous and discrete trajectories.
Dual-Attention Encoder (DAEncoder). DAEncoder adopts post schedule with two classic cross attention in parallel. Within each layer the spatial embedding attends to the temporal. then the temporal embedding attends to the updated spatial. The two directions use independent parameters. Variable lengths and missing steps handled with learned CLS token for each representation, padding masks derived from true lengths, and zeroing of padded positions after positional encoding. Each embedding later is summarized by Nonlinear Tanh-Projes Proj-Proeection Attention Pooling (NTAP), and the two summaries are concatenated to form the final representation. We do not employ gating or add, that spatial and temporal information contribute equally. The design is extensible to new attribute branches and multi-subjec trajectories that follow the same fusion and pooling pipeline.
Nonlinear Tanh-Profp Projectin Attention Pooling (NTAP). To obtain fixed-size representations from variable-length spatial or temporal sequences, we employ Nonlinear Fanh-Prooection Attention Pooling mechanism. Traditional masked attention pooling typically relies on a single linear pro- jection, which limits its capacity to capture higher-orde feature interactions. In contrast, our de- sign introduces non-linear transformation prior to computing the attention scores, leading to more expressive sequence representations for discrete trajectories and retain the same performance on continuous trajectories.
Let the input sequence be Z = (21, .2...., Zn) E Rddmm First, each feature vector is first projected by learnable matrix and passed through a tanh nonlinearity to capture nonlinear and higher-order interactions among feature dimensions, which are crucial for trajectory spatial and temporal data where dependencies are complex and dynamic, as
u = tanh W z.) E Rdam. W E Rdaamxxn xdam
(8)
The nonlinear projection u, preserves smooth variations, curved motion from continuous trajecto- ries, and highlights sharp turns, high-curvature event points from discrete trajectories.
Second, learnable context vector uw Rd captures the perspective from which the sequence should be summarized, models nonlinear interactions across dimensions to emphasize key spatial regions in the trajectory and prioritize critical time steps. Event-triggered points and return points in discrete movements are more easily emphasized. Third, with a mask score to handle variable sequence lengths, where padding masked by mask; as -o and otherwise. Lastly, with softmax as the score
a = softmar [u] u, + maski) E R, T TE R. mask; E [o.- Finally, the pooled representation that combined each feature vector according to the score as calculated
(9)
as z; E Rdu..
(10)

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4 EXPERIMENTS
4.1 EXPERIMENTS SETTING
We conduct experiments on the public Badminton, T-Drive, and Rome trajectory datasets. Parameter settings are in Appendix D.4 and full evaluation metrics are in Appendix D.4.I Each dataset split into 10,000 training trajectories, 4,000 for validation, and the remainder for testing. See Appendix D.I dataset details. Additionally, we perform experiments two larger-scale datasets, with results available in Appendix D.7
4.2 MODEL PERFORMANCE STUDY
Table 1: Framework effectiveness comparison on Badminton, T-Drive, Rome.
Badminton
T-Drive
Rome
Framework
HR @ 10 HR @5 R10@50
HR @ 10
HR @ 50
R10@50
HR@ 10
HR @ 50 R10@50
NeuTraj (ICDE)
0.009
0.036
0.041
0.177
0.222
0.270
0.123
0.217
0.261
ST2Vec (KDD)
0.073
0.178
0.212
0.496
0.593
0.844
0.479
0.598
0.830
TrajCL (ICDE)
0.084
0.243
0.197
0.241
0.369
0.359
0.094
0.293
0.260
ConDTC (IEEE)
0.009
0.021
0.047
0.139
0.348
0.343
0.091
0.279
0.323
Our method
0.190
0.347
0.484
0.564
0.661
0.916
0.535
0.664
0.913
To demonstrate the effectiveness of our model, we evaluate on three standard metrics, HR 10, HR @ 50, and R10@ 50, and compare against multiple baselines. As shown in Table [ after averag- ing performance across these three metries, our method exceeds the mean of all basclines by 27 971% on Badminton, 96% on T-Drive, and 127% on Rome. This demonstrates strong generalization across trajectories with diverse characteristics and highlights the stability of our approach. notable obser- vation is that ST2Vec and TrajCL achieve similar results on the Badminton dataset, suggesting that the contrastive positive and negative sampling strategy more adaptable to complex and rapidly changing trajectories. In contrast, the absence of anchors causes CL to underperform relative to ST2 Vee on the T-Drive and Rome datasets, which contain smoother trajectories, with a relative gap of 53.91 %. Anchors therefore crucial for continuous and smooth data, as also illustrated in Fig.
4.3 MODEL. EFFICIENCY STUDY
4.3.1 LEARNING PROCESS OF TRIDENT
To assess how the model learns trajectory representations, we used the converged model as refer- ence. We sampled 500 trajectories, computed their embeddings, clustered them via spectral cluster- ing, and fixed the resulting labels. The same trajectories were then passed through models at three stages (Initial, Middle, Convergent), and their embeddings were projected to 2D with t-Distributed Stochastic Neighbor Embedding (t-SNE) to visualize how the cluster structure evolved during train- ing. Fig. 3 shows embeddings progressing from densely overlapping clusters to partial separation and finally clearly groups. Combining triplet loss with distance regression preserves the native spatial layout and pairwise distances. In the initial Fig. 3a the pink and green groups are adjacent, are the light and dark blue groups. After optimization, the convergent Fig. 3c maintains these lo- cal neighborhoods while cleanly separating distinct populations. In contrast, alternatives such as CL and Se2Vec tend to overfit, collapsing points into single clump, which erases the original spatial structure.
4.3.2 ABLATION TESTS
reports the best performing combinations of loss functions and our encoder. The higher capacity of the Transformer made it prone to unstable or suboptimal convergence with limited data. The GRU underfit the complex temporal and spatial patterns. Despite reasonable results small and complex dataset, the BiL.TM proved insufficient. Across loss functions combined with our DA Encoder, it yielded the most stable training dynamics, and the design of the loss function was
Table 2a

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CSHE Comporent
-00
-30
-30
-3
-20
-10
(a) Initial stage
(b) Middle stage
(c) Convergent stage
SSKE Compinerez
Loke Component
-10
1o-
MS Component
+-NE Component
8-S4E Compotent
(d) Initial stage
(e) Middle stage
(f) Convergent stage
Figure 3: Visualization of learning process Badminton (a-c) and T-Drive (d-f).
Table 2: Left: loss and encoder combinations, Right: attention pooling ablation.
(a) Loss x Encoder
(b) Attention pooling
T-Drive dataset
T-Drive dataset
Loss
Encoder
HR@ 10
HR@50
R@ 10
Pooling HR@ 10 HR@50 R@ 10
DMT
Transformer
0.5038
0.6105 0.8688
None
0.5540
0.6539 0.9114
DMT
BiLSTM
0,5168
0.6118 0.8736
Std-AP 0.5506
0.6502 0.9038
DMT
GRU
0.4044
0.5298 0.7917
NTAP
0.5643
0.6610 0.9160
DMT
DAE
0,5643
0.6610
0.9160
Pairwise Logistic CL
DAE
0.5481
0.6319 0.9057
Badminton dataset
Siamese Contrastive Loss
DAE
0,2201
0.2487 0.4736
Pooling HR 10 HR@50 R@ 10
RBF-Triplet Loss
DAE
0.5094
0.6061
0.8688
None
0.1477
0.2898 0.4014
Sid-AP
0.1444
0.2805
0.3873
NTAP 0.1978
0.3596 0.5047
also critical. Table comppare models without attention. Simple pooling did not capture the salient features of the data. Our NTAP achieved the best trajectory modeling performance.
4.3.3 STABLE TESTS
Table 3: Generalization and data-efficiency evaluation (T-Drive).
(a) Cross domain
(b) Robustness to data diversity
Train
Test
Model
HR@ 10
HR
50
R 10
Reduction HR 10
HR@30
R@
T-Drive Rome
TRIDENT
0.1583
0.1934 0.3359
0%
0.5643
0.6610 0.9160
Rome
T-Drive TRIDENT
0.1642
0.1782
0.3557
10%
0.5585
0.6557
0.9031
T-Drive Rome ST2V
0.0019
0.0097
0.0099
30%
0.5438
0.6502
0.9156
Rome
T-Drive ST2V
0.1369
0.1638 0.3101
50%
0.5195
0.6230 0.8937
Our method demonstrates strong architecture-genraal capability by achieving stable performance across three distinct types of datasets (Badminton, T-Drive, and Rome) using the same model ar- chitecture and identical hyperparameter settings. To further evaluate its cross-domain behavior, additionally conduct transfer experiments across datasets: training on T-Drive and testing on Rome, and vice versa. The results show that method outperforms baselines, and in cross-city transfer it
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retains about 31% % of its original performance. Although performance drops, this is mainly due to in- trinsic differences the data rather than limitations of the model. Cities differ substantially in GPS coordinate distributions, road topology, traffic rules, and movement constraints, making trajectory distributions impossible to align directly. Furthermore, GPS trajectories and badminton trajectories follow entirely different motion mechanisms. Their spatio-temporal dynamics do not overlap, and therefore badminton is not included in this test. Overall, these additional experiments demonstrate that our framework remains competitive even under large domain gaps. Rome has longer trajectories while T-Drive trajectories are shorter, making it difficult for existing methods to evaluate between them. In contrast, our model can differentiate them, even though it was never trained on this type of data.
4.4
CASE STUDY
4.4.1
THREE TYPE OF TRAJECTORY CLUSTERING
To evaluate our trajectory clustering, visualize t-SNE learned embeddings from the Bad- minton dataset. We apply spectral clustering (k=6) and color the clusters the t-SNE plots. As shown in Fig. 4 our method preserves class cohesion and shows clearer inter-class separation than ST2Vec. Ve2 embeddings collapse into two coarse lobes, causing overlap and poor within-class resolution. For TrajCI., adding temporal features further mixes trajectories into single mass, likely because its loss lacks a triplet structure to anchor positives vs. negatives. Without an anchor that specifies which trajectories should be nearest and which should be farthest, the objective either (i) only pushes dissimilar pairs apart, or (ii) merely enforces higher similarity for positives than for neg- atives. Both cases fail to recover clean cluster boundaries. Preserving the intrinsic manifold structure also benefits downstream tasks: although our embedding appears more dispersed in t-SNE, this lay- out creates sufficient inter-cluster margins, clarifies temporal and spatial patterns, and enables more reliable clustering.
SSNE Compincre
Lobee Component
10
-30
-11
Component
50
-30
-30
-10
Component:
18
Companent:
33
(a) Visualization of ST2Vec
(b) Visualization of TrajCL.
(c) Visualization of our method
Figure Visualization of Badminton dataset different frameworks.
4.4.2 TOP-K T-DRIVE SIMILARITY QUERYING
We conducted top-k query experiments on the T-Drive dataset to evaluate the discriminative capa- bility of our framework. trajectory was randomly selected as the query trajectory. As shown in Fig. 5a the top-2 trajectories identified by our method are highly consistent with the ground truth: in particular, Our retrieved top-1 and top-2 match the ground-trut -ruth order. In addition to this random case, we also include deliberately chosen example with errors (Fig. [b). Notably, in Fig. [5b] although the retrieved nearest neighbor to the ground truth (by distance) is closest numerically, the spatial pattern exhibits hollow structure. Our method selects candidates based on global spatial consistency, thereby retrieving trajectories that are globally closer to the query.
4.4.3 K CLUSTER STRATEGY
We analyze only winner-ended rallies decided by a clean winner, excluding out, net contact, and other unforeed errors to isolate tactics and positioning. Using the embedding from TRIDENT, apply spectral clustering with = 6 (Appendix D.9). This self-supervised pipeline partitions com- plex trajectories into clusters and retrieves instances with closely matched temporal order and spatial

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Query Trajectory
Ground Top-2
Top-2
(a) Instance
Query Trajectory
Ground Truth Top-2
Our Top-2
(b) Instance
Figure 5: Top-k similarity querying (T-Drive).
positions, validating model effectiveness. The main text presents one representative cluster, with others in Appendix D.10 The cluster shows defensive failure pattern. The typical sequence is opponent net shot, net shot, opponent lift, smash, opponent net shot, losing point. We term it the net lift smash cycle. Fig. shows the three trajectories closest to the medoid, with numeric labels in- dicating movement order. Repeated front-to-back transitions let the attacker maintain initiative and disadvantage the defender. The recurrence of this sequence executed tactic rather than a clustering artifact. The result aligns with the front-to-back pressure prin- ciple, provides quantitative support for coaching knowledge. and demonstrates how the model links unlabeled trajectories to interpretable tactics, forming discovery-tor -everffcitiiiiiii loop. In contrast, naive clustering of raw time and coordinate data (or baselines lacking trajectory awareness) cannot surface patterns that simultaneously capture temporal order and continuous spatial movement. Our approach can.
the medoid indicates actually
Figure 6: Badminton strategy.
CONCLUSION
This work presents deep, self-supervised framework with a near-zero-hypernee renerrprthhy design that preserves spatial, temporal, and sequential structure during representation learning and clustering. We replace ranking losses with distance regression to align embedding distances with real-world proximity, retaining metric interpretability under compression. The model stabilizes each instance using its most informative neighbors and delineates cluster boundaries by tracking negatives. The approach captures sequential patterns in spatio-temporal data (shown on badminton singles trajec- tories). Future work includes cross-task evaluations (e.g., pedestrian trajectories, doubles rallies, basketball) and deeper interpretability.
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A APPENDIX
A.1
BANDMINTON TRAJECTORY
depict badminton singles player movement tra- jectories within three shots. The coordinate system fol- lows the same scale (1 meters per unit) and represents badminton half-court (6.1 6.7m). To extract more concentrated trajectories for better readability, we filtered specific shot sequences (Net shot Clear Smash) to ensure a structured pattern. We identified the most densely clustered location for step 1 and sampled trajec- tories within a 5-unit radius of that point, selecting ten representative trajectories. The magenta lines indicate the striking player's movement.
Fig.
Badminton Singles
Step
Step
Step
6
y axiss (m)
2
shown in Fig- S illustrates that the badminton counter strategy can be traced when the hitting location is re- stricted to limited area and the shot type remains the same. Under these conditions, there terns that can be observed in the data features. The most challenging aspect in this field has been the lack of ap- plicable model, and until recently, research focusing on mining strategies in the sports domain has been rare. The vast range of possibilities and uncertainties significantly increases the difficulty of exploration
identifiable pat-
2
4
6
axis (m)
Figure Movement patterns of consis- tent shot types and starting points in sin- gles.
5
5
axis (m)
E
axis (m)
4
3
3
>2
>2
>2
1
%
2
%
4
%
axis (m)
axis (m)
axis (m)
(a) Net shot
(b) Clear
(c) Smash
Figure Shot hit distribution in badminton singles.
B RELATED WORK
B.1 TRAJECTORY REPRESENTATION LEARNING
Trajectory clustering have introduced various techniques to enhance spatio-temporal pattern recog-
nition. Deep embedding methods that preserve temporal dependencies Yao et al. (2019) work well
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in smooth motion, but their accuracy degrades under irregular sampling, event-driven annotation, abrupt direction changes, and spatial attributes are usually not encoded jointly. Contrastive and end to end clustering approaches (Si et al., 2024 Fang et al. 202 improve discrimination, but they rely on sensitive augmentation choices and temperatures, and they optimize instance separation rather than calibrated cluster structure. Similarity oriented retrieval with learned metrics (Zhang) iet al., (2021) captures relationships defined by preset notion of similarity, but temporal order is only weakly enforced and transfer across datasets is limited. Spatially constrained similarity Yang et al. (202T) strengthens spatial matching, yet fusion with temporal order is often late, which can bias the metric toward space. Trajectory contrastive learning Chang et al. (2023b) sensitive to ordering and local window design, making long-range and multiscale dependencies difficult to learn in dynamic sports.
B.2 TRAJECTORY CLUSTERING TECHNIQUES
In prior work, trajectory clustering has played fundamental role in tasks such as trip classification (Lee et al. 2008). pattern detection (Chen et al., 2019). Its core objective is to group trajectories with similar movement characteristics that those within the same cluster are mutually similar and clearly distinet from other clusters. Traditional approaches often use raw trajectory representations (paths or discrete points), define similarity measures such as the Euclidean or Hausdorff distance, and apply clustering algorithns(k-means) Gaffney et al. (2007) constructed an EM framework based on the global Euclidean distance between full trajectories. Wang et al., (2023c) employ a two-layer isolation distribution kernel (IDK) to represent trajectories and measure similarity. This approach achieved promising results on vehicle data, but failed to capture both the discrete attributes (e.g., shot type) and the continuous attributes (e.g., event location). [Lee et al. (2007) segmented trajectories before clustering with the Hausdorff distance. However, such methods are sensitive to differing sampling rates, trajectory lengths, and noise. it may overlook crucial local features. To over- come the limitations of raw representations, deep trajectory embedding techniques have emerged. Fang et al. (2021b) introduces end-to-end deep clustering framework via self-training to address the reliance on handcrafted similarity measures and capture latent spatial dependencies. Wang et al. (2013) incorporated action labels (entry, exit, stop, move) into event detection. More recently, Yue et al. (2199) extracted points of interest (POIs) from trajectories for deep clustering. Although these deep methods reduce manual feature engineering, they still depend some extent on predefined labels or segments.
B.3 STRATEGY DISCOVERY IN TEAM SPORTS
This study focuses on temporal and spatial data, using trajectory clustering and mining to uncover previously undefined strategies. Knauf & Brefeld (2014) introduce novel spatiotemporal kernel that clusters multiple player trajectories simultaneously. This method captures strategic deviations between two teams. /aanakulasuriya et al. (2015) combines handcrafted features with the trajecto- ries of players on both teams to distinguish different characteristics associated with scoring oppor- tunities. Decroos et al. (2018) applies a similar technique to hockey identify attacking strategies that result in continuous scoring. Another line work focuses on ball movement. Although this ap- proach contributes to understanding overall team strategies, it struggles reveal fine iin-ggrined player trajectories. Trajectory mining and high-frequency spatio-temporal pattern mining both emphasize frequent patterns, though frequency does not necessarily imply strategic value.
C INPUT EMBEDDING MODULE
C. SPATIAL EMBEDDING WITH GCN
To capture both the structural information and the sequential nature of trajectories, trajectory em- bedding module that integrates graph convolutional networks (GCN) with a dedicated PAD node mechanism is used. Given the trajectory network G (L, E) defined in Section 2 each vertex E L is associated with a feature vector dg E Reda Using a GCN, the node features are encoded into the hidden space:
H GCN(X, A) RI4|xds
(11)
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where X IR A/xxd is the imput feature matrix, A is the adjacency matrix, and H is the resulting node embedding matrix.
When processing trajectories in batches, sequences of different lengths must be aligned. Conven- tional approaches pad shorter sequences with zero vectors, which makes difficult to distinguish padding positions from real nodes in the embedding space. A PAD node the node embedding matrix:
H' H: 0 R (
1)x4,
(12)
During batching, all padded positions are assigned this PAD node index, ensuring they are mantically separated from real nodes.
For a trajectory ($1, 82, ..... S,), its embedding sequence can then be obtained via direct indexing:
Ey B H H H', mxd,,
(13)
GCN encodes structural information from the trajectory network into node embeddings and the PAD node ensures that padding positions are clearly distinguished from real nodes, thereby enabling efficient batch computation without compromising semantic correctness.
C.2 TIME EMBEDDING
Temporal information is indispensable in trajectory modeling. A simple linear projection of tem- poral features into the hidden space often fails preserve fine-grained cues, such as hour-of-day, weekday, or periodic patterns. Although these raw signals are simple, they frequently carry deci- sive semantic meaning. To address this issue, we design a time embedding module with residual pathway, so that the model can learn high-level abstractions while progressively integrating raw tem- poral signals. Let each sampling point 8; (loc,, 44) consist of its location and timestamp, where f; E R denotes the temporal feature vector (e.g., date2vec encoding of the raw strole index or GPS time). We first apply a learnable linear projection:
2, = W bp-
E RRd
(14)
where Wp Rd dd and bp dd are learnable parameters. At the same time, maintains residual branch directly from the raw temporal input:
r; Wg4 + b..
(15)
with W. xdd, and by E Rd initialized zero This initialization ensures that during the early stages of training, the residual path has no effect, allowing the model to focus on learning training progresses, the parameters of the residual branch can gradually adjust, enabling the network to incorporate raw temporal cues when necessary. The final temporal embedding for step i obtained as
stable projected representations.
he Dropout( (24 ((((2(++)) +r)).
(16)
where (-)) denotes the ReLU activation. Through this design, the embedding able to capture high- level abstractions without discarding the influence of low-level raw features. When large-scale scale data available, the model benefits from learning complex temporal dynamics through the projection branch; when data scarce, the residual path ensures direct access to signals, thereby improving generalization and robustness. Preserve raw temporal cues while progressively integrating them into higher-level abstractions, ensuring stable and expressive temporal representations across data regimes.
D EXPERIMENTS
D.1 DATASET
Badminton WWagg et al., 2023a b) Movement trajectories from singles matches. After
handling missing data and removing rallies with <5 strokes, 12,667 rallies remain. Each trajectory uses the last five strokes, considering both players yields 25,334 trajectories. The test set contains 11, 11,334 trajectories.
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2011 The dataset contains approximately 15 million taxi GPS points collected from Feb to Feb 8, 2008. To enable road network- trajectory similarity analysis, raw GPS traces map-matched to OpenStreetMap and converted into time-ordered sequences of road vertices. Restricting to urban trips and filtering trajecto- ries with fewer than 10 samples leaves 29,830 trajectories. The test set contains 15,830 trajectories.
T-Drive (Yuan al. 2010
Rome (Bracciale et al. 2022) The dataset comprises 30,000 taxi trajectories from Rome, Italy, spanning over 30 days. Following the map matching and filtering procedures as applied to T-Drive. The test set contains 16,000 trajectories.
To enhance the completeness and reliability of our experiments, we incorporated two additional large-scale datasets from different cities. We conducted further training and evaluation. The details and experimental results of the added datasets are presented below and further in Appendix D.7
This dataset collects taxi GPS trajectories in Chengdu, China, during October and November 2016. The raw GPS traces are map-matched to the OpenStreetMap network and converted into time-ordered sequences of road segments. The dataset contains total of 84,251 trajectories, which are partitioned into a training set of 20,000, validation set of 15,000, and test of 49,251 trajectories.
Chengdu (DiDi Chuxing 2022
Xian (DiDi Chuxing 2022) This dataset comprises trajectories from Xian, China, span- ning the same period. Following the map-matching process to align traces with the road network, the dataset consists of 199,999 trajectories. We utilize 60,000 trajectories for training, 20,000 for validation, and the remaining 119,999 trajectories for testing.
D.2 DATASET GEOMETRY ANALYSIS
Table 4: Dataset CCR statistics.
Dataset
Count
Mean
Median
Std
Max
Badminton
24036
4.3894
3.2325
3.2921
17.4368
T-Drive
31681
1.6730
1.3136
1.0681
8.1906
Besides the mentioned discrete and continuous properties, we explain here the geometric differ- ences between the badminton and taxi trajectory datasets by analyzing Curvature Compression Ra- tio (CCR) from Yu et al. (2025). defined as the ratio of the trajectory are length to the straight-line distance between values due noise such identical starting points, apply quantile 0.95 to I truncation strat- egy, removing the top 5% outliers to preserve the intrinsic distribution of the data. The metric we use is:
start and end points. Since both datasets occasionally contain extreme CCR
N
La
Average CCR
Σ
(17)
N
D
where I, the length of the i-th segment, D; the distance between its start and end points, and N is the total number of segments.
When Average CCR the trajectory is perfectly straight.
When Average CCR > 1, the trajectory is more curved and deviates more from straight line; a larger value indicates higher tortuosity.
In Fig. 9 the badminton trajectories show significantly higher Average CCR in the histogram, with a longer tail to the right overall, reflecting higher curvature and greater diversity. In contrast, T-Drive trajectories have mean Average CCR around 1.9, consistent with the smoother movement patterns expected in continuous trajectories. This geometric difference supports the need to consider both high-tortuosity and low-tortuosity trajectory types when evaluating models.
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CCR Distriuttion
536mento 10rive
25
N002003e (6)
20
255
3.2
7.5
00.0
22.5
12.0
27.3
Average
Figure 9: CCR distributions of Badminton and T-Drive.
D.3. BASELINE
To comprehensively evaluate the effectiveness of our method in trajectory representation, we com- pare it against four representative models: ConDTC, ST2Vec, NeuTrajj, and TrajCL. These methods cover the major technical directions in current trajectory research:
ConDTC: Combines a Transformer encoder with location and temporal embeddings, to- gether with contrastive clustering, for fine-grained spatio-temporal pattern recogmition.
TrajCL: Utilizes multiple trajectory augmentation strategies, contrastive learning, and an improved dual-feature self-attention encoder, representing contrastive trajectory represen- tation learning approaches.
Employs triplet loss and spatio-temports co-attention mechanism to empha- size temporal-spatial details in similarity computation, serving as a key method for time- sensitive similarity modeling.
SeeVee:
NeuTraj: Uses RNN backbone with distance- distance-weighted ranking loss to preserve sim- ilarity order, and is often applied for approximating high-cost similarity measures such as Frechet and DTW.
These four models span different architectural paradigms (Transformer-basel) self-attention- based, and RNN-based encoders) and incorporate a range of loss functions, including contrastive, triplet, ranking, and distance-weighted ranking losses. Together, they form representative and balanced set of baselines for learning trajectory representation.
D.4 PARAMETER SETTINGS
In Our framework, the parameters are set with feature. size and embedding size = 64. Batch size, and learning rate are configured as 128, and 0.001. Training and evaluation processes were conducted on system running Python 3.10.18 .PyTorch 2.5.1 with cu121. Implementation was conducted in Tensorflow a system with an Intel(R) Core(TM) 19-14900KF CPU with an Nvidia RTX 4090 GPU. Choosing embedding size 64 avoids giving our method an unfair advantage from a larger latent space and keeps the computational cost low. For batch size and learning rate, adopted the values that produced the most stable training behavior on the T-Drive validation dataset. Batch sizes of 32, 64, and 128 reached nearly identical performance levels toward the end of the HR @ 10 training stage. Therefore, we selected a batch size of 128 to maximize training efficiency. Similarly, learning rates of 0.0005 (red line) and 0.001 (green line) both converged to comparable performance, indicating that either value would be suitable. We simply chose 0.001 as the learning rate.
D.4.1 EVALUATION METRICS
To evaluate the effectiveness of trajectory similarity learning, we adopt the top-k similarity search framework, following previous studies in this area (Fang al. 2022). Specifically, report on
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Batch Size 64
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Figure 10: Effects of different batch sizes.
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Figure 11: Effects of different learning rates.
three commonly used evaluation metrics: HR@ 10, HR@50, and RI0@50. HR@K (Hit Rate at k) measures the proportion of queries for which at least ground-truth trajectory appears in the top-k results returned by the learning- based method. Reflects the accuracy the retrieval at fixed cut-off. Rk@t (Recall at for top-k ground truth) evaluates the fraction of the top-k ground-truth trajectories that are retrieved within the top-t candidates by the model. A higher value indicates better retrieval performance and more effective similarity preservation in the learned embedding space. Ground-truth top-k similar trajectories are obtained using the TP (Trajectory Point) distance, and details are provided in Appendix D.5
D.5 GROUND TRUTH
We construct ground-truth top-k neighbors using the TP. For trajectory network G E), let de L x L + R20 U } be the shortest-path distance on G, where 1 denotes "unreachable". A trajectory is sequence of sampled nodes (optionally with timestamps): T, = ((1), 4)) T₃ = ((e), ())
The directed form
S ITP(T,,))
min ISRSM ISeS
max
ddli. tt)
(18)
The symmetric distance is the discrete Hausdorff form
STP (TP, max TTP(T, 2)), (T,, If there is no unreachable case the spatial implementation, then TP is
(19)
TP(T;, T3) [Sr (Tr Ty)]-
(20)
The advantages follows:
L1)sess Emphasis on Sequence Order: TP places less emphasis on strict sequence order. bad- minton strategy mining, data resemble taxi-route patterns where multiple paths may lead to the same
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goal; similarly, tactical applications can vary while the intent remains consistent. TP handles such non-sequential differences well.
C2)Sonsiders Both Spatial and Temporal Dimensions: The TP metric considers both spatial and temporal similarity, without requiring exact sequence order. This allows the TP metric to adjust more flexibly when dealing with complex spatiotemporal data, making it more adaptable to real- world scenarios.
Nearett Neighbor Matching: By aggregating via the maximum of nearest-neighbor distances, small local errors are tolerated but clear deviations are amplified, preserving both stability and sep- arability without extra decay/alignment hyperparameters.
D.6 LOSS AND ACCURACY
From the training curves, is clear that we did not stop early. For comparability with SOTA, we truncate training at epoch 326 in Fig. 12a In approach, convergence typically occurs between 290 and 330 epochs on the Badminton dataset, while on the T-Drive dataset, convergence occurs between 100 and 130 epochs. In comparison, Ve2 converges epoch 137 on Badminton and epoch 98 on T-Drive on average. Different methods define their losses with different scales and semantics, so we refrain from making comparisons between methods on raw loss values.
All experiments were conducted under identical CPU/GPU environments to ensure fair comparison. Our method demonstrates superior computational efficiency across both datasets: on Badminton, it achieves 3.4 seconds per epoch compared to ST2Vec's s 12.6 seconds, resulting in total training times of 17.8 minutes versus 27.3 minutes, respectively. This efficiency advantage even more pronounced on the T-Drive dataset, where our method requires only 9.7 seconds per epoch against ST2Vec's 39.7 seconds, leading to total training times of 17.8 minutes compared to 64.6 minutes. Notably, method maintains consistent training duration (about 18 minutes) across both datasets despite their different characteristics, highlighting the robustness of our computational approach (hardware and hyperparameter settings details in Sec. D.4]
We evaluated query efficiency on the Badminton validation set (4,000 trajectories) by measuring the time to compute similarity between query trajectories and all others in the dataset. Our method achieves 35.2 ms to query against all 4,000 trajectories. The SOTA method needs 38.4 ms in to- tal, representing 8.3% higher computation time. In contrast, traditional TP similarity requires 18.9 minutes, demonstrating the significant advantage of embedding- based approaches for large-scale trajectory analysis.
As shown in Fig. 12 because our model jointly optimizes the temporal and spatial dimensions and balance both, accuracy at a small hit radius @HR@ 10) is not expected to particularly high. However, since our model learns the spatiotemporal distance directly and minimizes localization error, the overall accuracy keeps improving over training and surpasses existing SOTA, and the advantage becomes more pronounced as the radius decreases.
D.7 SUPPLEMENTARY EXPERIMENTS ON LARGE DATASETS
Table 5: Results on the large-scale dataset.
Chengdu
Xian
Framework
HR@ 10 HR 50
R10@50
HR@ 10
HR@50
R10@ 50
ST2Vec (KDD)
0.5437
0.6166
0.8594
0.3944
0.3856
0.6823
Our method
0.6802
0.7681 0.9769
0.4647
0.5227
0.8108
To further validate the robustness and generalizability of our model, we additionally expanded the dataset. With the larger data set (21.1% in Chengdu, 24.1% in Xian), the performance improvements over the SOTA baselines substantially greater than those observed in the original data set (1 111.2% in T-Drive. 10.9% in Rome). These results not only confirm the model's 's stability under data scaling but also demonstrate its strong ability to generalize across diverse domains.
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(e) T-Drive Acc. comparison
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Figure 12: Visualization of accuracy and loss.
Table 6: Framework ranked effectiveness comparison Badminton, T-Drive, Rome.
Badminton
T-Drive
Rome
Framework
HR @
HR HR@S 50 R10@50
HR @ 10 HR @ 50 R10@50
HR @ 10 HR@50 R10@
NeuTraj
4.5
5
4
5
5
3
5
4
ST2Vec
2
2
2
2
2
TrIjCI.
3
3
4
3
ConDTC
4.5
4
5
4
4
5
4
3
Our method
1
1
1
1
1
D.8 FRAMEWORK SIGNIFICANCE TEST
Friedman Test. Table 6 shows that our model consistently achieves rank across all metrics and datasets compared to various baselines. To assess the statistical significance of these improvements, we first conducted the Friedman test to determine whether there are significant differences among the evaluated frameworks. The test yields a p-value of 1.078 10 which is well below the conven- tional threshold of 0.05, indicating the presence of statistically significant differences. Consequently, we further performed a Nemenyi post-hoc test to identify which frameworks differ significantly from our method.
Table 7: Nemenyi test (pairwise p-values).
Model
NeuTTaa
ST2Vec
TrajCI. ConDTC
NeuTraj
ST2Vec
0.02999
TrajCL
0.33728
0.83507
ConDTC
0.97568
0.14139
0.71
Our method
0.00002
0.38002
0.03727
0.0003
Nemenyi Test. Table 7 presents the pairwise p-values from the Nemenyi test based on ranks. The results indicate that our method significantly outperforms NeuTraj. TrajCL, and ConDTC (p-value < 0.05). While the comparison with ST2Vec shows a smaller level significance, our method still achieves higher numerical performance and better ranks across all datasets. Notably, Table
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demonstrates that our model outperforms ST2 on the badminton dataset, and Table Shoow shows that the performance gap increases on larger datasets.
D.9 CLUSTER NUMBER
To assess the influence of the number of clusters on segmentation quality, we conducted series of param- eter tests within the context of our proposed frame- work. This analysis was performed specifically on our cu- rated Badminton dataset, which contains rich spatiotem- poral patterns suitable for evaluating segmentation strate- Clusternn quality was assessed via the silhouette co- efficient across k E [3....,9) 9}: the best-scoring A was used in subsequent experiments. As reported in Fig. 13 the results consistently indicate that setting n cluster=6 yields the most favorable outcomes across all evaluation metrics. This suggests that a six-cluster configuration provides well-balanced representation of underlying tra- jectory structures in badminton rallies, effectively captur- ing their tactical diversity while avoiding both over- and under-segmenntttion.
3.12
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Figure 13: Silhouette scores on Bad-
minton.
D.10 CASE STUDY BADMINTON STRATEGIES
The clustering analysis reveals six distinct tactical patterns, each characterized by specific shot se- quences and strategic positioning implications. In Fig. [14 red points represent the starting positions of rally sequences, while yellow points indicate the ending positions where points are won or lost. Three distinct defensive failure patterns emerge from the analysis, each demonstrating how specific tactical weaknesses can be systematically exploited by opponents. In Fig. [4a the net lift smash cycle reveals defensive failure pattern where the sequence begins with opponent net shot, net shot, opponent lift, smash, opponent net shot, losing point. This pattern demonstrates how repeated front-to-back transitions allow the attacker to maintain initiative while placing the defender tinuous disadvantage. The flat defense collapse in Fig. 14b follows a similar trajectory, beginning with opponent smash, flat, opponent lift, smash, opponent net shot, losing point. In this sequence, the defender's 's flat return fails to regain initiative, enabling the opponent to sustain pressure through alternating rear and front court attacks, which aligns with the principle that passive flat defense fre- quently leads to forced errors or delayed responses at the net. The third pattern in Fig. [4c passive reset after attack, illustrates how conservative play can backfire even when initially positioned ad- vantageously. The sequence progresses from opponent lift, smash, opponent flat, clear, opponent smash, losing point, where although the attacker initiates with a smash, the defensive flat shot neu- tralizes momentum, yet the subsequent clear cedes initiative, allowing the opponent to deliver decisive counter-smash These patterns collectively illustrate how front-to-back pressure principles can be weaponized against defenders who fail to break the cycle regain tactical initiative.
con-
In contrast to the defensive failures, three successful offensive strategies demonstrate how tactical sophistication and tempo control can generate winning opportunities. The midcourt kill after rota- tion strategy in Fig. [4d] showcases how strategic positioning can create decisive openings, with the sequence unfolding as net shot, opponent lift, clear, opponent drop/smash, net shot, winning point. This pattern demonstrates that while both sides engage in front-back transitions, the eventual winner converts the rally through midcourt kill opportunity, highlighting the tactical value of combining carly net control with rear-court variation to open midcourt space. In Fig. [4c], the tempo shift to rear attack strategy emphasizes rhythm disruption as a pathway to offensive success, beginning with drop, opponent push, flat, opponent clear, smash, winning point. Here, the attacker employs drop to disrupt the rally's rhythm, forcing the opponent into a less aggressive push response and sub- sequently clear, which provides the opportunity to seize initiative with decisive rear-court smash. Finally, the net exchange turnaround strategy in Fig. 14f demonstrates the power of tactical re- covery, where the sequence moves from clear, opponent smash, net shot, opponent net shot, lift/net shot, winning point. Despite facing initial defensive pressure, the attacker successfully balances the rally through strategic net exchanges and ultimately regains initiative, confirming how proactive net
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play can transform defensive positions into offensive opportunities and supporting the fundamental principle of front-court dominance.
(a) Net lift smash cycle
(b) Flat defense collapse
(c) Passive reset after attack
(d) Midcourt kill after rotation
(e) Tempo shift to rear attack
(1) Net exchange turnaround
Figure 14: Visualization of badminton strategies.
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Reviews:
[{"review_id": "jsEx4779ue", "reviewer": "Reviewer_719K", "summary": "This paper proposes a unified trajectory representation learning framework designed to handle both continuous movement trajectories and discrete trajectories. Experiments on several datasets demonstrate the effectiveness of the proposed method for downstream tasks such as trajectory similarity search.", "questions": "1. What is the novelty of the proposed approach, given that most components of the model are already well studied in the literature? \n\n2. What is the fundamental difference between continuous and discrete movements? Why do they need to be treated separately, and which parts of the model explicitly account for this distinction? A clearer conceptual and architectural explanation would strengthen the contribution.\n\n3. Since discrete trajectories are the main focus of the paper and show the most performance improvement, the evaluation should include more relevant datasets, such as basketball or soccer trajectories, to better validate the generality of the proposed approach.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The paper tackles an interesting problem of learning representations that generalize across both continuous and discrete trajectory types.\n\n* The experimental results show that the proposed framework performs well on specific datasets, indicating potential for improving trajectory similarity search.", "weaknesses": "1. At first, I thought the proposed framework aimed to learn a unified trajectory representation by jointly learning from all types of trajectory data (e.g., both badminton and taxi datasets). However, it appears that the model is still trained separately for each data type. In that case, I do not see a clear advantage of this approach compared to existing methods. The overall model design (e.g., handling spatial and temporal embeddings separately and then combining them), together with the use of a triplet loss, is already widely adopted in trajectory representation learning.\n\n2. The paper should clearly define the fundamental difference between continuous and discrete movements. Is the distinction based on predictability or on sampling rate? If the latter, then directly resampling the taxi trajectories could produce a similar dataset, suggesting that the distinction may not be substantial. A formal definition and discussion are necessary to clarify this point and justify the motivation.\n\n3. The writing quality and organization need improvement. The introduction is overly long and occupies too much space, while the method section is too brief and lacks sufficient implementation details, making it difficult to reproduce the work. Additionally, the topic may be too narrow in scope and might not appeal to the broader ICLR readers.", "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": "e0HU75hgur", "reviewer": "Reviewer_M4PH", "summary": "This paper presents TRIDENT, a spatio-temporal trajectory representation learning framework based on a novel variant of contrastive learning. The approach is designed to be generalizable across different trajectory domains, demonstrated here on GPS trajectories and badminton (human movement) trajectories; the former being smooth and dense, and the latter sparse, abrupt, and event-driven.\n\nTo achieve this, the authors propose: (1) a GCN for spatial encoding and D2V for temporal embedding, (2) a Dual-Attention Encoder with tanh attention pooling to fuse spatio-temporal representations, and (3) a Distance-preserving Multi-kernel Triplet Loss (DMT). The DMT component is carefully designed to (a) preserve the geometric structure of the original trajectories and (b) leverage an anchor-positive-negative triplet loss setup, which is known to enhance the typical contrastive positive-negative pairs.\n\nExperimental results show that TRIDENT achieves substantial improvements over baseline methods across three datasets spanning two domains. Qualitatively, TRIDENT trajectory representations are distributed in a way that facilitates easier clustering.", "questions": "How was the badminton trajectory dataset collected? Providing more details on the data collection process would improve clarity and reproducibility.\n\nHow does the proposed method perform under cross-domain or cross-dataset transfer settings?\n\nDoes the method remain robust when trajectories are distorted or downsampled? For example, an analysis simulating GPS errors (such as those caused by urban canyons, as the authors mentioned in the introduction) would also help evaluate TRIDENT’s robustness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors carefully designed the model architecture and training strategy, tailored to capture the unique characteristics and complexities of spatio-temporal trajectories and their diverse domains due to varying collection methods and details.\n\nThe proposed DMT loss is an elegant adaptation of the triplet loss framework, which has proven effective in other domains such as text embeddings.\n\nThe model demonstrates strong cross-domain generalizability and achieves state-of-the-art performance compared to existing baselines.\n\nThe faster training/convergence is a nice benefit.", "weaknesses": "- The paper lacks cross-domain or cross-dataset transfer experiments. It would have been valuable to see pretraining on one domain (e.g., GPS) followed by evaluation or fine-tuning on another (e.g., badminton), OR training on one dataset (e.g., Beijing) and testing on another within the same domain (e.g., Rome).\n\n- The evaluation focuses primarily on trajectory retrieval. Incorporating additional evaluation tasks, e.g., robustness tests under trajectory downsampling or slight distortions, would strengthen the overall empirical validation.\n\n- Minor presentation issues:\nCitation formatting is occasionally incorrect; for example, missing brackets around in-line citations makes them harder to read.\nL55-66 “raw trajectories are embedding space” should be revised to “raw trajectories are encoded into the embedding space.”?\nThe title of Section 4.3.1 appears to contain a typographical error: “Learning Process of”…?", "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": "LxytmxsGH0", "reviewer": "Reviewer_LWGz", "summary": "This paper presents TRIDENT, a unified framework for learning spatio-temporal trajectory representations. It handles diverse data types—from smooth GPS routes to abrupt sports movements—within a single model. TRIDENT integrates: (1) a Graph Convolutional Network (GCN) for spatial features and a temporal embedding module fused via a Dual-Attention Encoder (DAEncoder); (2) a Distance-preserving Multi-kernel Triplet Loss (DMT) that maintains the metric geometry of the original trajectory space; and (3) extensive experiments on T-Drive, Rome, and Badminton datasets showing superior retrieval accuracy, efficiency, and cross-domain generalization. A case study on badminton rallies further demonstrates the model’s ability to reveal tactical patterns.", "questions": "Q1: Could you please clarify what \"D2V\" (line 129) refers to in the context of your temporal embedding module?\n\nQ2: What is the meaning of the “LEARNING PROCESS OF” in section3.1?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1: The proposed Distance-preserving Multi-kernel Triplet Loss (DMT) addresses a fundamental limitation of classic triplet loss by enforcing the preservation of metric structure, not just relative ordering.\n\nS2: TRIDENT effectively handles diverse trajectory types—from smooth taxi routes to abrupt sports movements—within a unified model.", "weaknesses": "W1: The overall architecture (“GCN + temporal model + attention fusion”) follows a well-established design pattern in spatiotemporal modeling, which somewhat limits the novelty of the framework.\n\nW2: The project's code and data are not publicly available, and the datasets used for training and testing the models are too small.\n\nW3: The paper lacks comparisons with general sequence representation learning models, which weakens its state-of-the-art (SOTA) conclusions.", "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": "vKFhF9zniF", "reviewer": "Reviewer_d5b8", "summary": "The paper presents a spatio-temporal representation framework that can handle perturbations in trajectories instead of assuming smooth and continuous motion. It also presents a loss function that preserves local order and robustness under noise. The framework was tested against 4 baselines using 3 datasets from different domains and demonstrated improvements.", "questions": "- In the training strategy (lines 220–222), the positive and negative sample are selected based on calculating a similarity measure. For that, there are two considerations:\n- Are there specific threshold to control these selections? If so, can you please elaborate on this? Without thresholds, it can happen that the positive sample is too close or identical to the anchor, and/or the negative sample is too far from the anchor. This hinders the effectiveness of contrastive triplet losses.\n- Is the computation cost considered? And are there are pre-computing steps to ensure that the same calculation between the exact same sample is not done many times? This is important for the scalability of the proposed method.\n- In lines 250–257, the authors discuss the values of k . However, there is no discussion nor mathematical notation showing how it is calculated. Can you please provide more details?\n- In lines 773–779, a brief description of setting the hyperparameters is presented. However, only a few hyperparameters are presented without the reasoning behind setting these values. Can you please elaborate on why setting these exact values and provide details and reasoning about other hyperparameters? \n- Did you consider automated approaches to optimise the hyperparameters of the model architecture? If so, please provide details. If not, why were such methods not considered?\n- On lines 859 – 873, the method of selecting the value of k used in the experiments is provided. The analysis was done on one specific dataset. However:\n- It is unclear whether the same K can be used for other datasets, or another analysis has to be conducted. If an analysis needs to be conducted each time, is it a process the user has to do manually or is it automated?\n- What is the motivation behind using the Silhouette Score for this assessment? Did you consider other scores such as Calinski-Harabasz index?\n- Did you consider clustering algorithms that do not require explicitly setting K and allow more flexible clustering?\n\nIf my questions and the points in the “Weaknesses” section are addressed, I am happy to raise my score.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- The paper addressed a relevant problem regarding extracting trajectories for perturbation data, specifically in sports-related data.\n- Very good introduction of the problem and the relevant research gap. The authors properly introduce the problem of not being able to simultaneously learn both point- and shape-based trajectories, in addition to the perturbation measures due to errors is measurement and sampling then highlight the particular challenges in sport trajectories and how they are relatively underexplored.\n- A scalable and unified solution. The paper utilities a unified architecture that can be trained on different data types, thus avoiding redesigning distances and model components.\n- Very good experiments and results. The authors conducted experiments spanning multiple trajectory types and showed improvements across all of them.", "weaknesses": "- Presentation issue (I recommend fixing these in case the paper proceeds to be accepted)\n- Incorrect usage of “\\citep” and “\\citet” across the paper.\n- The appendix section is not properly numbered. The appendix should have letter “A, B, …” instead of numbers.\nSome acronyms are used without being presented or before being fully presented. Examples: NTAP, GCN, \n- Very long introduction. The current introduction is very longs and includes both the introduction and the related work. It is better to have these sections separated for readability.\n- Unclear setting of hyperparameters. The authors do not provide details on how the hyperparameter of the model are set, nor the reasoning behind setting these hyperparameters. There is only a brief description in line 774-779.\n- The baseline methods are not properly presented. The authors compared their experiments with 4 baselines methods. However, the methods are not presented in details. It is important to present the details of these methods, regarding the architecture and the type of loss function they use. This assists in evaluating and understanding the experimental setup, in addition to clarifying the motivation of choosing these exact baselines.\n- When comparing the framework presented in the paper with the baselines, the results are presented in numerical forms without testing whether the improvements are statistically significant or not. The statistical significance of the claimed improvements in comparison with the baselines methods needs to be tested (using a Friedman test followed by Nemenyi post-hoc test, for example).\n- The code is not available for the review process. The authors mention in the paper abstract that “An anonymous repo with code and data is in the supplement.”. However, there is no such link in the appendix.", "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)