ScatterAD: Temporal-Topological Scattering Mechanism for Time Series Anomaly Detection
Authors: Tao Yin
Shaochen Fu
Zhibin Zhang
Li Huang
Xiaohong Zhang
Yiyuan Yang
Kaixiang Yang
Meng Yan
Venue: NeurIPS 2025 poster ()
Keywords: Time series anomaly detection
representation learning
mutual information
graph attention network
Primary area: deep_learning
TL;DR: ScatterAD leverages representation scattering as an inductive signal to jointly model temporal and topological patterns for effective multivariate time series anomaly detection.

Abstract: One main challenge in time series anomaly detection for industrial IoT lies in the complex spatio-temporal couplings within multivariate data. However, as traditional anomaly detection methods focus on modeling spatial or temporal dependencies independently, resulting in suboptimal representation learning and limited sensitivity to anomalous dispersion in high-dimensional spaces. In this work, we conduct an empirical analysis showing that both normal and anomalous samples tend to scatter in high-dimensional space, especially anomalous samples are markedly more dispersed. We formalize this dispersion phenomenon as scattering, quantified by the mean pairwise distance among sample representations, and leverage it as an inductive signal to enhance spatio-temporal anomaly detection. Technically, we propose ScatterAD to model representation scattering across temporal and topological dimensions. ScatterAD incorporates a topological encoder for capturing graph-structured scattering and a temporal encoder for constraining over-scattering through mean squared error minimization between neighboring time steps. We introduce a contrastive fusion mechanism to ensure the complementarity of the learned temporal and topological representations. Additionally, we theoretically show that maximizing the conditional mutual information between temporal and topological views improves cross-view consistency and enhances more discriminative representations. Extensive experiments on multiple public benchmarks show that ScatterAD achieves state-of-the-art performance on multivariate time series anomaly detection.

ScatterAD: Temporal-Topological Scattering Mechanism for Time Series Anomaly Detection
Tao Yin'
Xiaohong Zhang!-
Shaochen Fu'
Zhibin Zhang' Meng Yan
Li Huang'
Yiyuan Yang"
Kaixiang Yang*
*Chongging University
University of Oxford Sooth China University of Technologg
{yintao, fushaochenJostu.. equ. edw.eddu.cc. en, (xhongz, zbinz, lee.h, mengy)@cqu.ee edu. yiyuan. yangocs. yingans ox ac. uk, yangkx0scut mangxx@scut.edu.comm edu. cn
Abstract
One main challenge in time series anomaly detection for industrial IoT lies in the complex spatio-temporal couplings within multivariate data. However, traditional anomaly detection methods focus on modeling spatial temporal dependencies independently, resulting in suboptimal representation learning and limited sensitiv- ity anomalous dispersion high-dimensional spaces. In this work, we conduct an empirical analysis showing that both normal and anomalous samples tend to scatter in high-dimensional space, especially anomalous samples are markedly more dispersed. We formalize this dispersion phenomenon scattering, quanti- fied by the mean pairwise distance among sample representations, and leverage it as an inductive signal to enhance spatio-temporal anomaly detection. Techni- cally, we propose ScatterAD to model representation scattering across temporal and topological dimensions. ScatterAD incorporates a topological encoder for capturing graph-structured scattering and temporal encoder for constraining over- scattering through mean squared eгror minimization between neighboring time steps. We introduce contrastive fusion mechanism to ensure the complemen- tarity of the learned temporal and topological representations. Additionally, we theoretically show that maximizing the conditional mutual information between temporal and topological views improves cross cross-view consistency and enhances more discriminative representations. Extensive experiments on multiple pub- lie benchmarks show that ScatterAD achieves state-of-the-art performance multivariate time series anomaly detection. Code available at this repository: betts://githule cttttau.oooom/// http:///wwuu.ccmm///ooooo/////// commjix-sound//Saateeee /Satttee//
1 Introduction
Multivariate time series data in Industrial Internet of Things (IoT) systems exhibit intricate spatio- temporal relationships Tian al., 2023). and anomaly detection crucial to ensure the stable operation of the system EEfthymiuu et al. 20188, [Brannik]] 2020). Recognizing abnormal patterns from complex data has become a critical and widely studied problem both research and practical domains Mohammadi et al. 2018 Wollschlaeger et al. 2017). However, anomalies in multivariate time series frequently manilest synergistically in temporal and topological dimensions [Jin al. 2024 Yi et al. 2024 Zhao et al. [2224], presenting highly coupled complex characteristics, which pose a severe challenge to traditional anomaly detection methods.
*Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Scoteetgg (comn Ebeiidoon dituoo))
Narua Z
Kouesa seps
Contitrtis
(a) Temporal features
(b) Topological features (c) Combined features
(d) Scattering measurement
Figure I: The distribution patterns of seatter visualization and seatter measurement in time features, topological features, and combined camporal-topologiee features (obtained by simple linear combi- nation) on the SWaT dataset. Blue and red represent the embeddings of normal points and abnormal points respectively. Normal data exhibits clustering characteristics, abnormal data exhibits scattering characteristics, and combined features show more significant scattering differences. This provides a strong prior signal for anomaly detection.
Time series anomaly detection methods are broadly bifurcated into two principal categories: methods based on temporal pattern learning and methods based on topological structure modeling. Temporal pattern-based methods (such as Park et al. [0188] Su et al. 2019 Xu et al. 2021 [w et al.) (221]) primarily focus on the sequential dependency of time series data and identify data points that deviate from normal temporal patterns by constructing predictive or reconstructive models. However, recursive models for learning temporal patterns exhibit limitations in capturing pairwise dependencies between variables, limiting their ability to identify intricate anomalies [Zhao et al.) [2020] Xu et al.) 2021]. Conversely, topological structure-based methods (such as GDNiDeng and Hooi 2021], J,
GTAD(Guan et al. 2022]. TopoGDN (Liu et al. 2024] etc.) focus on modeling the graph structure relationship between variables and detecting anomalies by learning the mutual dependence between variables. While these methodologies can model spatial correlations. they often fail to fully leverage the dynamic characteristics of time series. Methodologies focusing exclusively on either temporal or spatial patterns often prove inadequate for modeling the intricate spatio-temport dynamics in multivariate time series, consequently leading to challenges in effectively capturing the "scattering" phenomenon of anomalies in the spatio-temporal dimension. As shown in Figure scattering patterns of temporal features, topological features, and their linear combination features on the SWaT dataset, using the average Euclidean distance quantify the degree of scattering between samples. It is evident from the scattering representation diagram and the scattering metric diagram that the combined feature space usually exhibits more pronounced scattering differences than the temporal feature space and the topological feature space. In essence, anomaly detection is the identification of samples outside the data distribution -uito--diittibution, OOD) Waag et al., 2020). Significant scattering differences can bring better anomaly separation capabilities to the model. In other words, when the scattering of anomalous samples is more pronounced, their feature representations are pushed further apart from the normal samples, creating a clearer boundary for anomaly detection. More recently, several studies have endeavored to consider both the temporal and spatial dimensions, such as MTAD- [Zhao et al.) 2020). However, MTAD-GAN GAT employ simple feature concatenation or serial processing methods and fail to address the challenge of collaborative optimization of spatio-temporal features in theory. In particular, they lack deep understanding and modeling of the complementarity of spatio-temporal features.
visualize the
The information bottleneck principle (rincppl(Tishbyy et al. 2000 offers a theoretical framework for address- ing such challenges of spatio-temporal feature complementarity. This principle aims to extract representation Z from the input variable X, such that Z preserves maximal information as possible about the target variable Y in X, while compressing redundant information in X that irrelevant to Y. In particular, when complementary representations need to be learned from different dimensions (such as temporal and topological), single bottleneck objective may fail to effectively align the relationship between the representations of each dimension [Federici et al. 2020]. Considering that anomalies Cemeeniio(Federiic multivariate time series typically manifest as abnormal scattering of spatio-temporal patterns, we propose a theoretical hypothesis: effective anomaly detection needs to capture the scattering represen- tation characteristics of both the temporal and topological dimensions and ensure the complementarity of these two features. Through thcoretical analysis, prove that under the condition of given graph

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structure G, maximizing the mutual information I 1[;; Zg G) between the temporal representation Zr and the topological representation Zg can facilitate effective anomaly detection. This theoretical result can be expressed as: max T(TT; Za G) s.t. T(X, 7T) rı, [(X, 7G) < r2, where r1 and r22 control the compression degree of the temporal and topological representations respec- tively. This shows that, subject to the constraint that the temporal and topological representations are fully compressed, maximizing their conditional mutual information can promote the synergistic complementarity of the two representations, thereby improving the performance of anomaly detection.
Specifically, we propose a novel anomaly detection approach ScatterAD, which enables effective representation learning of pempooltemporal representation through three principal mechanisms: first, the scattering mechanism designed to capture the scattering characteristics of the representation in representation; second, we further incorporate time
the feature dimension in the cempora-topollogee constraint mechanism to prevent excessive scattering of the representation to maintain the consistency of the temporal structure; finally, in order to align temporal and topological representations in the latent representation space, we introduce contrastive fuse mechanism to align the outputs of the temporal and topological encoders, maintaining the compression constraints while promoting the complementarity of the representations. Unlike prior work, which treats temporal and topological patterns separately, ScatterAD introduces a unified contrastive scattering mechanism that jointly optimizes temporal consistency and structural distinguishability.
The contribution of our paper is summarized as follows:
We introduce ScatterAD, novel anomaly detection approach that employs temporal- topological scattering mechanism to improve representational discriminability while pre- serving temporal structural consistency.
To the best of our knowledge, this is the first work introduce information bottleneck theory into multivariate time series anomaly detection, to theoretically reveal the complementarity between temporal and topological features. This leads to more discriminative representations and highlights the importance of integrating spatio-temporal information in future anomaly detection research.
Extensive experiments conducted on six public benchmark datasets and using twelve stan- dard evaluation metrics demonstrate that ScatterAD achieves state-of-the-art performance, validating both the theoretical foundations and practical effectiveness of our design.
2 Related Work
Anomaly Detection in Time Series Given the scarcity and imbalance of data labels, unsupervised anomaly detection methods have received widespread attention in recent years [Choi et al. 2021). Primarily focused on the reconstruction error-driven autoencoder model [Kingma et al. 2013 or its probabilistic extension, the variational autoencoder [Su et al., [019]. These methods capture local reconstruction patterns but struggle capture fine-scale dependencies and dynamic propagation across time points. Graph neural networks (GNNs) have been introduced to model the dependency structure between different variables in multivariate time series. Representative methods such as MTGNN [Wu et al.] [2020] and GDN (Deng and Hooi] 2021] typically regard each variable as a node in the graph and introduce graph convolution to learn spatial structure. However, these methods focus on spatial graph modeling between variables, neglecting the temporal graph structure. Consequently, they depend on topological associations between variables and have limited capacity to model the time dimension. More recently, several studies have initiated the exploration of graph structures in the time dimension. For instance, GANF [Dai and Chen 2022 and MTGFlow (Zhou et al. 2023 construct time series graph structures and combine density modeling methods to model the distribution of the entire sequence, but these methods still do not deeply explore the structural characteristics and interdependencies between time nodes. Although methods such as TopoGDN [Liu et al. 2024] and
CST-GL ZZeng et al. 2023] model spatio-temporal dependencies, they overlook representation scattering in high-dimensional spaces, which can potentially impede their sensitivity to the dispersion patterns characteristic of anomalies.
Information bottleneck and representation complementarity The classic information bottleneck principle (Tishby et al., (2000) explicitly articulates the objective of retaining task-relevant information while compressing redundant inputs. Based this foundation, Deep InfoMax [Hjelm et al. 2018
3

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and InfoGraph [Sun et al. 2019) employ a mutual information maximization strategy to align local and global representations structured data, including images and graphs. Models such as MVGRL. [Hassani and Khasahmadi [220)] and structured mutual information models (Yngg et al. 2025] promote complementarity between different representation views, which have the potential to substantially enhance the performance of downstream tasks.
3 Method
3.1
Overview
Time Consistent Learning
Contrastive Fuse Learning
Objjet
Multivariate time series
GAT Encoder
Seatrus: logo
Iades
Normalization
EMA
Tempural Graph
Construction
Temporal -oooologiall Scattering Representation Learning
Seattering
Learning
fo
GAT Encoder
Zuarget
Transy
(X. A)
Temporal Node Feature
Cononical Center
Scatter Center
Figure 2: The overall framework of ScatterAD employs temporal graphs to model the scattering patterns of nodes in multivariate time series. encodes temporal nodes graph vertices to capture structural scattering, constrains time consistency to avoid excessive scattering, and aligns temporal topological features through contrastive fusion, ultimately achieving effective detection through scattering representation learning.
As illustrated in Figure 2 ScatterAD employs an online encoder fo(:) and target encoder fo(-) ( to process the temporal graph G, where 0 and & denote the parameters of the respective encoders. (1) topological space is initially defined via Ln-normaliiing node features onto the unit hypersphere. with global scatter center c randomly initialized inside the unit ball (see Section 33.3] for details). Following the positive-only sampling strategy used in DCdetector (Yang et al.) 2023], all training samples are treated as positive samples. Within this embedding space, the target encoder f-(-) learns to model the compactness of normal samples, thereby capturing scattering representations and amplifying the scattering of anomalous samples during inference.
(2) The online encoder fo(-) yields houline and enforces temporal smoothness via a temporal constraint on adjacent temporal nodes, preventing excessive scattering in the target encoder (.). Only the online encoder is updated via backpropagation; the target encoder parameters é are updated using the exponential moving average (EMA) of the online encoder parameters 0 after each training step (see Section 3.2 for details). (3) The online and target encoders constitute a contrastive fusion framework to maximize the similarity between them, thereby fostering collaboration between the online and target encoders. (4) Regarding the anomaly judgment criterion, anomalies are evaluated based on scattering deviation and time consistency deviation (see Section 3.3 for details).
The definition of multivariate time series anomaly detection is as follows: Let X' {at) X' E R R N be multivariate time series with N dimensions and sliding window length of T. I; = [x] . ., x:N E N represent the N-dimensional observation vector at time t, and sequence length is T. The goal of anomaly detection is to predict the anomaly label )4 E {0. 1) for each timestamp t, where ) = indicates that time in an abnormal state, and ), = 0 indicates normal state.

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3.2 Temporal Graph Construction and Node Feature Extraction
Our framework employs a dual-encooo encoder architecture learn temporally and topologically aggregated representations of normal patterns at temporal nodes. Unlike prior graph- based time series models, where each node corrssondss node. Specifically, each graph node represents multidimensional time slice X E RN,, and the edge = (ut-k;10 14) E r], E [r +1,, 1, T), capturing the behavioral patterns across time through directed connections. Here, 7 is the look-back window size. In our experiments, we set r = 2. To extract temporal dynamics from the input sequence, we apply a multi-scale causal convolutional encoder. Let h())
sensor or univariate stream, we consider each time step as distinct
set is defined
X denote the input time series:
k-1
1==")= for t k K....,T
h PReLU
BN
Σ W
(1)
where We( denotes the i-th convolution kernel in k-length causal convolution, BB(-)) signifies batch normalization [loffe and Szegedy 2013]. To further integrate topological dependencies across temporal nodes, we employ multi-head GAT layer to compute attention coefficients across nodes:
exp (Leaky (W/hh/WWW)))
EKEN: exp (LeakyReLU(e,,, (W,,s]|[W,s))
a denotes the normalized attention coefficients of the h-th head. h5s hy (t) represent the input feature vectors of nodes and Wh is the head-specific parameter matrix and the aj is the attention coefficient vector.
H +
Whh )
K = ELU
Σ
softmax (0
h=1
JEN;
Subsequently, the node features are aggregated with topological features and temporal features, yielding 2; N² + h0) 2; IR. Txduu denotes the stacked temporal and topological representations.
3.3 Optimization Target
Time Consistent Learning The loss of time consistency is introduced to node features Zontine. E 24 to reinforce the temporal dependency adjacent nodes and prevent them from excessive dispersion in the feature subspace, thereby preserving the key temporal features of the nodes. The formulation is follows:
T-1 Σ
Lieme T
Rontine,t++ Zontine, 112
here, each row Rontine, & Rdo corresponds to the node representation at time step and Ltime penalizes abrupt representation shifts between consecutive encoding states.
Temporal Topological Scattering Representation Learning We propose a graph scattering mechanism to explicitly guide the target encoder in learning decentralized representations nodes. For node representation, we initially define regular subspace S and scattering center For the subspace @ transformation function Trans(-) is employed to transform the representation from the original space E.d to $ Specifically, we apply Le Lo-ommnalg each row vector 2 within the matrix Ztarget,ti
Starget,1 = T Transgs (Ztarget,))
Ztarget,t
(zargeet | 1}, (5)
[target,
where Etarget.t E & is the target representation on the unit sphere (normalized), as defined in the formula, the representations of all nodes are distributed in the hypersphere $* This mapping prevents arbitrary scattering of representations in the space, thereby mitigating instability and optimization difficulties during training. Afterwards, we initialize a global seattering center, c, which serves fixed anchor point within the latent space. This center is randomly sampled to reside strictly
5

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inside the unit hypersphere. The initialization is two-step process: first, an intermediate vector c' is sampled from a standard multivariate normal distribution. Second, this vector L2-normalized to project it onto the unit hypersphere and then scaled by random scalar E (0, 1). This procedure can be formally expressed as: c' ~ I),, scattering center c, we then introduce the representation scattering loss, wetter designed pull the representations of normal samples towards it:
Te where c ~ ((0, 1). With this fixed
Lscatter
1
target.t
(6)
Starget, ||/
By maximizing this loss function, the learned target features remain close to the semantic center while preserving directional diversity, which is crucial for amplifying the scattering behavior of anomalous nodes during inference.
Contrastive Fuse Learning To ensure that the nodes maintain consistent directions in both the embedding space and the projected hypersphere space, while preserving local structures and global geometric properties. We maximize the similarity between positive sample pairs from both encoders, thereby ensuring that the embedding space preserves local structural relationships while the scattering space maintains global node relationships. The formulation is as follows:
Lcontrast
Σ (a.d)EE
log a (cos (2onltinee
(d) Zlarget.t)
Here, & denotes the set of edges of graph and a is the sigmoid activation function, (s, d) representing the origin and terminus of the corresponding edge, constituting positive sample pair, condine, and (d) Starget,t denote the feature vectors generated by the encoder fo(-) and the encoder f.(.).
Exponential Moving Average Mechanism During the training process, the parameters 0 of the online encoder are updated via gradient descent, and the parameters
updated through the Exponential Moving Average mechanism [Cai et al.) 2021] by the online encoder. Compared with direct alignment constraints and centralized representations, the prediction target serves as a buffer mechanism, enabling the target encoder to adaptively learn local time constraint information and global topological scattering representation. The exponential moving average mechanism is applied at the end of each training epoch. The formula is as follows:
+- m0 + (1 (1-m) m.
Among them, (0 m 1), this process facilitates the integration of graph time structure consistency into the node scattering representation process, effectively alleviating the confrontation brought about by direct optimization of alignment constraints and decentralized representations. Representation scattering is used to enhance the global distinguishability of information, and time consistency loss ensures the consistency of local information. The interaction of these two mechanisms enables the target encoder to learn global topological scattering features more stably while preserving local temporal information.
Loss Function
The overall training objective of ScatterAD integrates three loss components:
Liotal Lscatter + Ltime + Leontrasts
Cscatter captures the scattering representation by modeling the compactness of normal samples within the topological space, thereby implicitly amplifying the contrast with anomalous patterns and enhancing feature discriminability. Lisme enforces time consistency by penalizing abrupt changes between adjacent time steps, mitigating over-scattering. Ccontrast aligns temporal and topological representations through cross-view contrastive loss based on cosine similarity, fostering their complementarity. This joint optimization encourages the model to learn discriminative, temporally consistent, and cross-view complementary representations for robust anomaly detection.
3.4 Anomaly Criterion
To jointly evaluate anomalies from both the topological scattering and time consistency perspectives, define unified anomaly scoring function that captures two complementary forms of deviation.

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For the anomaly score of the input sequence X, it is initially transformed into node representation 2, = fo (X), 2, E xgxx The scattering deviation score (X*), which quantifies how far the target node representation deviates from the learned center in the normalized representation space, and the time inconsistency score L(X)) which penalizes abrupt representation shifts between consecutive time steps. The ultimate anomaly score is formulated as follows:
)
AnomalyScoro({)- = =(()) min D (2),c) + ((X):
2
(10)
k
==1
Scattering
devriation
inconsistency
This is a point-level anomaly detection approach, where anomalous points are expected to receive
higher anomaly scores. Following prior work (Xu al. 2021] Yang al. 2023 we introduce
threshold 5 to convert scores into binary labels: a point is labeled as anomalous (y = 1) if its score exceeds 6, and normal (y 0) otherwise.
4
Experiments
4.1
Experimental Setup
Datasets
We conducted evaluations on six real-world multivariate time series datasets, including (1) PSM (Pooled Server (etrics)(Abdulaal Abdulaal et al., 2027]; (2) MSL. (Mars Science Laboratory) (Hundmann
al. 2018: (3) (Seuree Water Treatment) (Mthhur and Tippenhauer, 2016]; (4) WADI (Water
Distribution Ahmed et al. 2017; : (5) NIPS- TOPSS-S--WANLL. et al. 2021]; (6) NIPS-TS-GECCO Lai
ct al. 2021]. See Appendix for dataset details and statistics.
Metrics We evaluate detection performance using both label-based and score-based metrics. For label-based metrics, we report the point-adjusted F1 score (PA-F), which considers an anomaly segment detected if any timestamp within it is flagged. Although it may overestimate performance, it widely used [Xu al. [021] Guan al. 2022 Song et al. 2023). To ensure fair comparisons, we also report the Affiliated FT score (Aff-F) [Huet et al., 2022 which measures alignment quality between predicted and ground-truth anomaly segments. Additionally, we include score-based metries: Area under the ROC Curve (A-ROC) (Fawcett) (006) and Area under the Precision-Reche Curve (A-PR) (Boyd et al. 2013]. In summation, we report results across twelve evaluation metries, with comprehensive reports detailed reports in Appendix C
Baselines We compare our method against a diverse of baselines, including recent state-of-the- art graph-based and transformer- based models: Sub-Adjacent Transformer (S. T.) Yue et al.
Yue
2024].
TopoGDN (L. et al.) (2024), Memto [Song al. 2023). DuoGAT (D.G.) Lee al.
2023
MTGFlow (MTG)
et al. 2023). iTransformer (iT.) Liu et al., 2023). DCdetector (DC.) Yang
jt al. 2023), J. Anomaly Transformer (A.T.) et al.) 202 J, and GANF Dai and Chen 2022 We
also include classical methods such as Modern'l ICN (M.TCN)(Lup and Wang 2024]. Variational Au-
toencoder (VAn)ePineeiroo Cinelli et al.) (022], Isolation Forest (IF) et al. 2008 and PCA (Shyu
ct al.
2003 for comprehensive evaluation.
4.2 Main Results
We evaluate ScatterAD on six real-world multivariate time series datasets utilizing four standard evaluation metrics: Affiliated-F1 (Aff-F), Point-Adjusted-FF1 (PA-F), AUC-ROC (A-ROC), and AUC-PR (A-PR). As demonstrated in Table ScatterAD consistently achieves the best performance. ranking first in 15 out of evaluation settings. Notably, ScatterAD achieves the highest Affiliated-FI scores across all datasets, demonstrating its superior ability localize anomalies level. While performance on the NIPS-TS-SWAN (N-T-S) dataset is mixed, with the highest scores in PA-F (0.736), A-ROC (0.792), and A-PR (0.716), but notably low Aff-F (0.038), we attribute this to the irregular and bursty nature of anomalies in the dataset, which makes segment- level matching particularly challenging. Nonetheless, the robust Point-Adjusted-FF suggests robust detection capabilities under noisy and complex conditions. Furthermore, ScatterAD consistently achieves the highest AUC-ROC values across all datasets, indicating a strong trade-off between true positive and false positive rates under varying threshold settings.
the segment
7

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Table I: Performance comparison on six real- datasets. Evaluation Metrics: Aff-F (affiliated FI), PA-F point-addjustd FI), A-RO- (AUC-ROC), and A-PR (AUC-PR). Higher values indicate better performance. All reported results are averaged over multiple independent runs to ensure robustness. Bold indicates the optimal performance; Underline indicates the suboptimal performance.
Datasel
Metric
Ours
S.T.
T.G.
Memto
DC.
M.TCN
IT.
MTG
A.T.
D.G.
GANF
VAE
IF
PCA
Aff.P
0.867
0.673
0.674
0.674
0:209
0.673
0.677
MSL
PA-F
0.964
0.863
0.714
0.664
0.9:7
0.807
0.659
0.648
0.94
0.652
0.089
0.512
0.596
0.441
A-ROC
0.986
0.751
0.794
0:951
0.961
0.627
0.604
0.857
0.70
0.788
0.678
0.833
0.673
0.732
A-PR
0.932
0.754
0.727
0.902
0.891
0.739
0.781
0.703
AFF-F
0.797
0.786
0.689
0.659
0.653
0.701
0.652
0.436
0.657
0.624
0.329
0.524
0.569
0.477
PSM
PA-F
0.981
0.941
0.801
0.977
0.977
0.965
0.926
0.797
0.978
0.780
0.788
0.596
0.543
0.467
A-ROC
0.986
0.731
0.830
0.981
0.928
0.581
0.074
0.825
0.758
0.634
0.908
A-PR
0.969
0.806
0.963
0.542
0.866
0.629
0.751
0.753
0.961
0.245
0.745
0.443
0.344
0.687
AE-F
0.701
0.631
0.591
0.592
0.687
0.685
0.619
0.563
SWaT
PA-F
0.951
0.863
0.788
0.756
0.939
0.884
0.916
0,500
0.943
0.822
0.244
0.526
0,472
0.548
A-ROC
0.982
0.956
0.766
0.811
0.876
0.675
0.662
0.769
0.989
0.094
0.574
0.532
0.501
0.502
A-PR
0.909
0.824
0.592
0.851
0.503
AI-FF
0.605
0.756
0.532
0.701
0.725
0.558
0.671
0.673
0.708
0.636
0.667
0.556
0.555
0.477
WADI
0962
0.873
0.834
0.807
0.731
0.266
0,738
0.705
0.681
0.512
A-ROC
0.960
0.915
0.612
0.693
0.829
0.831
0.814
0.782
0.982
0.372
0,782
0.501
0.860
0.502
A-PR
0.765
0.743
0.532
0.691
0.651
0.697
0.627
0.523
0.637
0.470
0.508
0.384
0.406
0.357
Aff-P
0.038
0.805
0.685
0.484
0.533
0.099
0.659
NIPS-TS-SWAN
PA.F
0.736
0.714
0.660
0.687
0.733
0.731
0.729
0.591
0.095
0.503
0.630
0.497
0.522
0.526
A-ROC
0.792
0.570
0.595
0.791
0.655
0.562
0.537
0.788
0.787
0.671
0.778
0,522
0.729
0.198
A-PR
0.216
0.946
0.527
0.702
0.626
0.545
0.531
0.699
0.509
0.714
0,712
0.326
0,473
0.326
AIT-F
0.825
0.476
0.589
0.424
0.648
01.466
0.435
0.348
0.658
0.667
0.268
0.481
0.531
0.509
NIPS-TS-GECCO
PA-F
0.784
0.491
0.620
0.504
0.472
0.357
0.625
0.491
A-ROC
0.969
0.536
0.787
0.875
0.715
03946
0.817
0.732
0.685
0.690
0.898
0.868
0.684
0.763
A-PR
0.633
0.318
0.495
0.519
0.523
0.615
0.474
0.552
0.870
0.535
0.337
0.443
0.418
0.543
15
4
Figure jilustrates anomaly score distributions SWaT. As shown in Figures 3a competing methods exhibit significant overlap between normal and anomalous scores, limiting detection pre- cision. In contrast, ScatterAD (Figure 3d) exhibits a clear separation, and the anomaly scores for normal points are sharply concentrated near zero, while anomalies are widely spread across higher scores, forming multiple distinguishable modes. This distinct gap between normal and anomalous distributions greatly reduces the false positive rate and enhances detection precision.
Andiraly Sibre
Anbhayy Store
Anormaly Store
Anornaly Store
(a) Anomaly Trans
(b) Sub-Adjacent Trans
(c) MTGFLLW
(d) Ours
Figure 3: The distribution of anomaly scores presented by different frameworks on the SWaT dataset.
4.3
Ablation Studies
To validate the effectiveness of ScatterAD's components, we conduct an ablation study on its dual encoder architecture and contrastive fusion mechanism (Table Z). Removing the temporal online encoder ("w/o T-Enc "") impairs the model's ability to capture temporal variations, reducing detection performance. Excluding the topological target encoder ("w/o S-Ene ") leads to significant drops, particularly on WADI and SWAT, underscoring the need to model global structural dependencies. Eliminating the contrastive fusion module ("w/o C-Fuse") results consistent performance degra- dation across all datasets, confirming its importance in aligning temporal and topological cues. Removing the Exponential Moving Average mechanism ("w/o EMA") destabilizes target encoder updates, degrading performance by limiting adaptation to temporal and structural patterns.
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Table 2: Ablation analysis of key components of ScatterAD across four real-world datasets. Eval- uation Metries: AF (Affiliated-FI), AR (Area Under the ROC Curve). Bold indicates the optimal performance; Underline indicates the suboptimal performance.
MSL
PSM
WADI
SWaT
Variation
#1 Count
w/o T-Ene
0.764
0.927
0.810
0.951
0.568
0.958
0.659
0.944
1
w/o S-Ene
0.841
0.963
0.799
0.985
0.488
0.921
0.643
0.974
0
w/o C-Fuse
0.795
0.932
0.788
0,946
0,549
0.953
0.701
0.975
0
w/o EMA
0.805
0.986
0.792
0.979
0.582
0.955
0.683
0.957
ScatterAD
0.867
0.983
0.794
0.986
0.587
0.960
0.704
0.982
6
4.4
Model Analysis
Analysis of Scattering Mechanism To validate our hypothesis that dynamically balancing scatter- ing and structure consistency enables learning node representations with both discriminability and temporal dependency, we analyze the evolution of scattering during training (Figure 4). When using only the time consistency loss Lume the heat map shows smooth but low-amplitude scattering, indicating strong temporal continuity but limited anomaly sensitivity. contrast, using only the seat- tering loss Lscatier 4b increases node dispersion and discriminability but disrupts temporal coherence. Our full method 4c which combines Lime, Lawatter and Leontrast. achieves a balanced scattering pattern that preserves structure while enhancing anomaly separability. Compared to single-loss variants, yields more stable training and more effective representations for temporal anomaly detection.
to-
Ngde index
jgde Boe
Training peridd
12
Training prrianionisperiad
Training herations) period
(a) Ltime
(b) Lecatter
(c) Call
Figure 4: Scattering of different time nodes during training on the MSL dataset, measured by the average Euclidean distance between sample representations. (a) Optimized with only Cime- (b) Optimized with only Lecatte- (c) Our method. Darker colors indicate nodes with greater average distances from others. Compared to (a) and (b), our loss function preserves temporal dependencies while promoting more discriminative node representations during training.
Analysis of Latent Discrepancy in Layer-Wise Evolution The embedding space transforms significantly across GAT layers, as shown in Figures []] Initially, normal and anomalous sample embeddings are entangled (Figure 5ay. After passing through the GAT layers (Figures (5 and [cc), structured separation emerges. Normal sample embeddings form cohesive clusters, while anomalous ones scatter outwards. This progression reveals that our model effectively increases the inter-class embedding discrepancy. Rather than explicitly enforcing margin- based separation, the model implicitly learns to emphasize this discrepancy by capturing richer spatio-temporal interactions through attention mechanisms. This behavior aligns with our notion of spatio-temporal scattering, wherein the joint modeling of temporal dynamics and variable interdependencies causes anomalies to naturally drift away from the normal data manifold in the representation space. Crucially, by amplifying the dissimilarity between normal and anomalous embeddings in both local neighborhoods and global structure, the model enhances its sensitivity to disstriiiiti (OOD) patterns.
Analysis of Anomaly Criteria Figure visually compares the anomaly scores of different methods. Our approach demonstrates superior detection across various anomaly types compared to baselines. Specifically, MTGFlow suffers from high false positives, while Anomaly momalyyTrans Trans has noticeable false negatives. Sub-Adjie Anj-Adjacent Trans, despite detecting all anomaly types, also generates significant false

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(a) Random Init
(b) GAT-Layer
(c) GAT-Layer 2
Figure 5: t-SNE embeddings of GAT on the PSM dataset. Blue and yellow dots denote perturbed negative and positive samples, respectively. The embeddings evolve from an initial entangled state to increasingly disentangled clusters in deeper encoder layers.
alarms, particularly for contextual and shapelet anomalies, where it incorrectly assigns high scores to normal regions. In contrast, our method not only highlights anomalies more effectively but also reduces false positives, leading to improved accuracy and robustness.
Oite hare
Nort Detecde
Deracto
Cobal Detecioo
Cretantul PRE beted
MM MWy
MON MA MA
WW
W
W
TELS
WW WW
MMN MMMAN
WMMMA
(a) Anomaly Series
(b) A.T.
(c) MTGFlow
(d) S.T.
(e) Ours
Figure 6: Visualization comparison of ground truth anomalies and anomaly scores across different types. The pink segments are marked as anomalies of the pattern.
5 Conclusion
In this work, we propose ScatterAD, novel approach integrating topological and temporal contrastive learning. ScatterAD enhances the discriminability of representations through a scattering mechanism, followed by the application of a topological constraint mechanism to prevent over-scattering and maintain temporal consistency, and introduces contrastive fusion strategy foster the synergistic complementarity of temporal and topological representations to improve anomaly detection perfor- mance. Comprehensive experimentation demonstrates that ScatterAD attains strong performance across diverse benchmarks. The proposed temporal and topological representation fusion mechanism significantly improves the model's ability to distinguish normal and anomalous patterns, providing valuable insights for developing more effective anomaly detection systems.
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Acknowledgments
This work is supported by the National Key Research and Development Project of China (No. 2024YFB3309900), the Fundamental Research Funds for the Central Universities (No. 2025CDIZKKYJH--09) the National Natural Science Foundation of China (No. 62476101), the Chongqing Technology Innovation and Application Development Project (No. CSTB2023TIAD-STXO00 and the Guangdong Basic and Applied Basic Research Foundation (No.
2024A1515140137).
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Answer: [Yes]
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A Proof of Theorems
Mutual Information and Representation Learning: In representation learning, we focus on how the representation Z extracted from the original data X retains information related to the target variable Y. Mutual information I (2;Y) measures the amount of information about Y contained in
1(2))) = J.J p12, log
p(z.9) p(z)p(u) dy dz.
(11)
For discrete variables, the above integral becomes a summation. Equivalently, (2:Y)= = H(Y) - H(Y) Z).
(12)
Representation learning aims to balance information retention and representation complexity:
[(Z) ((Z XC(Z),
(13)
where C(Z) measures complexity and À 0 controls the tradeoff. We consider temporal encoder Ex and a topological encoder Eg:
= (X)), ZG = Ex()), Z = S(Zr Za G).
(14)
Here f(- G¹) is constructed from domain knowledge and is independent of both the target Y and the temporal representation Zy:
fusion mechanism. Assumptions. We assume the graph G is
I(G;Y) =0 I[G; Zz) = 0.
(15)
Theorem (Graph-Conditioned Information Bound)
Let 7 = [(Zr, Zg G) be the fused representation. Then
I(Z; Y) min( I(Zr, |
(16)
Moreover, under the above assumptions, obtains
I(Z:Y) I ZT).
(17)
By the Data Processing Inequality (DPI) and the Markov chain X + (ZT, Lc, G) + Z, we have
I(Z:Y) $ {():Y).
(18)
Similarly,
I(Z;Y) & I(Zr, Zo. G;Y).
(19)
Applying the chain rule to I(Zr, Zo. G;Y) gives
T(Zr, To, G:Y) = I(GYY) + [(Zr, Zo; G).
(20)
Under (5). [[G = 0, hence
T(Z:Y) < T(ZT, 7o; G).
(21)
Next, by chain rule decomposition,
Zo; G) = I(Zr;; G) + IZZ::Y Zr,
(22)
By the independence [(;; Zr) = 0 and the fact that conditioning cannot increase mutual information,
I(Zr;Y G) = ((TrYY) I(Zo:Y Zr, < (Zo;; Y Zr).
(23)
Combining yields 117 Finally, merging with (8) gives the bound (16).
Remark. The contrastive loss in our model maximizes lower bound on [(ZT; Za G), encouraging complementary information extraction under graph constraints.
Theorem (Graph-Conditioned Mutual Information Maximization with Asymmetric Encoders)
21

---

Let Zy = (o(((( and Za = sg(Ec((X) be the temporal and topological representations under
graph G, where pe a predictor and sg(-) denotes stop- nop-gradient. The contrastive loss:
-Blo.din log exp(sa ZT |)) (d(d))
Crontrast
(24)
Ed' exp(so (or Zo (d'))
B
achieves:
I(2r Zc|G) 2 (B) Coontraat
(25)
Proof:
The proof builds on the variational lower bound for mutual information and its connection contrastive learning. Variational Lower Bound of Conditional MI. Following the variational
framework for mutual information estimation [Oord et al. 2018 Poole et al. 2019]. then define
conditional mutual information:
p(Zrls): G) p p(Zrss] G) p(zll cy]
I(Z;; Zo G) (G.d)~ E [zr Zald log
(26)
Introduce variational distribution (Z Zr[a], G) to approximate the true conditional distribution. According to the variational lower bound, we get:
[Zold [r G) p(Zald| G)
I[Z Za G) 2 B E Ezr(s 2ol4 log
(27)
Then, maximize the lower bound by contrast loss: Assume the scoring function is normalized cosine similarity:
(Zr [s]), [dld)
so [r]ss) Za(d)) 7 ((pp((r(s)) [Z]ld||
(28)
Define variational distribution:
<ZZold Zr(s), = exp
(sp (Z Za(d))
(29)
exp(so(24 Zz [s], Zald'))
dEBB
Substituting into the variational lower bound we get:
T(r;; Za
G) 2 E B( so [Zr(s), Zcld) - log Σ
(30)
d'eB
Contrast loss Ccontrast It can be rewritten as:
Ccontrast [(s (e (Zr Zo[d)] + E, log
(xp(s)(Zrls),
(31)
d'E8
Combining the two equations, we can get the final lower bound:
I ((Zr Za
G) > log([B) -
Leontrast-
(32)
Therefore, Minimizing the contrastive loss Coontrast is equivalent maximizing lower bound on the conditional mutual information / ((r,, ZG G), and this optimization inherently encourages the temporal embedding Zr and the topological embedding Zg share as much graph-constrained information as possible.
22

---

B Experimental details
B.1 Datasets
Our model ScatterAD evaluated on six real world multivariate datasets. The specific descriptions of the datasets shown in the following table.
Table 3: Dataset statistics used in our experiments. "Dims" denotes the number of dimensions (variables). "Train" and "Test" indicate the number of time points in training and labeled test sets, respectively, "AR" denotes the anomaly rate (%)
Benchmark
Source
Dims
Train
Test (labeled)
AR (%)
MSL
NASA Space Sensors
55
58,317
73,729
10.48
PSM SWaT
eBay Server Machine
25
132,481
87,941
27.76
Infrastructure System
51
495,000
449,919
12.14
WADI
Infrastructure System
123
1,209,601
172,801
5.71
NIPS-TS-GECCO NIPS-TS-SWAN
Water Quality for IoT
9
69,260
69,260
1.10
Solar Weather (Space)
38
60,000
60,000
32.60
The SWaT and WADI datasets be obtained by filling the following form: https
7 //docs. (doocl.goggl.... ele http:////wwooo/oo//////////////. vie viewf couform?ediis. rtt.eedddttreeeeteeeettrteeeeeee
B.2 Baselines
For all baseline comparisons, we rely on their official codebases and adopt the hyperparameter settings as suggested in the respective publications. The corresponding source codes are publicly accessible the URLs provided below.
Sub-Trans httpa///wh////g////////////////a ModernTCN \CC441//1hh///////hhoooo//////hr
iTransformer (ICLR'24 (CCe:///atp////htho.oooooo///rrr
TopoGDN (CIKM'24) ( http::////hhhh.ooo/ooo/////////a
DCdetector
MEMTO (NeurlPS 23) http:://gihibb.ommmmmmy3////11/1
DuoGAT (CIKM'23) http:////ggttttooooooooo/o//ataa
MTGFlow (AAAI'23)
GANE (ICLR'22) http:://gggtttbb.oo/aaaaaaaaaaap
inomalyTrans hhnn://iw...kR///ee//////oo/o/oa
VAE http:///2hhhh.oo//00/00////////t
IF http:///hhhh.ooaaa000//////////t
PCA http:////hhhcooooooo//////oooost
B.3 Evaluation Metrics
In anomaly detection evaluation, to comprehensively assess model performance across diverse anomaly detection scenarios, adopt unified evaluation protocol inspired by Liu al. [Liu and Paparrizos 2024), covering both label-based and score-based metrics.
For label-based evaluation We use the and Affiliated-F1 (Aff-F) (Huet et al. 2022], which aim to better reflect the proximity between predicted and true anomaly ranges. For completeness, we also include the widely used yet imperfect point-adjusted metric, including Point-Adjusted-Preeti Precision (PA-P), Point-Adjustdd-Ress Recall (PA-R) and Point-Adjusted-F (PA-F).
(Aff-P), Affiliated-Recat (Aff-R),
23

---

score-based evaluation We employ conventional metrics such as (AUC-PR) and (AUC-ROC), along with their range-based extensions, Range-AUC-PP (R-A-P) and Range-AUC-ROC (R-A-R) Pa parrizos et al. 2022]. which consider anomaly duration and sequential nature to more comprehensively reflect model performance. Additionally, we incorporate the Volume Under the Surface measures, VUS-ROC(V-ROC) and VUS-PR(V-PR) [Paparrizos et al.] 2022], which enhance anomaly score reliability and interpretability by introducing tolerance buffers around anomaly boundaries and using continuous labels.
B.4 Implementation Details
All experiments were performed in Python 3.9 using PyTorch and on NVIDIA Tesla-A800 GPUs. We used the Adam optimizerduring training. Initially, the batch size was set to 128, and the time window size was uniformly set to 110 for all datasets except NIPS-TS- GECCOON-TWW and NIP-TS-Swan(N- T-S), which were set to 90. The GAT in ScatterAD has H=4 attention heads per layer and a hidden dimension of 512. For hyperparameter tuning, the training set was temporarily divided into 80% for training and 20% for validation. For each dataset, the learning rate was uniformly set to 0.0001.
C Additional Experiments
Due to space constraints in the main text, we present additional experimental results in this section. In particular, we further evaluate the performance using the following metries: Kange-AUC-ROC and Range-AUC-PR, VUS-ROC and VUS-PR, Affiliated Precision, Affiliated Recall and Affiliated FI, as well point-adjusted metrics including PA-P, PA-R and PA-F. The reported results are averaged over multiple runs to ensure reliability.
Table 4: Quantitative comparison of four evaluation metrics. R-A-R and R-A-P are Range-AUC-ROC and Range-AUCC AUC-PR. V-ROC and V-PR are volumes under the surfaces created based on the ROC curve and PR curve. Bold indicates the optimal performance; Underline indicates the suboptimal performance.
Dataset
Metrie
Ours
S.T.
T.G.
Memto
DC:
M.TCN
IT.
MTG
A.T.
D.G.
GANF
R-A-R
0.920
0.750
0.695
0.701
0.913
0.756
0.655
0.748
0.891
0.613
0,846
0.725
0.641
0.633
MSL
R-A-P
0.919
0.754
0.876
0.643
0.891
0.745
0.638
0.753
0.875
0.574
0.733
0.630
0.654
0.61
V-ROC
0.872
0.745
0,705
0.601
0.904
0.718
0.647
0.618
0.884
0.512
0.644
0.626
V-PR
0,885
0.752
0.785
0.639
01883
0.737
0.631
0.727
0.864
0.573
0.678
0.631
0.6%
0.652
R-A-R
0.889
0.731
0.761
0.820
0
0.781
0.738
0,703
0.874
0.588
0.686
0.611
0.692
0.616
R-A-P
0:918
0.925
0.808
0.865
0.629
0.821
V-ROC
0.879
0.729
0.611
0.814
0.861
0.784
0.741
0.707
0.811
0.583
0.687
0.574
0.652
0,516
V-PR
0.911
0.798
0.795
0.867
0.888
0.842
0.808
0.825
0.904
0.525
0.813
0.738
0.659
0.569
R-A-R
0.912
0.956
0.734
0,710
0.967
0.791
0.931
0.539
0.952
0.563
0,568
0.732
0.638
0.663
SWaT
R-A-P
0.915
0.868
0.646
0.678
0.941
0.772
0,886
0.551
0.795
0.484
0.424
0.648
0.674
0.694
V-8OC
0.915
0,938
0,776
0.970
0.791
0.924
0,510
0.951
0.322
0.563
0,842
0.307
V-PR
0.917
0.870
0.637
0.681
0.944
0.771
0.881
0.551
0.796
0.383
0.414
0.656
0.678
0.689
R-A-R
0.927
0.945
0.623
0.866
0.912
0.561
0.629
0.614
0.955
0.413
0.617
0.692
0.714
0.677
0.773
0.265
0.359
0.637
0.794
V-ROC
0.877
0.943
0.627
0.758
0.937
0.573
0.631
0.618
0.956
0.412
0.620
0.692
0.719
0.669
V-PR
0.799
0.854
0.576
0.659
0.786
0.276
0.359
0.634
0.796
0.441
0.634
0.549
0.439
0.542
R-A-R
0.858
0.870
0,607
0.882
0.881
0.872
0.866
0.843
0.878
0.729
0.748
0.807
0.702
0.794
NIPS-TS-SWAN
R-A-P
0.941
0.936
0.587
0.851
0.948
0.946
0.944
0.879
0.929
0.705
0.781
0.639
0.646
0.743
V-ROC
0.829
0
0.607
0.861
0.853
0.847
0.805
0.866
0,607
0.807
0.755
0.681
V-PR
0.932
0.925
0,587
0.929
0.935
0.933
0.931
0,858
0.918
0.687
0.739
0.629
0.637
0.633
R-A-R
0.721
0.536
0.599
0.523
0.629
0.522
0.523
0.566
0.581
0.522
0.564
0.614
0.635
0.626
NIPS-TS-GBCCO
0.587
0.318
0,581
0,183
0.345
0.311
0.367
0.566
0.319
0.519
1.561.00
0.522
0,517
V-8OC
0.748
0.541
0.589
0,527
0,623
0.521
0.521
0.368
0.576
0.513
0.565
0.669
0,665
0.624
V-PR
0.611
0.323
0.583
0.179
0.339
0.314
0.353
0.546
0.443
0.512
0.505
0.997
0.582
0.554
#1 Count
12
0
0
2
0
24

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Table 5: Performance comparison across six benchmark datasets using affinity- -ased precision (Aff-P),, recall (Aff-R), and FI score (Aff-F). Our method consistently outperforms others, especially in Aff-F. Bold indicates the optimal performance; Underline indicates the suboptimal performance.
Dataset
Metric
Ours
S.T.
Memnto
DC.
M.TCN
MTG
A.T.
D.G.
GANF
VAE
Aff-P
0.994
0.549
0.709
0.515
0.516
0,588
0.559
0.921
0,518
0.512
0,762
0.587
0.423
0.694
MSL.
A
0.782
0.871
0.642
0.704
0.972
0.893
0.785
0.234
0
0.998
0.205
0.714
0.342
0.512
AB-F
0.867
0.673
0.674
0.595
0.674
0.709
0.652
0.374
0.673
0.677
0.323
0.642
0.574
0.591
Af-P
0.746
0,774
0.627
0.641
0.561
0.664
0.644
0.967
0.551
0.465
0.961
0.755
0.489
0.389
AfF-R
0.846
0.798
0.764
0.677
0.783
0.743
0.662
0.282
0.814
0.945
0.198
0.412
0.731
0.312
AB-F
0.797
0,786
0.689
0.659
0.653
0.701
0.652
0.436
0.657
0.624
0,329
0.524
0.589
0.437
Aft-P
0.567
0.582
0.538
0.584
0.531
0.586
0.581
0.662
0.571
0.478
0.552
0.499
0.743
0.387
SWaT
AfF-R
0.926
0.689
0.981
0.827
0.934
0.356
0.683
0.578
0.676
0.642
0.725
Af--F
0.704
0.631
0.594
0.592
0.687
0.685
0.716
0.465
0,622
0.619
0.608
0,563
0,502
0.537
Aft-P
0.714
0.632
0.652
0.545
0.619
0.546
0.575
0.771
0.555
0.476
0.751
0.456
0.658
0.601
WATH
0.525
0.956
0,429
0.981
0.874
0.571
0.809
0.508
0,985
0.957
0,601
0.726
0,422
Aft-P
0.605
0.756
0,532
0.701
0.728
0.558
0.571
0.673
0.708
0.636
0.667
0.556
0.555
0.477
AIT-P
0.754
0.798
0.521
0.525
0.531
0.515
0.491
0
0.672
0.509
0.435
0.374
0.711
0.645
NIPS-TS-SWAN
AfE-R
0:019
0.824
0.997
0.017
0.445
0.554
0.526
0.002
0.054
0.935
0.003
0.396
0.318
0.731
AIT-F
0,038
0.805
0.685
0.033
0.484
0.533
0.507
0.003
0.099
0.659
0.005
0.385
0.438
0.680
ATT-P
0.829
0.805
0.508
0.638
0.913
0.555
0.541
0.706
0.533
0.501
0.592
0.732
0,402
0.539
NIPS-TS-GECCO
AR--R
0.820
0.337
0,702
0.317
0.883
0.373
0.365
0.231
0,859
0.998
0.173
0.362
0.698
0.481
AT--P
0.825
0.476
0.589
0.424
0.648
0.446
0.438
0.348
0.658
0.667
0.268
0.481
0.531
0.509
Count
Table 6: Performance comparison in terms of PA PA-Precision (PA-P), PA-Recall (PA-R), and PA-FI (PA-F) on six benchmark datasets. Bold indicates the optimal performance; Underline indicates the suboptimal performance.
Dalasel
Metric
Ours
S.T.
T.G.
Memroo
DC.
M.TCN
IT.
MTG
A.T.
D.G.
GANF
VAE
IF
PCA
PA-P
0.933
0
0.971
0.922
0.884
0.825
0.744
0.912
0.742
0.422
0.421
MSL
PA-R
0.997
0.803
0.836
0.504
0.971
0.743
0.518
0.574
0,951
0.582
0,584
0.635
0.723
0.116
PA-F
0.964
0,863
0.714
0.664
0.942
0,807
0.659
0.648
0.934
0.652
0.489
0,512
0.598
0.444
PA-P
0.985
0,980
0.969
0.971
0.975
0.962
0.992
0.973
0.837
0.647
PSM
BA-R
0.977
0.904
0.819
0.966
0.984
0.954
0.893
0.666
0.983
0.731
0.652
0.524
0.739
0.337
DA-F
0.981
0.911
0.801
0.977
0.977
0.965
0.926
0.797
0.978
0.780
0.788
0.596
0.531
0.167
DA-P
0,931
0.931
0.894
0.874
0,952
0.892
0.786
0.548
0.585
SWIT
PA-R
0.973
0.923
0.754
0.659
0.949
0.875
0.962
0.338
0.992
0.863
0.157
0.477
0.328
0.221
PA-F
0.951
0.863
0.788
0.756
0.939
0.884
0.916
0.500
0.943
0.822
0.244
0.526
0.472
0.548
PA-P
0.813
0.856
0.841
0,751
0.607
0.291
0.355
0.875
0,637
0.966
0.847
0.658
0.572
0.547
WADI
PA-R
0.917
0.890
0.829
0.874
0.917
0,372
0,511
0,570
0,947
0.595
0.570
0.116
0.612
0.723
PA-F
0.862
0.873
0.807
0.731
0.326
0.419
0.690
0.738
0,705
0.681
0.512
PA-P P
0.991
0.997
0.694
0.988
0.966
0.954
0.952
0.662
0.972
0.591
0.695
0.633
0.364
0.699
NIPS-TS-SWAN
PA-R
0.585
0.556
0.629
0.526
0.591
0.592
0.591
0.533
0,540
0.438
0.576
0.395
0.738
0.403
DX-F
0.736
0,7L1
0.687
0.733
0.731
0.729
0.591
0.695
0.503
0.630
0.497
PA-P
0.669
0.524
0.582
0,889
0.391
0.404
0.471
0.371
0.277
0.539
0.346
0.982
0.548
0.466
NIPS-TS-GECCO
PA-R
0.944
0.463
0.789
0,352
0.597
0.321
0.321
0.391
0.391
0,745
0.364
0,673
0,422
0.735
PA-F
0.784
0.491
0.620
0.504
0.472
0.357
0.381
0.381
0.325
0.025
0.355
0.491
0.81
0.583
#1 Count
6
2
2
0
0
0
2
3
0
25

---

D Analysis on the Universality and Robustness of the Scattering Phenomenon
To investigate the universality of the scattering phenomenon and evaluate the model's robustness, we conduct a quantitative analysis across all six benchmark datasets under varying levels of noise. Specifically, we inject additive Gaussian noise with four different intensities E {0, 0.5, 1.0, 2.0)) into the normalized test data, where 0.0 serves the no-noise baseline. We then quantify the model's internal discriminative power by calculating
average scattering score for both normal and anomalous samples. This score is defined as the mean Euclidean distance of samples to their respective class center. The separation ratio between these scores (Anomalous/Norma is analyzed to measure class distinguishabiiity, with results presented in Table 7
Table Analysis of scattering scores and their separation ratio under different Gaussian noise levels (o). A separation ratio greater than 1.0 suggests that anomalous samples are, average, farther from their class center than the normal samples, confirming the scattering phenomenon.
Noise Level
Scattering Score (Normal)
Scattering Score (Anomalous)
Separation Ratio %)
Dataset
(σ)
0.0
16.0
18.2
1.14
0.5
21.6
31.6
1.46
MSL
1.37
2.0
65.9
84.6
1.28
0.0
31,5
37.8
1.20
0.5
33.1
40.3
1.22
1.0
41.2
47.4
1.15
2.0
61.1
65.7
1.08
0.0
73.10
94.09
1.29
0.5
75.63
95.71
1.27
SWaT
1.0
79.22
98.03
1.24
2.0
84.50
102.11
1.21
0.0
0.78
1.16
1.49
0.5
34,9
43.4
1.24
WADI
1.0
87.1
72.4
0.83
2.0
166.7
137.0
0.82
0.0
20.2
28.4
1.41
0.5
21.8
32.4
1.49
NIPS-TS-SWAN
1.0
27.7
42.2
1.52
2.0
42.1
65.5
1.56
0.0
21.60
95,97
4.44
0.5
26.10
105.33
4.04
NIPS-TS NIPS-TS-GECHO TS-GECCO
1.0
40.70
41.40
1.02
2.0
68.60
197.30
2.88
The results presented in Table 7 clearly reveal consistent anomaly scattering pattern across the six diverse, real-world datasets. Notably, for the WADI dataset, the scattering score of normal samples surpasses that of anomalies under medium-to-high noise conditions (o > 1.0), resulting in separation ratio less than 1.0. We attribute this phenomenon to WADI's high dimensionality (123 dimensions), where strong noise may have compounding effect and lead to complex changes in inter-variable correlations. This finding indicates that while our method is generally robust across various noisy environments, its performance may be sensitive strong noise when applied to extremely high-dimensional datasets that feature subtle anomaly patterns. This analysis helps to comprehensively characterize the performance boundaries of our method.
E Parameter Sensitivity Analysis
We analyze the sensitivity of ScatterAD to key hyperparameters across six datasets (Figure 7 The model performs consistently across varying window sizes, demonstrating robustness both short- and long-range temporal contexts. A window size of 110 yields the highest AUC-ROC and is thus
26

---

used in all experiments. Therefore, this brief window size is selected in our experiments. We further studied the sensitivity of the other 2 parameters: the dimension of the hidden layer and the number of encoder layers. For encoder depth, two layers consistently achieve the best performance while keeping the model lightweight. Increasing the hidden dimension improves performance, and we choose 512 as a balanced default to maintain model efficiency.
1.00
AUC-RCC
0.90
AUC-RD
MS1
PSM
0.8
N-T-10
6.10
0.75
10
98
100
110
120
Window Säze
Layer Number
1.00
8.55
AUC-ROC
0.00
AUC--OC
MSE
PS3
8,85
0.90
0.75
200
410
680
800
1003
0,5
1.0
D. model Size
EMA Decay (m)
Figure 7: Parameter sensitivity analysis in ScatterAD.
Anomaly threshold is a important hyperparameter, which may affect the determination of anomaly or not. We have default value for all benchmarks. As shown Table when is the range of 0.5 to I, it has little effect the final model performance. PSM and SMAP are also more robust to anomaly thresholds than MSL. For the three benchmarks, its best results appear when a equals 0.8 or 1.
Table 8: Sensitivity analysis of the hyperparameter å on six datasets. Metrics: AF (Affiliated-FI). AR (AUC-ROC). Best values per column are highlighted bold.
Dataset
MSL
PSM
SWaT
WADI
N-T-S
N-T-W
Metric
AF
AR
AR
AR
AR
8 0.2
0.863
0.986
0.789
0.955
0.656
0.517
0.929
0.944
0.029
0.786
0.799
0.954
6 = 0.4
0.861
0.982
0.793
0.952
0.698
0.521
0.934
0,974
0.038
0.792
0,803
0.964
ő 0.6
0.843
0.978
0.783
0.946
0.704
0.586
0.961
0.975
0.035
0.790
0.814
0.965
8 0.8
0.855
0.981
0.794
0.969
0.701
0.587
0.960
0.957
0.038
0.792
0.818
0.959
1.0 1.0
0.867
0.986
0.792
0.969
0.702
0.583
0.955
0.957
0.037
0.789
0.823
0.969
27

---

F Analysis of Performance on the NIPS-TS-SWAN Dataset
notable performance discrepancy is observed the NIPS-TS-SWAY dataset, specifically con- cerning the Affiliated-F1 I (Aff-FI) metric. It is important to first establish that our proposed model, ScatterAD, successfully generates separable anomaly scores for this dataset. The high Point-Adjusted Fl (PA-FI) score of 0.736 (as reported in Table demonstrates that anomalous events are correctly assigned higher scores than normal ones the point level.
To investigate the low Aff-Fl score, we hypothesize that the metric itself may be overly sensitive to minor localization errors on this particular dataset. To test this, we devised a controlled simulation experiment, independent of our model, to evaluate the intrinsic robustness of the Aff-F1 metric. In this simulation, assume perfect model whose predictions exactly match the ground truth (achieving an F1 score of 1.0). We then introduce a minor, realistic localization error by shifting the entire prediction sequence by small number of timesteps (At) and observe the resulting changes in both Aff-F1 and traditional point-wise Fl score. The experiment is also conducted on the MSL dataset for comparison.
Table 9: Sensitivity analysis of F1 metrics temporal localization errors. A perfect prediction sequence is shifted by a few timesteps, and the degradation of segment-level (Aff-FI) and point-level Fl scores measured.
Localization Error (Shift in steps)
Aff-F1 Score (Segment-level)
Point-wise F1 (Point-level)
Dataset
steps
1.000
1.000
1 step
0.065 (93.5%.)
0.686
NIPS-TS-SWAN
2 steps
0.174
0.697
5 steps
0.223
0.704
10 steps
0.226
0.709
steps step
1.000
1.000
1.000
0.995
MSL
2 steps
1.000
0.991
5 steps
0.972 (2.8%L)
0.977
10 steps
0.889
0.954
The results in Table are revealing. For the NIPS-TS-SWI SWAN dataset, minuscule localization shift of just one timestep causes the Aff-FI score to plummet from 1.0 to 0.065, a 93.5% reduction. In stark contrast, the same shift the MSL dataset has impact the Aff-F1 score, and even 5-step shift results in only a minor 2.8% drop. This demonstrates the extreme sensitivity of the Aff-F1 metric specifically on the SIPS-TS-SWAY dataset. This inherent characteristic of the metric and dataset interaction explains why our model, along with many other state-of-the-ee methods (as shown in Table D. exhibits poor performance on this particular metric. Conversely, the more traditional point-wise F1 score shows much greater resilience. This justifies our decision to report both scores for a balanced and comprehensive evaluation.
To further validate our model's 's detection capability at a macro level, we compiled overall prediction statistics for ScatterAD on the NIPS-TS-SWAN dataset, shown in Table T0
Table 10: Macro-level prediction statistics for ScatterAD on the NIPS-TS-ST SWAN dataset.
Statistic
Ground Truth
Our Prediction
Total Anomaly Ratio
32.6%
32. %
Number of Anomaly Segments
551,937
843,596
The statistics show that the total proportion of anomalies predicted by ScatterAD (32.1%) closely aligns with the ground truth (32.6%), demonstrating its accurate high-level detection capability. However, the model identifies higher number of individual anomaly segments. This suggests that the
28

---

model may tend to partition a single continuous ground-trung truth anomaly into several smaller, consecutive segments. Consequently, due such minor, unavoidable localization offsets, its performance is significantly underestimated by the strict segment-matching criteria of the Aff-FI metric under these specific data conditions.
G Analysis of the Scattering Center Initialization
The initialization strategy for the scattering center, c, is key design consideration impacting model stability and the potential for learning suboptimal solutions. In our framework, the scattering center c is initialized once the beginning of each training run and remains fixed throughout the training process: it is not a learnable parameter updated via backpropagation. The role of this fixed center is twofold:
The center offers a stable, convergent target direction for the representations of normal samples. This guides the model to learn mapping that projects these representations into a compact region on the latent hypersphere.
Provide Fixed Anchor Point:
Break Symmetry and Regularize: fixed, predefined center (e.g., a canonical basis vector like [1, O, .... O)) might introduce an undesirable bias, potentially inducing the model learn a trivial solution where all normal samples map to a specific axis. By randomly initializing we avoid this risk and compel the model to learn more generalizable, rotation- invariant representations. This can be viewed as form of implicit stochastic regularization, preventing the model from overfitting predefined direction in the latent space.
G.1 Stability Analysis
To empirically validate the stability of this random initialization strategy, we conducted an experiment by repeating the full training and evaluation process with different random seeds on the MSL and PSM datasets. A different seed results in a different initialization for the scattering center c. As shown in Table T1 the standard deviation of the performance metrics is extremely low, confirming that the model's 's performance is robust and not sensitive to the specific random choice of the center.
Table 11: Stability analysis of the random scattering center initialization across 5 runs with different random seeds. The low standard deviation indicates high stability.
Dataset
Metric
Original
Mean
Std. Dev.
Aff-F1
0.865
0.865
+ 0.003
MSL
A-ROC
0.986
0.986
+ 0.001
Aff-FI
0.797
0.797
+ 0.002
PSM
A-ROC
0.986
0.986
+ 0.003
G.2 Ablation Study on Initialization Strategies
Furthermore, to demonstrate the effectiveness of chosen approach, performed ablation study comparing several alternative center initialization strategies:
Zero: Initialize the center the origin (zero vector) to test the simplest symmetric starting point.
Fixed-Raae -adeu-: Initialize the center on hypersphere with fixed radius to test the model's sensitivity the initial magnitude.
Multi-center: Use multiple independent scattering centers test the model's ability to capture potentially multi-modal distributions of normal data.
The results, presented Table T2 compare these strategies with the random_ in_ ball approach (randomly selecting point within the unit hypersphere) proposed in our paper. The findings confirm
29

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that our model is not sensitive to the specific initialization strategy of c. The performance remains consistently high across different methods, indicating that the random_in_ robust, and effective choice that avoids the need for additional hyperparameter tuning (e.g., setting radius the number of centers).
in ia__iinn_llllll__ strategy a simple,
Table 12: Ablation study different scattering center initialization strategies. Performance metrics (Aff-FI and AUC ROC) remain stable across various strategies, validating the robustness of our approach.
Dataset
Center Strategy
Num Centers
Radius
Aff-F1
AUC-ROC
random_ in ball (Ours)
N/A
0.867
0.986
zero
0.0
0.866
0.986
MSL
fixed_radius
0.3
0.866
0.986
fixed_radius
0.7
0.864
0.986
multi-center
3
N/A
0.869
0.986
ball random_ine in_llllllalllllll (Ours)
N/A
0.797
0.986
zero
0.0
0.797
0.986
PSM
fixed_radius
0.3
0.797
0.986
fixed_radius
0.7
0.796
0.986
multi-center
N/A
0.792
0.980
random_ in ball (Ours)
N/A
0.704
0.982
zero
0.0
0.704
0.982
SWaT
fixed_radius
0.3
0.698
0.977
fixed_radius
0.7
0.702
0.979
multi-center
3
N/A
0.702
0.980
H
Robustness to Graph Topology
To evaluate the model's sensitivity and robustness to the graph structure, we conducted an experiment to simulate dynamically changing graph topologies. We extended our data loading process to generate a different graph structure for each input time window during both training and inference. This contrasts with our main approach of using single, static graph learned from the training data. We designed two dynamic topology generation strategies:
Random Topology: For each time window, connections between nodes are randomly established with fixed probability bloo_p33.. To ensure basic graph connectivity, the connections between physically adjacent nodes are always preserved. This strategy simulates scenarios where inter-sensor correlations are irregular and appear randomly over time.
K-Nearest Neighbors (KNN) Topology: For each time window, a topology is constructed by computing the Fuclidean distance between the time series of each node and all other nodes within that window. Each node is then connected to its k-nearest neighbors (we use & = 3).
The performance and efficiency these dynamic strategies were compared against our original static graph approach the MSL and PSM datasets. The results are presented in Table 13
The results in Table indicate that the model's detection performance highly robust the underly- ing graph structure. While dynamic topology strategies lead to marginal performance improvements some specific cases (e.g., Random Topology on MSL and KNN Topology on PSM), the overall accuracy metrics remain remarkably stable across all three approaches. However, these dynamic strategies introduce significant computational overhead, as evidenced by the substantially lower throughput and higher average inference times. This analysis demonstrates that our proposed model
30

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Table 13: Performance and efficiency comparison between static (Original) and dynamic graph topology strategies, While dynamic strategies offer marginal performance changes, they incur significant computational overhead, reducing throughput and increasing inference time.
Throughput (samples/sec)
Avg. Inference Time (ms)
Dataset
Strategy
Afr-F
PA-F
A-ROC
A-PR
Original (Ours)
0.866
0.964
0.986
0.932
60.35
16.57
MSL.
Random Topology
0.867
0.967
0.987
0.937
38.33
26.09
KNN Topology
0.862
0.961
0.984
0.927
49.17
20.34
Original (Ours)
0.797
0,981
0.986
0.969
62.17
16.09
PSM
Random Topology
0.791
0.975
0.980
0.961
31.77
31.47
KNN Topology
0.803
0,966
0.971
0.948
25.66
38.98
exhibits strong robustness graph topology, and employing static graph derived from the training data strikes an excellent balance between detection effectiveness and computational efficiency.
I Convergence analysis
1.4
333
1.800
1450
2000
(a) MSL.
(b) PSM
(c) SWaT
14
(d) WADI
N-T-S
(f) N-T-W
Figure 8: The convergence performance of our model's is loss six different datasets is shown in the figure.
The convergence performance of the total loss (Equation 9) of our model six different datasets is shown the figure. Several consistent features can be observed from these curves: In all experiments, the loss function shows a similar convergence pattern, rapidly decreasing from an initial value of about 1.6 and gradually stabilizing. Specifically, within the first 100 iterations, the loss value drops rapidly to about 0.4, indicating that the model can effectively capture the key features of the data. Subsequently, the loss curve enters gentle decline phase and converges to stable value of about 0.3-0.4. is worth noting that although the total number of iterations each experiment different (300, 800, 2000 and 3000 respectively). they all show similar convergence curve shapes, proving that the temporal topology constraint model we proposed has stable optimization characteristies and can effectively learn the main patterns of the data in small number of iterations (about 100-200 iterations). This consistent convergence behavior further verifies the robustness and scalability of our model on different datasets.
31

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Run Times
To evaluate the practicality of ScatterAD in production environments, we comprehensively compared the training time, inference time, and GPU memory usage of various neural network models on the SWaT dataset. The results are shown in the table.
Table 14: Comparison of training time, inference time, and GPU memory consumption the SWaT dataset.
Method
Training Time (s/iter)
Inference Time (s)
GPU Memory (GB)
GANF
0.177
0.834
8,202
DuoGAT
0.283
239.608
6,050
AnomalyTrans
0.378
86.134
12.424
MTGFlow
0.378
2.119
9,458
iTransformer
0,027
59.515
2.061
MCdernTCN
0.012
59.882
1.923
DCdetector
0.012
59.882
1.923
MEMTO
0.621
8.582
19.91
TopoGDN
0.131
66.146
2.830
Sub-Adjacent Trans
0.107
1.762
5.396
ScatterAD
0.042
2.124
3,458
Table 14 reports the training and inference time per iteration, along with GPU memory usage on the SWaT dataset. Our method achieves the fastest training speed (0.0421s/iter). It also offers competitive inference efficiency and low GPU memory consumption. Compared DuoGAT and Anomaly Trans, which suffer from extremely long inference times and high memory overhead, our model demonstrates superior scalability and deployment practicality.
K Limitations and Future Work
While ScatterAD shows strong performance across diverse real-world datasets, there remain few avenues for future improvement. ScatterAD effectively captures spatial-tempores dependencies, and incorporating explicit modeling of inter-series interactions may offer additional benefits in specialized applications such as industrial monitoring or financial forecasting. For future work, we aim explore adaptive graph construction that evolves with streaming data to improve robustness in non-stationary environments. In addition, integrating causal discovery modules could enhance interpretability and detection precision. Another promising direction is extending ScatterAD to semi- supervised or active learning setting, where limited anomaly labels can further guide representation learning.
32


Reviews:
[{"review_id": "Gv7mSvzmaZ", "reviewer": "Reviewer_xfeK", "summary": "This paper presents ScatterAD, a novel anomaly detection framework for multivariate time series in industrial IoT, which links representation learning to the information bottleneck principle, proving that maximizing conditional mutual information between temporal and topological views improves anomaly discrimination. ScatterAD jointly models temporal dynamics and topological techniques via the scattering loss, temporal constraint, and contrastive fusion. Extensive experiments on six benchmarks show state-of-the-art results of ScatterAD.", "questions": "1. The temporal graph is fixed during training/inference. I  wonder how the performance of ScatterAD would be if the topology were to change suddenly.\n2. Using mean pairwise Euclidean distance instead of dispersion metrics like Mahalanobis distance or others needs clarification.  Does this choice affect sensitivity to feature scaling?\n3. PA-F of the method was high, but the AF-F in NIPS-TS-SWAN dropped sharply, although the authors stated the possible reasons in Sec. 4.2. More analysis could reveal method limitations. Or does scattering fail for bursty patterns?", "limitations": "1. Although the authors acknowledge static topology and lack of causality modeling in Appendix H. However, these aspects need more expansion.\n2. More testing of the proposed method for Real-world noisy data will help explore the robustness.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: This paper addresses a critical gap in spatio-temporal anomaly detection by unifying temporal consistency and topological dispersion. Achieves impressive results across diverse datasets. It is novel to formalize scattering as a learning signal and integrate information bottleneck theory into multivariate time series anomaly detection. The paper is well-structured, with clear method descriptions and precise theoretical proofs in the Appendix. \nAuthors provide a technically rigorous study with thorough ablation studies, sensitivity analysis, and visualizations. The proposed scattering mechanism is empirically validated via feature space analysis and training dynamics.\n\nWeaknesses: While inference speed is competitive, the need for careful hyperparameter tuning may hinder plug-and-play adoption in other testing domains. The mutual information bound relies on strong independence assumptions, which may not hold if the graph structure correlates with anomalies. Assuming a static graph structure, it seems to limit its applicability to systems with evolutionary dependencies.", "was_revised": true, "final_justification": "The concerns have been mostly addressed by the authors.", "rebuttal": "We sincerely thank Reviewer xfeK for the detailed and constructive comments, which are instrumental in further improving our paper.\n\n**W:** Regarding hyperparameter sensitivity, theoretical assumptions, and their impact on generalization.\n\nTo systematically evaluate the robustness of our model, we conduct a detailed sensitivity analysis in Appendix E of our paper on key hyperparameters, including the time window size, number of encoder layers, hidden dimension, EMA decay rate, and the anomaly threshold $δ$. The results show that ScatterAD maintains stable performance across a wide range of parameter settings. And, we detail the impact of changes in graph structure in Q1.\n\n**Q1:** Regarding the performance of a static graph when the topology changes.\n\nTo simulate dynamically changing graph topologies, we extend our data loading process to generate different graph structures for each input time window during training and inference. We design two dynamic topology generation strategies:\n\n(1) **Random Topology**: For each time window, we randomly establish connections between nodes with a certain probability (edge_prob=0.3). To ensure basic graph connectivity, we preserve the connections between adjacent nodes. This strategy simulates scenarios where connections are irregular and appear randomly.\n\n(2) **K-Nearest Neighbours (KNN) Topology**: For the topology constructed within each time window, we compute the Euclidean distance between each node and all other nodes, connecting it to its k-nearest neighbours ($k=3$).\n\n| **MSL**             |   **AFF_F**   |   **PA-F**   |   **A-ROC**   |   **A-PR**   |   **Throughput (samples/second)**   |   **Avg. Inference Time (ms)**   |\n| :-------------- | :-------: | :------: | :-------: | :------: | :--------------------------: | :--------------------------: |\n| Original (Ours)        |   0.866   |  0.964   |   0.986   |  0.932   |            **60.35**             |            **16.571**            |\n| Random Topology |   **0.867**   |  **0.967**   |   **0.987**   |  **0.937**   |            38.33             |            26.091            |\n| KNN Topology    |   0.862   |  0.961   |   0.984   |  0.927   |            49.17             |            20.338            |\n| **PSM**         | **AFF_F** | **PA-F** | **A-ROC** | **A-PR** | **Throughput (samples/second)** | **Avg. Inference Time (ms)** |\n| Original (Ours)         |   0.797   |  **0.981**   |   **0.986**   |  **0.969**   |            **62.17**             |            **16.085**            |\n| Random Topology |   0.791   |  0.975   |   0.980   |  0.961   |            31.77             |            31.472            |\n| KNN Topology    |   **0.803**   |  0.966   |   0.971   |  0.948   |            25.66             |            38.976            |\n\nThe results indicate that while dynamic topology strategies lead to marginal performance improvements in some cases (e.g., Random Topology on MSL and KNN Topology on PSM), they introduce additional computational overhead. Therefore, ScatterAD exhibits strong robustness to graph topology, and employing a static graph strikes a good balance between efficiency and effectiveness.\n\n**Q2:** Clarification on the choice of Euclidean distance and its sensitivity to feature scaling.\n\nOur goal is to find a simple yet effective inductive bias. Compared to complex methods that require covariance estimation and rely on distributional assumptions, Euclidean distance, being a computationally simple and assumption-free geometric metric, is a more direct and robust choice to achieve our objective. To address the potential issue of feature scaling,  we first apply standard Z-score normalisation to all input data. Second, we perform L2 normalisation on the target representation $h_{target}$ before calculating the scatter loss ($L_{scatter}$) (Equation 5), ensuring robustness to feature scales.\n\nTo address the concerns regarding different distance metrics, we conduct a new experiment comparing our method with two other classic distance metric variants: **Mahalanobis distance** and **Kullback-Leibler (KL) Divergence**. In the Mahalanobis distance variant, we estimate the global covariance matrix from the representations of the training set, following standard practice. We report the performance on three representative datasets below.\n\n| Dataset | Metric | ScatterAD (Euclidean - OURS) | ScatterAD (Mahalanobis) | ScatterAD (KL Divergence) |\n| :------ | :----- | :--------------------------: | :---------------------: | :-----------------------: |\n| MSL     | Aff-F1 |          **0.867**           |          0.851          |           0.789           |\n|         | PA-F   |          **0.964**           |          0.956          |           0.928           |\n|         | A-ROC  |          **0.986**           |          0.986          |           0.972           |\n|         | A-PR   |            0.932             |        **0.934**        |           0.866           |\n| PSM     | Aff-F1 |          **0.797**           |          0.792          |           0.129           |\n|         | PA-F   |          **0.981**           |          0.976          |           0.739           |\n|         | A-ROC  |          **0.986**           |          0.980          |           0.793           |\n|         | A-PR   |          **0.969**           |          0.962          |           0.701           |\n\n**Q3:** In-depth analysis of performance discrepancy on the NIPS-TS-SWAN dataset.\n\nFirst, we would like to clarify that ScatterAD successfully generates separable anomaly scores for the NIPS-TS-SWAN dataset. The high PA-F1 score (0.736) demonstrates that anomalous events are assigned higher scores than normal ones. Furthermore, to objectively analyse the characteristics of the Aff-F1 and Point-wise F1 metrics, we design a new experiment to simulate the metric sensitivity, which is independent of our model and aims to test the robustness of the Aff-F1 metric itself on the NIPS-TS-SWAN dataset. We assume a model whose predictions can perfectly match the ground truth. We then simulate a minor, realistic localisation error by shifting the entire prediction sequence by $n$ timesteps and observe how the Aff-F1 and Point-wise F1 scores change.\n\n| Dataset      | Localization Error (Shift) | **Aff-F1 Score (Segment-level)** | **Point-wise F1 (Point-level)** |\n| :----------- | :------------------------- | :------------------------------: | :-----------------------------: |\n| NIPS-TS-SWAN | 0 steps                    |            **1.000**             |              1.000              |\n|              | 1 step                     |          0.065 (93.5%↓)          |              0.686              |\n|              | 2 steps                    |              0.174               |              0.697              |\n|              | 5 steps                    |              0.223               |              0.704              |\n|              | 10 steps                   |              0.226               |              0.709              |\n| MSL          | 0 steps                    |            **1.000**             |              1.000              |\n|              | 1 step                     |              1.000               |              0.995              |\n|              | 2 steps                    |              1.000               |              0.991              |\n|              | 5 steps                    |          0.972 (2.8%↓)           |              0.977              |\n|              | 10 steps                   |              0.889               |              0.954              |\n\nThe results show that for the NIPS-TS-SWAN dataset, a localisation shift of just one timestep can cause the Aff-F1 score of the model to catastrophically collapse from 1.0 to 0.065. This explains why many other SOTA models (as shown in Table 1 of our paper) also perform poorly on this metric. In contrast, the more traditional Point-wise F1 score shows only a modest decline. This is why we report both scores.\n\nTo further validate our model's detection capabilities at a macro level, we compiled overall prediction statistics for ScatterAD on the NIPS-TS-SWAN dataset.\n\n| Statistic                  | Ground Truth | Our Prediction |\n| :------------------------- | :----------: | :------------: |\n| Total Anomaly Ratio        |  **32.6%**   |   **32.1%**    |\n| Number of Anomaly Segments |   551,937    |    843,596     |\n\nThe statistics show that the total proportion of anomalies predicted by ScatterAD (32.1%) closely aligns with the ground truth (32.6%), demonstrating its accurate detection capability. However, the higher number of predicted segments suggests that the model may tend to respond to a single continuous anomaly as several smaller segments. Consequently, its performance is underestimated by the Aff-F1 metric under these specific data conditions due to such minor, unavoidable localisation offsets."}, {"review_id": "acWgj6LUik", "reviewer": "Reviewer_9v8x", "summary": "This paper proposes ScatterAD, a novel approach for multivariate time series anomaly detection that leverages temporal-topological scattering mechanisms. The method combines a temporal encoder and a topological encoder with contrastive fusion to learn complementary spatio-temporal representations. The core insight is that anomalous samples exhibit more pronounced scattering patterns in high-dimensional space compared to normal samples. The authors provide theoretical justification using information bottleneck theory and demonstrate state-of-the-art performance across six real-world datasets.", "questions": "(1) Could you provide detailed architectural specifications for the online encoder $f_θ(·)$ and target encoder $f_ϕ(·)$? Additionally, please clarify how h_online and h_target are obtained from these encoders - are they direct outputs or derived through additional processing steps?\n\n(2) Many critical parameters lack proper definition, including $l, k$ in Section 3.2, $τ$ in equation (1), and ξ in equation (8). Could you provide precise definitions of these parameters and their typical value ranges? Furthermore, what are the specific values of parameters such as $l$ you used in your experiments?\n\n(3) How does the theoretical result $I(Z_T; Z_G | G)$ connect to your actual implementation? What specific graph structure $G$ is used in practice, and how does it relate to the conditional mutual information maximization in your theoretical framework?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n(1) The paper introduces conditional mutual information maximization based on information bottleneck theory into multivariate time series anomaly detection, providing theoretical proof that maximizing  $I(Z_T; Z_G \\mid G) $ can effectively enhance cross-view consistency and improve representation discriminability.\n\n(2) The method jointly considers temporal features and topological features, significantly enhancing the model's ability to capture complex spatio-temporal dependencies and anomaly patterns.\n\n(3)The paper employs multiple optimization objectives including scattering loss $L_{scatter}$, temporal consistency loss $L_{time}$, and contrastive fusion loss $L_{contrast}$. The contrastive fuse learning design can effectively prevent over-scattering of representations while maintaining temporal structural consistency, simultaneously improving feature discriminability and representation stability.\n\n**Weaknesses**\n\n(1) The paper states that $h_i$  \"represent the input feature vectors\" in equation (2), but subsequent text indicates that these symbols denote different data representations $h_i = h'_i + h^{(l)}$ while using identical notation. This creates ambiguity regarding what specific data each variable represents.\n\n(2) Section 3.1 introduces the online encoder $f_θ (∙)$ and target encoder $f_ϕ (∙)$ without providing concrete model architecture details. The paper lacks a clear explanation of how $h_{online,t}$ and $h_{target,t}$ are obtained, making it difficult to understand the specific implementation and reproduce the results.\n\n(3) Numerous formulas contain unexplained parameters, including $τ$ in Section 3.2, $l,k$ in equation (1), and ξ in equation (8). Additionally, critical parameter values such as the range or default value of $l,τ$ are not specified.\n\n(4) Section 3.2 states \"the node features are aggregated with topological features and temporal features, yielding $h_i = h'_i + h^{(l)}$\". \nThis description lacks clarity regarding how many values are being aggregated and what specific representations correspond to \"node features,\" \"topological features,\" and \"temporal features,\" respectively.\n\n(5) Section 3.3 \"Temporal Topological Scattering Representation Learning\" presents $c = \\text{Randn}(\\cdot), c = \\frac{c}{\\|c\\|} \\cdot \\epsilon$, what appears to be two equations without clearly describing their sequential relationship or dependencies.\n\n(6) The experimental comparison could benefit from including more classic and recent methods such as ModernTCN [Ref1], iTransformer [Ref2], and DCDetector [Ref3].\n\n- [Ref1] Moderntcn: A modern pure convolution structure for general time series analysis, ICLR. 2024.\n\n- [Ref2] Itransformer: Inverted transformers are effective for time series forecasting, ICLR 2023.\n\n- [Ref3]Dcdetector: Dual attention contrastive representation learning for time series anomaly detection, KDD. 2023.", "was_revised": true, "final_justification": "This paper provides a comprehensive clarification during rebuttal that addresses most of my concerns.", "rebuttal": "We sincerely thank Reviewer 9v8x for the insightful comments and valuable suggestions. We will address your concerns one by one below and will integrate these clarifications into the final version of our paper.\n\n**W1:** Ambiguity of the notation $h_i$.\n\nThank you for pointing out this ambiguity. We will use $z_i = h_t^t + h_i^{(l)}$ to denote the final representation and will update this notation consistently throughout the rest of the revised manuscript.\n\n**W2:** Lack of specific model architecture details.\n\nTo clarify, the online encoder $f_\\theta(·)$ and the target encoder $f_\\phi(·)$ share the exact same backbone architecture, which consists of a causal convolution module (Eq. 1 in our paper) followed sequentially by a Graph Attention Network (GAT) layer (Eqs. 2 and 3 in our paper). However, the outputs of these two encoders undergo **asymmetric processing steps** before being used in the final loss computation, as detailed below:\n\n| Processing Stage         | Online Path                                   | Target Path                                             |\n| :----------------------- | :-------------------------------------------- | :------------------------------------------------------ |\n| 1. Encoding              | $h_{pred} = f_\\theta(X)$                      | $h_{target} = f_\\phi(X)$                                |\n| 2. Asymmetric Processing | $h_{online} = \\text{Predictor}(h_{pred})$     | $h'\\_{target} = L2{Normalize}(h_{target})$ |\n| 3. Loss Contribution     | Used to compute $L_{time}$ and $L_{contrast}$ | Used to compute $L_{scatter}$ and $L_{contrast}$        |\n\nThis **asymmetric design** is a well-established and highly effective strategy in momentum contrastive learning frameworks (Kaiming et al., 2020; Byeongchan et al., 2023). It prevents the model from collapsing by breaking the symmetry through an additional learning task (predicting the target representation). We will update the methods section in the final version of our paper with a detailed process description and diagram to ensure the clarity and full reproducibility of our work. \n\n**W3:** Unexplained parameters.\n\nAs per your suggestion, we will define all parameters upon their first appearance in the revised manuscript and include their default values. We provide the clarifications here:\n\n*   (1) $\\tau$ in Section 3.2: This is the look-back window size of the temporal graph, which defines neighbourhood relations as $(u_{t-k}, u_t)$, where $k \\in [1, \\tau]$. In our experiments, we use $\\tau=2$.\n*   (2) $l$, $k$ in Equation (1): $l$ represents the layer index of the causal convolution, and $k$ is the kernel size. We use two two-layer ($l=2$) with a kernel size of $k=3$.\n*   (3) $\\xi$ in Equation (8): This represents the parameters of the target encoder $f_\\phi(·)$. We use $\\xi$ to maintain consistency with the momentum update literature, where $\\theta$ is often used for the online encoder. For clarity, we will unify the notation to use $\\phi$.\n\n**W4:** Ambiguity of feature aggregation.\n\nIt is important to note that the aggregation operation $h_i = h_t^t + h_i^{(l)}$ is a simple **element-wise addition**. To be precise:\n\n*   (1) $h_t^t$ (\"temporal features\") is the output of the initial **causal convolution encoder** (from Eq. 1).\n*   (2) $h_i^{(l)}$ (\"topological features\") are the attention-weighted features learned from neighbouring nodes via the **GAT layer** (from Eq. 3).\n*   (3) $h_i$ (\"node features\") is the final representation obtained by the **element-wise addition** of these two components.\n\nWe will revise the relevant text in Section 3.2 in the revised manuscript, following your suggestion, to explicitly state that this is an element-wise addition and to clarify the origin of each term. Furthermore, as per your suggestion in W1, we will resolve the ambiguity caused by the notation $h_i$.\n\n---\n\n**W5:** Relationship between the formulas for $c$.\n\nThank you for requesting clarification on this. The two parts of this formula describe a **two-step sequential process** for initializing the scattering center $c$.\n\n*   (1) Step 1: $c'  = \\text{Randn}(\\cdot)$: First, a vector $c'$ is sampled from a standard normal distribution.\n*   (2) Step 2: $c = \\frac{c'}{\\||c'\\||} \\cdot \\epsilon$: The sampled vector $c'$ is then L2-normalized, projecting it onto the unit hypersphere, and subsequently multiplied by a random scalar $\\epsilon \\in [0, 1)$. This ensures that $c$ is strictly located inside the unit ball.\n\n**W6**: Comparison with classic baselines.\n\nFollowing your suggestion, we provide extensive comparisons with classic baselines such as ModernTCN, iTransformer, and DCDetector. The results for all baseline models are reproduced by strictly following the configurations and guidelines from their official open-source codebases. Due to space constraints, we present four key scores here; the complete results will be updated in the revised manuscript.\n\n| Dataset       | Metric | ScatterAD | DCDetector | ModernTCN | iTransformer |\n| :------------ | :----- | :-------: | :--------: | :-------: | :----------: |\n| MSL           | Aff-F  | **0.867** |   0.674    |   0.709   |    0.652     |\n|               | PA-F   | **0.964** |   0.957    |   0.807   |    0.659     |\n|               | A-ROC  | **0.986** |   0.961    |   0.627   |    0.604     |\n|               | A-PR   | **0.932** |   0.891    |   0.739   |    0.721     |\n| PSM           | Aff-F  | **0.797** |   0.653    |   0.701   |    0.652     |\n|               | PA-F   | **0.981** |   0.977    |   0.965   |    0.926     |\n|               | A-ROC  | **0.986** |   0.948    |   0.581   |    0.687     |\n|               | A-PR   | **0.969** |   0.866    |   0.629   |    0.751     |\n| SWaT          | Aff-F  |   0.704   |   0.687    |   0.685   |  **0.716**   |\n|               | PA-F   | **0.951** |   0.939    |   0.884   |    0.916     |\n|               | A-ROC  | **0.982** |   0.876    |   0.675   |    0.662     |\n|               | A-PR   | **0.909** |   0.824    |   0.592   |    0.619     |\n| WADI          | Aff-F  |   0.605   | **0.725**  |   0.558   |    0.671     |\n|               | PA-F   | **0.862** |   0.731    |   0.766   |    0.754     |\n|               | A-ROC  | **0.960** |   0.829    |   0.831   |    0.814     |\n|               | A-PR   | **0.765** |   0.651    |   0.697   |    0.627     |\n| NIPS-TS-SWAN  | Aff-F  |   0.038   |   0.484    | **0.533** |    0.507     |\n|               | PA-F   | **0.736** |   0.733    |   0.731   |    0.729     |\n|               | A-ROC  | **0.792** |   0.655    |   0.562   |    0.537     |\n|               | A-PR   | **0.716** |   0.626    |   0.545   |    0.531     |\n| NIPS-TS-GECCO | Aff-F  | **0.825** |   0.648    |   0.446   |    0.435     |\n|               | PA-F   | **0.784** |   0.472    |   0.357   |    0.381     |\n|               | A-ROC  | **0.969** |   0.715    |   0.946   |    0.817     |\n|               | A-PR   | **0.633** |   0.523    |   0.615   |    0.474     |\n\n---\n\n**Q1:** Encoder architecture specifications and the origin of $h_{online}$, $h_{target}$.\n\nWe provide detailed specifications in our response to **W2**. We will also explicitly state these details in Section 3 of the revised manuscript to improve clarity and reproducibility.\n\n**Q2:** Definitions and values of key parameters.\n\nPlease see our response to **W3**. We will ensure that all parameters are clearly defined upon their first appearance and will add a table of key hyperparameters in the appendix of the revised manuscript. \n\n**Q3:** Connection between the theoretical result $I(Z_T; Z_G | G)$ and the practical implementation.\n\nThank you for this crucial question regarding the connection between our mutual information theory and its practical implementation. The link is as follows:\n\n(1) **Theoretical Goal**: Our theory (Appendix A) shows that maximizing the conditional mutual information $I(Z_T; Z_G | G)$ is a principled way to learn complementary representations from the temporal ($Z_T$) and topological ($Z_G$) views, given the graph $G$.\n\n(2) **Practical Implementation**: The **contrastive fusion loss** $L_{contrast}$ (Eq. 7) serves as a tractable, differentiable lower bound for this mutual information. By minimising $L_{contrast}$, we effectively maximise this lower bound, thus driving the model to achieve the theoretical objective. In our implementation, the graph structure $G$ is the **temporal graph described in Section 3.2**, where nodes are time points at timestep $t$, and edges connect temporally adjacent nodes (e.g., $(t-1, t)$). This graph $G$ constrains the entire process, defining the neighbourhood for the GAT's aggregation operation and providing positive pairs for the contrastive loss.\n\n**References**\n\nHe, Kaiming, et al. \"Momentum contrast for unsupervised visual representation learning.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2020.\n\nLee, Byeongchan, and Sehyun Lee. \"Implicit contrastive representation learning with guided stop-gradient.\" Advances in Neural Information Processing Systems 36 (2023): 30885-30897."}, {"review_id": "eHmMew5fLx", "reviewer": "Reviewer_WWNK", "summary": "The authors insightfully propose ScatterAD, a spatiotemporal anomaly detection framework for industrial IoT, introducing representation dispersion as an inductive bias to model the scattered nature of anomalies in high-dimensional space. ScatterAD combines a topological encoder and a temporal encoder to extract complementary features, and detects anomalies via scattering deviation and temporal inconsistency. The authors also prove that maximizing conditional mutual information is equivalent to a contrastive loss objective. Experiments on six benchmarks confirm its effectiveness through strong performance, visual analysis, and ablation studies.", "questions": "Please see weakness.", "limitations": "The authors have sufficiently addressed the limitations of their study. I have no additional comments.", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe strengths of the work can be summarized as follows: \n\n1. Well written and clear. The paper introduces representation scattering as a novel perspective for high-dimensional anomaly detection, with a logical flow from dispersion to feature separation and detection.\n2. The paper presents a novel MTSAD approach inspired by the scattering phenomenon in industrial data, demonstrating clear insight.\n3. The effectiveness of the method is demonstrated on multiple MTSAD benchmarks with comprehensive evaluations.\n4. The authors’ theory unifies contrastive learning and conditional mutual information to enhance spatiotemporal representation complementarity.\n5. The authors provide the code implementation in the supplementary materials, which facilitates reproducibility.\n\nWeaknesses:\n\n1. A potential weakness lies in the initialization of the scattering center 𝑐 in Equation (6), which is randomly sampled within the unit sphere. This strategy may lead to suboptimal local minima and instability during training. For instance, poor initialization is a well-documented problem in machine learning, often causing slower convergence or trapping the model in unfavorable local optima, which degrades performance[1, 2].\n2. The scattering mechanism fundamentally depends on cosine similarity between samples and a reference center. However, the manuscript does not discuss the well-documented limitations of cosine similarity in high-dimensional settings, which may affect the robustness of the method. \n\n[1] Glorot, Xavier, and Yoshua Bengio. \"Understanding the difficulty of training deep feedforward neural networks.\" Proceedings of the thirteenth international conference on artificial intelligence and statistics. JMLR Workshop and Conference Proceedings, 2010.\n\n[2] Sutskever, Ilya, et al. \"On the importance of initialization and momentum in deep learning.\" International conference on machine learning. PMLR, 2013.", "was_revised": true, "final_justification": "I carefully read the author's reply and was pleased to find that the author addressed my concerns. Therefore, I am happy to further raise my score. I have no other additional questions.", "rebuttal": "We greatly appreciate Reviewer WWNK for the comprehensive and insightful comments.\n\n**W1**: Regarding the random initialization strategy for the scattering center $c$.\n\nYour concerns about training stability and suboptimal solutions are entirely valid, and these are key considerations in our model design.\n\nFirst, we wish to clarify the core role of $c$. In our framework, $c$ is initialized only once at the beginning of each training run and **remains fixed throughout the training process**. It is not a parameter learned via backpropagation. Its purpose is twofold:\n\n*   **(1) Provide a fixed anchor point**: $c$ offers a convergent target direction for the representations of normal samples, enabling the model to learn to map these representations into a compact region on the hypersphere.\n*   **(2) Break symmetry and regularize**: If we fix $c$ to a constant (e.g., [1, 0, ..., 0]), it might introduce an undesirable bias, inducing the model to learn a trivial solution where all normal samples map to a specific axis. Randomly initializing $c$ avoids this risk, compelling the model to learn more generalizable representations. This can be viewed as a form of implicit stochastic regularization, preventing the model from overfitting to a pre-defined direction in the latent space.\n\nTo empirically address your concerns about instability and suboptimal solutions, we conduct a new experiment: we repeat our full training and evaluation process with 5 different random seeds on the MSL and PSM datasets. A different random seed results in a different scattering center $c$. We report the mean and standard deviation of the Aff-F1 and A-ROC metrics.\n\n| Dataset | Metric | Original | **Mean**  | **Std. Dev.** |\n| :------ | :----- | :------: | :-------: | :-----------: |\n| **MSL** | Aff-F1 |  0.865   | 0.865|    ± 0.003    |\n|         | A-ROC  |  0.986   | 0.986 |    ± 0.001    |\n| **PSM** | Aff-F1 |  0.797   | 0.797 |    ± 0.002    |\n|         | A-ROC  |  0.986   | 0.986 |    ± 0.003    |\n\nFurthermore, we design several variants to compare the performance of **different center initialization strategies**, including:\n\n*   (1) **Zero**: Initialize the center at the origin (zero vector) to test the simplest symmetric starting point.\n*   (2) **Fixed-Radius**: Initialize the center on a hypersphere with a fixed radius to test the model's sensitivity to the initial magnitude.\n*   (3) **Multi-center**: Use multiple independent scattering centers to test the model's ability to capture potentially multi-modal normal data.\n\nIn the table below, Num Centers denotes the number of scattering centers used, and Radius specifies the initial magnitude for the fixed_radius strategy. We compare these strategies with the Random-in-Ball (randomly selecting a point within the unit hypersphere) strategy proposed in our paper.\n\n| Dataset | Center Strategy           | Num Centers | Radius |  Aff-F1   |  AUC ROC  |\n| :------ | :------------------------ | :---------: | :----: | :-------: | :-------: |\n| MSL     | **random_in_ball (Ours)** |      1      |  N/A   | 0.867 | **0.986** |\n|         | zero                      |      1      |  0.0   |   0.866   |   0.986   |\n|         | fixed_radius              |      1      |  0.3   |   0.866   |   0.986   |\n|         | fixed_radius              |      1      |  0.7   |   0.864   |   0.986   |\n|         | Multi-center              |      3      |  N/A   |   **0.869**   |   0.986   |\n| PSM     | **random_in_ball (Ours)** |      1      |  N/A   | **0.797** | **0.986** |\n|         | zero                      |      1      |  0.0   |   0.797   |   0.986   |\n|         | fixed_radius              |      1      |  0.3   |   0.797   |   0.986   |\n|         | fixed_radius              |      1      |  0.7   |   0.796   |   0.986   |\n|         | Multi-center              |      3      |  N/A   |   0.792   |   0.980   |\n| SWaT    | **random_in_ball (Ours)** |      1      |  N/A   | **0.704** | **0.982** |\n|         | zero                      |      1      |  0.0   |   0.704   |   0.982   |\n|         | fixed_radius              |      1      |  0.3   |   0.698   |   0.977   |\n|         | fixed_radius              |      1      |  0.7   |   0.702   |   0.979   |\n|         | Multi-center              |      3      |  N/A   |   0.702   |   0.980   |\n\nThe results confirm that for most datasets, ScatterAD is not sensitive to the specific initialisation strategy of $c$, and the  **random_in_ball** strategy proposed in our paper is a simple and effective choice.\n\n**W2**: Discussion on the effectiveness of cosine similarity in high-dimensional representation space.\n\nYour point that distances between random vectors in high-dimensional space tend to converge is entirely valid for unprocessed or randomly distributed high-dimensional data. However, the core of our method is to avoid processing such data directly. Instead, the basic task of ScatterAD is to learn a nonlinear mapping from the original high-dimensional space to another well-structured high-dimensional representation space. In this learned space, the distribution of vectors is not random, specifically:\n\n*   (1) $L_{scatter}$: Actively pulls the representations of normal samples towards the same fixed center $c$, forming a compact cluster locally.\n*   (2) $L_{time}$: Enforces temporal continuity in the representation space by minimising the distance between representations of adjacent timesteps.\n*   (3) $L_{contrast}$: Ensures that the representations from different views ($online$  and $target$) remain structurally consistent, further reinforcing the geometry of the representation space.\n\nTherefore, we are not relying on the properties of cosine similarity in a random high-dimensional space; rather, we leverage our model to create a representation space that possesses both spatial structure and temporal continuity. Furthermore, we provide a comparison of our anomaly scoring strategy with two variants based on classic distance metrics:\n\n| Dataset | Metric | ScatterAD (Euclidean - OURS) | ScatterAD (Mahalanobis) | ScatterAD (KL Divergence) |\n| :------ | :----- | :--------------------------: | :---------------------: | :-----------------------: |\n| MSL     | Aff-F1 |          **0.867**           |          0.851          |           0.789           |\n|         | A-ROC  |          **0.986**           |          0.986          |           0.972           |\n| PSM     | Aff-F1 |          **0.797**           |          0.792          |           0.129           |\n|         | A-ROC  |          **0.986**           |          0.980          |           0.793           |\n\nThe results show that our simple inductive bias based on Euclidean distance is more effective than methods that attempt to fit complex correlations (Mahalanobis distance) or probability distributions (KL Divergence). This also serves as evidence that our model successfully learns a well-structured representation space."}, {"review_id": "Nko7GBbxd3", "reviewer": "Reviewer_jjez", "summary": "This paper introduces a novel anomaly detection framework, ScatterAD, which jointly enhances  feature discriminability and temporal consistency by using a temporal-topological scattering mechanism. Specifically, ScatterAD employs a dual-encoder architecture, consisting of an online encoder and a target encoder, where the online encoder is updated via backpropagation and the target encoder is updated using an exponential moving average (EMA) of the online encoder's parameters. The online encoder enforces temporal consistency across adjacent time steps, while the target encoder captures compact and consistent scattering representations in a projected hypersphere space. Moreover, a contrastive fusion mechanism is employed to align temporal and topological representations through a cross-view contrastive loss. The effectiveness of the paper is validated through extensive experiments.", "questions": "(a) Could the authors provide further justification or quantitative analysis to confirm that the observed scattering behavior is consistent and not occasional?\n\n(b) Could the authors clarify whether the reported experimental results are based on a single run or averaged over multiple runs?\n\n(c) The global scattering center c in the paper is randomly initialized inside the unit ball. Could the authors clarify whether this random initialization significantly affects the model performance or stability? Have you ever considered using alternative strategies?", "limitations": "Yes.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nThe paper formalize the dispersion phenomenon of normal and anomalous samples in high-dimensional space as scattering, quantified by the mean pairwise distance among sample representations. The authors further leverage this insight as an inductive signal to enhance spatio-temporal anomaly detection and propose a novel anomaly detection framework, ScatterAD, providing  valuable insights for multivariate time series anomaly detection. The overall paper is well-structured and easy to follow. The motivation of each component in the proposed framework is well explained. The effectiveness of the proposed framework is validated through both  theoretical analysis and extensive experiments across diverse datasets.\n\nWeakness:\nWhile the paper provides empirical observations that the scattering of anomalous samples tend to be more pronounced in high-dimensional space, it remains unclear whether such a phenomenon generalizes well across datasets with different types of anomalies or varying noise levels. The paper could benefit from systematically examining the scattering phenomenon across more datasets. Additionally, the experimental results should report the mean and standard deviation, with multiple repetitions to ensure statistical reliability.", "was_revised": true, "final_justification": "The authors have addressed my concerns; however, the novelty and technical quality still do not warrant a higher score, so I am maintaining my original rating.", "rebuttal": "We sincerely thank Reviewer jjez for the detailed comments and insightful questions.\n\n**Q1:** Analysis of the universality of the scattering phenomenon.\n\nTo address your concerns about the universality of the scattering phenomenon, we present the quantitative results on the scattering phenomena for all six benchmark datasets used in our study. Furthermore, to evaluate the model's robustness, we inject additive Gaussian noise of three different intensities (σ =0, 0.5, 1.0, 2.0) into the normalized test data (with σ=0.0 as the no-noise baseline). We quantify the impact of noise on the model's internal discriminative power by calculating the average scattering score for normal and anomalous samples, which we define as the mean Euclidean distance of samples to their respective class center, and analyzing their separation ratio.\n\n| Dataset | Noise Level (σ) | Scattering Score (Normal) | Scattering Score (Anomalous) | **Separation Ratio (Anomalous/Normal)** |\n| :------ | :-------------: | :--------------------: | :-----------------------: | :-------------------------------------: |\n| MSL     |       0.0       |          16.0          |           18.2            |                **1.14**                 |\n|         |       0.5       |          21.6          |           31.6            |                **1.46**                 |\n|         |       1.0       |          36.0          |           49.3            |                **1.37**                 |\n|         |       2.0       |          65.9          |           84.6            |                **1.28**                 |\n| PSM     |       0.0       |          31.5          |           37.8            |                **1.20**                 |\n|         |       0.5       |          33.1          |           40.3            |                **1.22**                 |\n|         |       1.0       |          41.2          |           47.4            |                **1.15**                 |\n|         |       2.0       |          61.1          |           65.7            |                **1.08**                 |\n| SWaT    |       0.0       |         73.10          |           94.09           |                **1.29**                 |\n|         |       0.5       |         75.63          |           95.71           |                **1.27**                 |\n|         |       1.0       |         79.22          |           98.03           |                **1.24**                 |\n|         |       2.0       |         84.50          |          102.11           |                **1.21**                 |\n| WADI    |       0.0       |          0.78          |           1.16            |                **1.49**                 |\n|         |       0.5       |          34.9          |           43.4            |                **1.24**                 |\n|         |       1.0       |          87.1          |           72.4            |                  0.83                   |\n|         |       2.0       |         166.7          |           137.0           |                  0.82                   |\n| N-T-S   |       0.0       |          20.2          |           28.4            |                **1.41**                 |\n|         |       0.5       |          21.8          |           32.4            |                **1.49**                 |\n|         |       1.0       |          27.7          |           42.2            |                **1.52**                 |\n|         |       2.0       |          42.1          |           65.5            |                **1.56**                 |\n| N-T-W   |       0.0       |         21.60          |           95.97           |                **4.44**                 |\n|         |       0.5       |         26.10          |          105.33           |                **4.04**                 |\n|         |       1.0       |         40.70          |           41.40           |                **1.02**                 |\n|         |       2.0       |         68.60          |          197.30           |                **2.88**                 |\n\nThe results in the table above clearly reveal a consistent anomaly scattering pattern across six different real-world datasets. In addition to this quantitative evidence, we would like to draw the reviewer's attention to Figure 3 in our paper, which qualitatively supports the consistency of the scattering behavior from another perspective. Furthermore, we wish to highlight the case of the WADI dataset, where under medium-to-high noise, the scattering score of normal samples surpasses that of anomalies. We attribute this phenomenon primarily to WADI's high dimensionality (123 dimensions), where strong noise may have a compounding effect and lead to complex changes in inter-variable correlations. This also indicates that while ScatterAD is robust in various noisy environments, its performance can be sensitive to strong noise when dealing with extremely high-dimensional datasets with subtle anomaly patterns. Thank you again for this insightful question, which helps us to more comprehensively characterize the performance boundaries of our method.\n\n**Q2:** Clarification on the reproducibility of experimental results.\n\nThank you for this question, which is crucial for ensuring the rigor and reproducibility of our work. The experimental results reported in our paper, including the baseline evaluations, are based on the average of three independent runs. We will add standard deviations for all core experimental results in the revised version to more comprehensively demonstrate the stability and statistical reliability of our findings.\n\n**Q3:** Regarding the random initialization strategy for the scattering center $c$.\n\nWe sincerely thank you for your question regarding the initialization strategy for the global scattering center $c$. This insightful question is also raised by **Reviewer WWNK**, and we kindly invite you to refer to our detailed response in **Section W1**.  In brief, our experiments show that the random initialization of $c$ is a simple yet effective strategy. For a detailed analysis, including stability tests, we kindly direct you to the **Section W1**."}]

Author rebuttal:

Meta-review: This paper makes solid contributions to multivariate time series anomaly detection through: (1) a novel and well-motivated approach using representation scattering, (2) theoretical grounding through information bottleneck theory, (3) strong empirical results with comprehensive evaluation, and (4) clear practical applicability to industrial IoT scenarios. While initial presentation issues were raised, the authors demonstrated commitment to addressing all concerns in the final version. The work represents a meaningful advance in the field with both theoretical insights and practical impact.

The paper received positive evaluations from all four reviewers with final ratings of 5 (Accept), 4 (Borderline Accept), 6 (Strong Accept), and 4 (Borderline Accept). The reviewers appreciated the novel approach of leveraging representation scattering as an inductive signal for multivariate time series anomaly detection, the theoretical grounding through information bottleneck theory, and the comprehensive experimental validation.

While the work is technically solid with good experimental results, it represents an incremental advance in anomaly detection methodology rather than a breakthrough warranting oral/spotlight presentation. The contributions are more suited for detailed technical discussions at a poster session.
Decision: Accept (poster)