KAN-AD: Time Series Anomaly Detection with Kolmogorov–Arnold Networks
Authors: Quan Zhou
Changhua Pei
Fei Sun
HanJing
Zhengwei Gao
Haiming Zhang
Gaogang Xie
Dan Pei
Jianhui li
Venue: ICML 2025 poster (2025)
Keywords: anomaly detection
time series analysis
Primary area: general_machine_learning->sequential_network_and_time_series_modeling

Abstract: Time series anomaly detection (TSAD) underpins real-time monitoring in cloud services and web systems, allowing rapid identification of anomalies to prevent costly failures. Most TSAD methods driven by forecasting models tend to overfit by emphasizing minor fluctuations. Our analysis reveals that effective TSAD should focus on modeling "normal" behavior through smooth local patterns. To achieve this, we reformulate time series modeling as approximating the series with smooth univariate functions. The local smoothness of each univariate function ensures that the fitted time series remains resilient against local disturbances. However, a direct KAN implementation proves susceptible to these disturbances due to the inherently localized characteristics of B-spline functions. We thus propose KAN-AD, replacing B-splines with truncated Fourier expansions and introducing a novel lightweight learning mechanism that emphasizes global patterns while staying robust to local disturbances. On four popular TSAD benchmarks, KAN-AD achieves an average 15% improvement in detection accuracy (with peaks exceeding 27%) over state-of-the-art baselines. Remarkably, it requires fewer than 1,000 trainable parameters, resulting in a 50% faster inference speed compared to the original KAN, demonstrating the approach's efficiency and practical viability.

KAN-AD:
Time Series Anomaly Detection with Kolmogoroy-Arnold Networks
Quan Zhou 12 Changhua Pei
Fei Sun
Jing Han
Zhengwei Gao Haiming Zhang' Gaogang Xie
Dan Pei Jianhui Li
Abstract
Local Peaks
Local Drop
Local Drop
Time series anomaly detection (TSAD) underpins real-time monitoring cloud services and web systems, allowing rapid identification of anoma- lies prevent costly failures. Most TSAD meth- ods driven by forecasting models tend to overfit by emphasizing minor fluctuations. Our analy- sis reveals that effective TSAD should focus on modeling "normal" behavior through smooth lo- cal patterns. To achieve this, we reformulate time series modeling as approximating the series with smooth univariate functions. The local smooth- ness of each univariate function ensures that the fitted time series remains resilient against local disturbances. However, direct KAN implemen- tation proves susceptible to these disturbances due to the inherently localized characteristics of B-spline functions. We thus propose KAN-AD, replacing B-splines with truncated Fourier expan- sions and introducing novel lightweight learning mechanism that emphasizes global patterns while staying robust to local disturbances. On four pop- ular TSAD benchmarks, KAN-AD achieves an average 15% improvement detection accuracy (with peaks exceeding 27%) over state-of-the- baselines. Remarkably, it requires fewer than 1,000 trainable parameters, resulting in 50% faster inference speed compared to the original KAN, demonstrating the approach's 's efficiency and practical viability.
Figure I. Illustration of local drops and peaks.
Time Series
Anomaly Score
Clean Triin Sample
Sample
Timeshet
KAN AD
Nample
Figure 2. Comparison of anomaly detection performance. Top: All methods successfully detect anomalies when trained on clean data (black curve, anomalous segments in pink). Bottom: TimesNet and KAN fail to detect anomalics when trained on noisy data. Blue markers indicate local drops and peaks; red curve shows anomaly scores.
1. Introduction
Time Series Anomaly Detection (TSAD) serves as a criti- cal component in modern IT infrastructure (Li et al., 2019; Qu et al., 2024) and manufacturing systems (Zhan et al., 2021; Wang et al., 2022), enabling rapid identification of potential anomalies and providing sufficient clues for fault localization (Sun et al., 2024; Kieu et al., 2022). The emer- gence of deep learning- based forecasting approaches (Xu et al., 2022; Wu et al., 2023; Zhou et al., 2023) have super- seded traditional rule-based methods (Breunig et al., 2000; Siffer et al., 2017), establishing new state-of-the-art perfor- mance through their capacity fit historical data and detect anomalies via prediction--obseraaton comparisons.
However, the effectiveness of the precassting-bsed ap- proach declines when encountering time series with local- ized disturbances. As illustrated Figure 1, time series data frequently exhibit local peaks and drops that can sig- nificantly impact model learning. Existing deep learning methods (Tuli al., 2022; Wu al., 2023) often overfit to these local disturbances, compromising their ability to detect anomalies effectively. From the third column of Fig- ure 2, we can observe that compared to training with clean data, TimesNet (Wu et al., 2023) trained on noisy data fails to detect anomalies in the samples.
'Computer Network Information Center, Chinese Academy of Sciences 'University of the Chinese Academy of Sciences 3 Hangzhhuu Institute for Advanced Study, University of the Chi- nese Academy of Sciences "Institute of Computing Technology, Chinese Academy of Sciences 'ZTE 'Department of Computer Science and Technology. Tsinghua University "School of Fron- tier Seiences, Nanjing University. Correspondence to: Jianhui Li <lijh @ www..du..nn.... Changhua Pei cnn....@cii..c.....
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Our experimental analysis reveals that forecasting-

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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
TSAD methods suffer performance degradation by attempt- ing model every detailed patterns in raw time series data. While these methods aim to identify anomalies through comparison with predicted behavior, such detailed model- ing proves unnecessary and potentially detrimental, espe- cially given that real-world time series typically contain various forms of anomalies and irrelevant disturbances, pre- senting two significant challenges: firstly, the difficulty in establishing universal criterion for filtering these dis- turbances, and secondly, developing another model to en- sure the forecasting model's input is free of local distur- bances is resource-intensive. Given these inherent limita- tions in both filtering- basedd and dual- modeling approaches, researchers have explored VAE-based approaches to address the challenge of local disturbance mitigation. VAE-based approaches (Xu al., 2018; Wang et al., 2024) assume that normal patterns in time series cluster in a low-dimensional latent space and can be effectively reconstructed, thereby overcoming interference from data perturbations. Never- theless, as demonstrated in FCVAE (Wang et al., 2024), VAE-based approaches struggle with underfitting, which impairs their ability to reconstruct the original time series and limits their effectiveness.
terms which is computation intensive, we only use the first N terms of Fourier series. To overcome the limitation that the first N terms of Fourier series can only model periodic no smaller than 1 N. we designed an alternative index- based univariate function to capture the fine-scale periodic missing from the first N terms. Third, we incorporated differencing to isolate time series trend effects on coefficient estima- tion, leading to improved modeling accuracy through more precise coefficients.
Our comprehensive evaluation demonstrates that KAN-AD achieves 15% higher FI accuracy while being 50% faster than the original KAN architecture. Our code is publicly available at https: / /ggttubtt thub. . wwm http/SCCCooddOps.// KAN- AD. Our contributions are as follows:
We reformulate the problem to assist deep learning- based forecasting models for time series anomaly de- tection (TSAD) tasks by minimizing overfitting to local perturbations.
We introduce KAN-AD, an innovative TSAD approach. KAN-AD, built meticulously on the KAN backbone, exhibits substantial improvements in both detection precision and inference efficiency.
To mitigate local disturbances, reformulate TSAD by ap- proximating time series using smooth univariate functions, building on the theoretical foundation that normal sequences exhibit greater local smoothness than abnormal ones (Xu et al., 2022). To achieve this formulation, Kolmogorov- Amold Networks (KANs) (Liu et al., 2025) offer a promis- ing direction by decomposing complex objectives into com- binations of Iearnable univariate functions based on the Kolmogorov-Arnold representation theorem (Kolmogorov, 1957). This decomposition approach has shown remarkable effectiveness in various domains (Yu et al., 2024; Bodner et al., 2024). However, direct application of KAN to TSAD presents significant challenges. From the fourth column in the upper part of Figure 2, it can be observed that mod- els trained clean training samples can detect anomalies in the test samples. But we find that KAN fails to detect anomalies when the training samples contain noisy samples The main reason is that, although KAN can specify uni- variate functions, i.e., B-spine function, these functions are not specifically designed for time series and can still overfit local features, failing to completely eliminate the impact of local peaks or drops.
We performed comprehensive experiments on four pub- licly available datasets, verifying the effectiveness and efficiency against state-of-the-art TSAD benchmarks.
2. Preliminaries and Problem Formulation
2.1. Problem Statement
This paper primarily addresses the issue of anomaly de- tection in single time series curves, also known as uni- variate time series (UTS). To elaborate on the problem more comprehensively, consider the following UTS observa- tional data: fo. = (20, 21, 22, 22. and anomaly labels C = {co, C2, 02, ..,,..,,,.,,, ct.. where 2a E R, c E {0, 1}, and E N. Here, Z0xtt represents the entire observed time series, and C denotes the temporal anomaly labels.
Given a UTS x = (zo: I1, 22, 22...., Ve),, the objective of UTS anoomaly detection is to utilize the data [Ta, .i,..., Fil preceding each point x; to predict c4.
To address these challenges, we propose KAN-AD, adopt- ing KAN as our backbone. By considering the characteris- tics of time series, we redesign KAN in three aspects. First, we replace the B-spine function with Fourier series. Fourier series have local smoothness compared to spline functions, while their natural periodicity allows for better modeling of global patterns (Dym & HP, 1972; Stein & Shakarchi, 201 I). Second, as the Fourier series contains unlimited
2.2. Kolmogorov-Armold Networks
1.2.1. THEORETICAL FOUNDATION
The Kolmogorov-Armold representation theorem demon- strates that any multivariate continuous function can be decomposed into a finite sum of univariate functions, as shown in Equation (I), where Pq. are univariate functions that map each input variable Tpp and b, are continuous
2

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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
functions.
= (ioo(
cos( cos(n)ras) sinnnss)sinnn(tttt sin(reros))
2n+1
12
22,, (n)) = Σ
pop(sp))
(sros) = [A1, B1. Ag. B An. Bn] #6 = Ao + (zo::) x H
g=1
pml
(4)
KAN(2)= = (0)-11 2 .0... Bo) (2)
Formally, we employ Fourier series for normal pattern rep- resentation, motivated by two key advantages over alter- native approaches such as B-spline functions. First, the constituent sine and cosine functions exhibit superior lo- cal smoothness, avoiding the potential overfitting to local noise. Second, Fourier series naturally capture global pat- terns, particularly excelling at modeling periodic behaviors in time series. Following this motivation, we introduce the function deconstruction (FD) mechanism, where f. the mapping between the historical window 20:1 and its next behavior Zi+,, can be expanded as shown in Equation (3), The normal pattern can be represented by the finite N terms of the series (Kolmogorov, 1957), denoted as g(z), while the terms beyond N capture the stochastic observational noise c. The normal pattern Jh.s can then be expressed as in Equation (4), where H denotes the univariate function matrix. This decomposition combined with learnable coeffi- cients filters out potential noise and significantly simplifies the construction of normal patterns.
2.2.2. NETWORK ARCHITECTURE AND FUNCTION REPRESENTATION
KAN consists of a series of interconnected univariate sub- networks, each responsible for learning distinet features of the data. Unlike traditional multi-layer perceptrons (MIPss, which employ fixed activation functions at each node, KAN replaces each weight parameter with univariate function. The resulting functional form for deeper KAN can be ex- pressed as Equation (2), where each Φ represents a layer of univariate functions applied to the input or intermediate outputs. The vanilla KAN (Liu et al., 2025) implements these univariate functions using B-splines (De Boor, 1978), which provide localized function approximation capabili- ties. However, this localization property presents a notable consideration in anomaly detection contexts. Since anoma- lous patterns typically manifest as localized features (Xu et al., 2022), B-splines may inadvertently fit these outliers, potentially compromising model accuracy.
3.2. Mapping Phase
3. Methodology
As shown in Figure 3b, the primary purpose of the mapping phase is to transform the original time series signal Toss R into multiple new sets of values To:i E RT*((+NN through a series of univariate functions. Here, T is the size of the sliding window. The first N represents the number of sine series univariate functions, and the other N repre- sents the number of cosine series univariate functions. The detailed calculation method is shown in Equation (3). No- tably, besides the univariate function terms, an A term representing the average value within the sliding window is also present, which varies across different windows. To mitigate the impact of fluctuating Ao on coefficient fitting, a constant term elimination module is employed.
The core challenge in time series anomaly detection (TSAD) lies in establishing accurate normal patterns while maintain- ing robustness to local disturbances (Li et al., 2021). Tradi- tional approaches that directly predict based on historical data inevitably incorporate local noise into their learned patterns. Building on the observation that normal sequences exhibit greater smoothness than abnormal ones, we pro- pose KAN-AD, novel anomaly detection framework that leverages this smoothing feature to identify anomalies in complex time series data.
3.1. Design of KAN-AD
The pipeline of KAN-AD consists of three main stages: mapping. reducing, and projection. In the mapping phase, we decompose the input time window into multiple uni- variate functions. The reducing phase then combines these functions through learned coefficients to reconstruct the "normal" pattern. Finally, the projection phase leverages this pattern to predict future behavior, enabling anomaly detection through comparison with real-time observations.
Constant Term Elimination: In Fourier series, As repre- sents the mean value of the function. Although normaliza- tion ensures that the entire time series has mean of zero, individual time windows may still exhibit significant fluctu- ations in their means due to the presence of a trend. These variations in the constant term ultimately affect the model's accurate estimation of Fourier coefficients, leading to biases in the construction of the normal pattern.
N
To mitigate the impact of mean fluctuations on the model's approximation of normal time series patterns, we employ first-order differencing during data preprocessing to min- imize the residual trend component in the data and sub- sequently renormalize the differenced data, This strategy
(fo::) = Ao +
An cos(n + Bn sin(no)o:)) +e
9(
3

---

KAN-AD: Time Series Anomaly Detection with Kolmogorov-Armold Networks
KAN
learnahle univariate functions on edges
Univariate Functions (cos(ar), coa(2x), (oe() [sin(a), sin(2s), sim(sa)]
A. B, B
A.
eogf (2)
1D CNN Kernel
operations nodes
(ine (P)
ooe (2)
Value
Ingut Bliding Window'ss sin (max)
2
Qutpet
KAN-AD (ours)
A2
fixed activation functions nodes
Cool paa))
B2
leurnable weights
on edges
fixed univariate functions on exgess
Time
B
Mapping
Reducing
Projection
(a) Illustration of learning components in KAN KAN-AD. KAN-AD learns the coefficients on edges with fixed univari- ate functions, and performs weighted sum operations nodes. Blue lines indicate edges with weights.
(b) Illustration of the KAN-AD process using sliding window approach. During the mapping phase, raw time windows are transformed into multiple univariate functions. the reducing phase, one-dimensional convolutional kernel learns coeflicients for these univariate functions, aggregating them into normal pattern for the current time window. In the projection phase, single-layer MLP predicts future normal patters.
Figure 3. Illustration of KAN-AD.
allows the model to focus on estimating Fourier coefficients A1 and Bin: thereby avoiding the need to learn frequently changing constant terms. After this differential strategy, the normal pattern 2/ can be expressed as Z (#na)
The function deconstruction (FD) mechanism addresses this challenge by transforming the modeling of normal patterns into weighted combination univariate functions. This transformation substantially reduces the model's parameter quantity instead of requiring numerous parameters for fine- grained feature modeling, FD mechanism achieves efficient representation through estimating coefficients of small number of univariate functions.
Periodic-Enhaneee KAN-AD: Fourier series of finite N terms cannot model period smaller than &. which limits KAN-AD's ability to express time series containing more subtle periods.
To address this limitation and enhance the model's 's ability to capture periodic patterns time series, we introduce ad- ditional univariate functions with different periods. Specifi- cally, incorporate trigonometric components cos( (2mss and sin( 2mmz where denotes the window index, with efficients learned through one-dimensional convolution net- works. Our implementation utilizes three complementary univariate functions shown in Equation (5): the raw time variable X, the Fourier series S,, and the sine-cosine wave P,. This integration of multi multi-periode periodic univariate functions enhances KAN-AD 's capacity to model temporal patterns.
H (0) StttttccKXX S1, PI...., Px), ANE [I...., N] (6)
O (1) CNN(CNN(H(( vI E 11,2.... L.]
(7)
2
Conv(H) - Σ>> We[m] Helj + m
(8)
CNN(H) GELUUBN(Conn(HH
(9)
To effectively estimate these univariate function coefficients, employ stacked one-dimensional convolutional neural network (ID CNN). This architecture choice is motivated by two key factors: ID CNNs excel at sequence modeling through temporal dimension traversal, while their convolu- tional kernels naturally capture the diverse features intro- duced by the FD mechanism. As shown in Equation (6). KAN-AD first constructs univariate function matrix ()) by combining the required functions for given time win- dow. This matrix is then processed through multiple stacked ID convolutional layers with a kernel size of 3, progres- sively approximating the normal pattern through coefficient learning, as expressed in Equation (7). Here, L denotes the number of CNN blocks, with the network CNN(H) and
X Fo.i
S, = {sin (naro:)) (os(oo)):))
(5)
2
2Tni
P. = {sin( cos(
T
3.3. Reducing Phase
Another challenge in real-world time series anomaly de- tection the high computational cost. Existing methods often sacrifice efficiency for accuracy, making them im- practical resource-constriined or large-scale settings.

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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Armold Networks
convolution operation (H) defined in Equations (8) and (9). The convolution operation in Equation (8) applies
Table 1. Dataset Statistics.
Dataset Curves
Train Train Ano%
Test Test Ano%
kernel We to each channel He, where indices and represent positions within the convolutional kernel and time window, respectively.
KPI
29 3,073,567
2.70%
3,073,556
1.85%
TODS
15
75,000
5.32%
75,000
6.38%
WSD
3,829,373
2.43%
3,829,537
0.76%
UCR
203
3,572.316
0.00%
7.782,539
0.47%
To ensure training stability and reduce internal covariate shift, we apply batch normalization (loffe & Szegedy, 2015) after each convolutional layer (Equation (9)), followed by Gaussian Error Linear Units (GELUs) (Hendrycks & Gim- pel, 2016) for activation. The final stage of the reducing phase employs residual connection (He et al., 2016) be- tween the hidden state HL and the original input H))) to maintain numerical stability, as shown in Equation (10). Fi- nally, a 1 1-width convolutional kernel reduces the dimension- ality of HL)) to generate the normal pattern approximation To within the current time window:
WSD (Zhang et al., 2022), and UCR (Wu & Keogh, 2021).
Dataset characteristics are summarized in Table 1, includ- ing curve counts, sizes, and anomaly rates. The anomaly interval length distributions, shown in Figure 6, reveal that while most anomalies span less than 10 points, WSD and UCR contain extended anomaly segments exceeding 300 points, enabling comprehensive evaluation. Detailed dataset descriptions are provided in Appendix A.1.
4.1.2. MODEL TRAINING AND INFERENCE
H = H (D) (0)
(10)
We implement a systematic experimental protocol for both our method and baseline approaches. For each time series, we train dedicated KAN-AD models using consistent hyper- parameters: batch size 1024, learning rate 0.01, and maxi- mum 100 epochs. The validation strategy varies by dataset, with UCR reserving 20% of training data and other datasets employing a 4:1:5 ratio for training, validation, and testing splits. To ensure fair comparison, we faithfully replicate all baseline methods following their original implementations and hyperparameter settings as specified in their respective papers. During inference, we standardize the batch size to across all methods for comparable efficiency assessment. Results presented in Table report means and standard de- viations from five independent trials with different random sceds.
#0. GELU (DOWUn
(11)
Here, DownConv(H) (H)
Da W. Holi] denotes the con-
volution operation for reducing dimensions.
3.4. Projection Phase
After obtaining the current window's 's normal mode approxi- mation "'... we predict the future behavior r++1 through single-layer MLP, leveraging KAN-AD's accurate approxi- mation capability:
I++1
W . T0. +
(12)
where W and b denote the weight matrix and bias term of the linear layer.
4.1.3. BASELINES
4. Evaluation
We conducted comparative experiments with ten state- of-the-art time series anomaly detection methods: LST- MAD (Malhotra et al., 2015), FCVAE (Wang et al., 2024), SRCNN (Ren al., 2019), FITS (Xu al., 2024), TimesNet (Wu al., 2023), OFA (Zhou et al., 2023), TranAD (Tuli al., 2022), SubLOF (Breunig et al., 2000), Anomaly Transformer (Xu et al., 2022) (abbreviated as An- oTrans in the tables), KAN (Liu et al., 2025) and SAND (Bo- niol et al., 2021). Detailed descriptions of these methods can be found in Appendix A.2. For datasets not featured in the baseline literature, we meticulously tuned hyperparameters via grid search to optimize the performance of the baseline method on the respective evaluation metrics.
In this section, we conduct comprehensive experiments pri- marily aimed at answering the following research questions.
RQI: How does KAN-AD compare to state-of-the-art
anomaly detection methods in performance and efficiency?
RQ2: How sensitive is KAN-AD to hyperparameters?
RQ3: How effective is each design choice in KAN-AD?
RQ4: How sensitive is KAN-AD to anomalies in the train- ing data?
In addition, we also evaluate our method on a multivariate time series anomaly detection dataset to demonstrate the application potential of KAN-AD in more scenarios.
4.1.4. EVALUATION METRICS
4.1. Experimental settings
In practical applications, operations teams are less con- cerned with point-wise anomalies (i.e., whether individual data points are classified as anomalous) and more focused on detecting sustained anomalous segments within time se-
1. DATASET
We evaluate KAN-AD on four publicly available UTS
datasets: KPI (Competition, 2018). TODS (Lai et al., 2021),
5

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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
Table 2. Performance comparison. Best scores are highlighted in hold, and second hest scores are highlighted in bold and underlined.
Metries include F1 (Best F1). F1, (Event FI), FLa (Delay FI), AUPRC (area under the precision- recalll curve) and Avg Fl. (average Fl. score across four datasets).
KPI
TODS
WSD
UCR
Method
Aug Fi,
F1
F1,
Fla
AUPRC
F1
F1,
Fla
AUPRC
F1
F1,
Fl, AUPRO
F1
F1,
F14
AUPRC
SRCNN
0.4137
0.0994
0.2266 0.3355 0.6239
0.1918 0.4399 0.6076 0,4092
0.1185
0.1951 0,3080 0.5964
0.1369
0.1656 0,5109 0.1367
SAND
0.2710
0.0397 0.1097 0.2022 0.5372 0.1879 0.5103 0.5145 0.1761 0.0839 0.1267 0.1238 0.7044 0,5108 0.5116 0.6550 0.2056
AnoTrans
0.6103
0.3020 0.3623 0.5676 0.4875 0.1915 0.2918 0.4148 0,4348 0,2311 0.1517 0.3527 0.6135 0.1696 0.1084
0.5458 0.2236
TranAD
0.7553
0,5611 0.6399 0.7399 0.5035 0.2460 0.3619 0,4501 0.7570 0.6338 0,4158 0.7106 0.5278 0.1840 0.1554 0.4599 0,4062
SubLOF
0.7273 0,2805 0.4994 0.7015 0.7997 0.4795 0.7169 0.7809 0.8683 0.6585 0,4917 0.8353 0.8468 0.4772 0,4151 0.8001 0,4739
TimesNet
0.8022
0.6363 0.6995 0.8166 0.6232 0.3327 0,4495 0.6031 0,9406 0.8444 0.6170 0.9376 0.5273 0.1805 0.1439 0,4536 0,4985
FITS
0.9083
0.6353 0.8175 0,9359 0.7773 0.5416 0.6312 0.7725 0.9732 0,8391 0.6535 0.9771 0.6664 0.2926 0.2912 0.5969 0.5772
OFA
0.8810 0.6150 0.7952 0,9009 0.6928 0.5811 0,5588
0.7206 0.9564 0.8344 0.6250 0,9615 0.6294 0.3176 0.1503 0.5699 0,5870
FCVAE
0.9398
0.7556 0.8624 0,9572 0,8652 0.6995 0.7482
0,8798
0.9650 0,8610 0.6583 0.9653 0,7651 0.3812 0.2857 0.71450.6744
LSTMAD
0.9376
0.7742 0.8782 0,9624
0,3633
0.6981
0.7655
0,8740
0,9866
0,9028 0.6743 0,9849 0,7040 0.3482 0.3121 0.6432 0.6808
KAN
0.9411
0,7816 86666 0,9664
0,8109 0,6466
0,7518
0,8286
0,9879 0,8939 0.6650 0,9881 0,8016 0.4120 0,3971 0.7489 0.6835
0,9442
0.7989
0.8755
0,9693
0,9425
0.8940
0,8391
0,9716
0,9888
0.8927 0.6623 0.9868 0,8554 0,5335 0,5177 0,8188 0.7798
KAN-AD
+0.0007
+0,0054
+0,0024
+0,000
+0,0040
+0.0022
+0.0055 +0,003
+0.0005 +0,0025 +0.0022 +0,0009 +0,040 +0,0066 +0.0042 +0.0041
Ground Trush
Anumaaly psint
Table 3. Efficiency comparison on UCR dataset.
Tree Pootiive
Desered Resalt
Fisse Positive
Method
GPU Time
CPU Time
Parameters
F1,
Fulse: Nagative
SAND
5637s
0.5108
D.Dcayy PA
SubLOF
299s
0.4772
Poire-wise PA
OFA
220s
3087s
81.920 M
0.3176
AnoTrans
201s
1152s
4.752
0.1696
Fomnt-wior PA
FCVAE
2327s
1743s
1,414 M
0.3812
TimesNet
182s
259g
73,449
0,1805
LSTMAD
73s
267s
10,421
0,3482
Figure 4. Illustration of the adjustment strategy. Point-wise PA gives an inflated score when some anomaly segments persist for a long duration. Event-wise PA treats each anomaly segment as an event, completely disregarding the length of the anomaly segment. k-delaay PA considers only anomalies detected within the first points after the anomaly onset, treating any detected later as undetected.
KAN
66s
34s
1,360
0.4120
FITS
32s
17s
624
0,2926
TranAD
113s
62s
369
0,1840
KAN-AD
42s
36s
274
0.5335
with results summarized in Table 2. Our analysis focuses on three key dimensions: detection accuracy, model effi- ciency, and computational requirements. Across diverse ex- perimental settings, KAN-AD demonstrates consistent and robust performance advantages. In the TODS dataset, where training data contains a substantial proportion of anomalies, KAN-AD significantly outperforms SOTA by 27% on Event F1, highlighting its robust learning capabilities in handling noisy training data. For datasets exhibiting strong periodic characteristics (WSD and KPI), KAN-AD achieves com- parable or superior performance relative to statc-of-the-an approaches. Even in the challenging UCR dataset scenario, where the training set lacks anomaly samples and contains significant periodic variations, KAN-AD effectively cap- tures normal patterns, whereas baseline methods show re- duced effectiveness in pattern recognition. Quantitatively, KAN-AD achieves more than a 15% improvement in av- erage Event F1 score compared to existing state-of-the-an methods.
ries data. Furthermore, due to the potential impact of such segments, early identification is crucial. Previous work (Xu et al., 2018) proposed the Best F1 metric, which iterates over all thresholds and applies a point adjustment strategy to calculate the Fl score. However, has been criticized for performance inflation (Lai et al., 2021; Wu & Keogh, 2021).
To address this, we also adopt Delay F1 (Ren et al., 2019) and Event F1. Delay Fl is similar to Best F1 but uses delay point adjustment strategy. As shown in Figure 4, the second anomaly was missed because the detection delay exceeded the threshold of five time intervals. In all experi- ments, a delay threshold of five was used across all datasets. Event FI, on the other hand, treats anomalies of varying lengths as anomalies with length of 1, minimizing per- formance inflation caused by excessively long anomalous segments. For the sake of convenience, unless otherwise stated, we use Event Fl as the primary metric, as it is more alignment with the need for real-time anomaly detection in real-world situations.
The computational efficiency analysis, presented in Table 3. reveals another distinctive advantage of KAN-AD. We note that several baseline methods are excluded from this compar- ison due to implementation constraints: SAND's ' CPU-only execution requirement and SubLOF's limited multi-core utilization capabilities preclude fair comparison in modern hardware acceleration contexts. Among the other mod-
4.2. RQL. Performance and Efficiency Comparison
We present a comprehensive evaluation of KAN-AD across multiple time series anomaly detection (TSAD) experiments,
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
tions among normal samples across cycles lead to periodic false alarms during normal segments, consistent with our observations in Figure 2. Among the methods listed, while OFA, LSTMAD, SubLOF, and FITS can detect anomalies, their high anomaly scores during normal segments indi- cate excessive sensitivity to minor fluctuations in normal data. In contrast, KAN-AD excels in identifying anomalies while maintaining minimal anomaly scores during normal segments.
Original Sample
Su
PITS
FOVAE
KAN
TimesNet
TranAD
SAND
Anomaly Transformer
LSTMAD
KAN-AD (ours)
Figure 5. Case study on UCR InternalBleeding 10. The black curve represents the original sample, the red curve represents the anomaly scores provided by the method, and the true anomaly segments are highlighted in pink.
4.3. RQ2. Hyperparameter sensitivity
The KAN-AD model incorporates two key hyperparameters: the number of terms in univariate functions N and the win- dow size T. To investigate the ultimate impact of these pa- rameters on model performance, we conducted experiments on the UCR dataset while holding all other parameters con- stant. As findings summarized in Figure 7, a larger window size facilitates more accurate learning of normal patterns when N is fixed, leading to improved performance. When T is fixed, insufficient univariate functions limit KAN-AD's expressive power, while excessive N can lead to overfit- ting. Overall, KAN-AD achieved its best performance with T = 96 and N = 2. Notably, even with suboptimal hyper- parameter settings like T = 16 and N = 1, we surpassed SOTA methods on the UCR dataset.
1.0
0.54
01.82
2
KPI
TODS
0.78
0.7
WSD>
0.76
B ba
0.6
10
Asmmdy Lengh
2
16
Figure 6. Anomalous lengths distribution.
Figure 7. Model performance under different hyperparameters.
els, we observe wide spectrum of model complexities, with parameter counts ranging from millions to hundreds. Large-scale models like OFA utilize 81.92M parameters, while established approaches such as Anomaly Transformer, FCAVE, and TimesNet employ between 73k and 4.75M parameters. contrast, KAN-AD achieves competitive per- formance with remarkable efficiency, requiring only 274 parameters, 25% reduction compared to TranAD, the next most compact model in our comparison.
4.4. RQ3. Ablation Studies
In this section, we investigated the impact of constant term elimination modules, different univariate function selec- tions on algorithm performance and the influence of the function deconstruction mechanism.
1.4.1. CONSTANT TERM BLIMINATION MODULE
These empirical findings underscore KAN-AD's excep- in TSAD tasks. By achiev- ing state-of-the-art or near state-of-the-art performance while significantly reducing the parameter footprint, KAN-AD demonstrates the effectiveness of our design prin- ciples in creating efficient, practical solutions. This com- bination of high detection accuracy and minimal computa- tional requirements positions KAN-AD as an ideal choice for resource-constrained or cost-sensitive deployment sce- narios, offering a compelling balance between model com- plexity and detection capabilities.
We employed a constant term elimination (CTE) module during data preprocessing to mitigate the influence of the offset term Ao in Equation (3). Further experiments were conducted across all datasets to evaluate the impact of in- corporating CTE within the preprocessing pipeline. As pre- sented in Figure 8, the impact of CTE varies across datasets, reflecting inherent data characteristics. For datasets with pronounced periodicity or strong temporal stability (e.g.. WSD), the benefits of CTE are less apparent. Conversely, for datasets exhibiting larger value fluctuations or trends (e.g., KPI, TODS and UCR), CTE yields significant im- provements.
tional
4.2.1. CASE STUDY
4.4.2. SELECTION OF UNIVARIATE FUNCTIONS
We analyzed anomaly detection performance on UCR dataset samples to illustrate how various methods respond to identical anomalies, as shown in Figure 5. The selected sam- ple displayed pattern anomalies, marked by significant devi- ations from typical behavior. Both TranAD and TimesNet exhibit difficulty establishing normal patterns. Minor varia-
To assess the impact of different univariate functions on model performance, we conducted experiments using com- mon univariate functions listed in Table 4. In our imple- mentations, due to varying input range requirements across
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
CITY
Tyler
Chrlyeber
Chibydder der
0.9
CTE
0.5
0.6 -+- SAX AD
1STMAD
0.7
2.3
-8- TrAnAU
POWAE
SRONN
KAN
0.4
TimesNet
AsoomalyTranffcmar
0.5
0.4
0.6
0.2
0.3
KPI
TODS
WSD
UCR
10%
Dataset
Figure 8. Model performance under different preprocessing.
Figure Model performance under diffcrent univariate function.
Figure 10. Model performance under
different anomaly ratios in training.
Table Commonly used univariate functions for time series ap- proximation.
Table 5. Model performance UCR dataset under different func-
tion deconstruction strategies.
Name
4 7e (z)
Variation KAN-AD
F1
F1d
AUPRC
0.5335
0.5177
0,8188
Taylor Series
Fourier Series
(os( ) + sin (n.))
w/o
0.5153
0.4974
0.8066
Chebyshev Polynomial I
cos(n arccos (x)
w/o P
0.5081
0,4810
0.8007
Chebyshev Polynomial II
sin((x+1) arccos x))
w/o
0.5056
0.5113
0.7998
(in(arccos(x:))
w w/o X&P
0.4737
0.4583
0.7872
/o X&SS
0.4698
0.4610
0.7767
/o S&P
0.4561
0.4637
0.7595
univariate functions, appropriate normalization techniques are employed. Specifically, min-max scaling to the range x e [-1, was utilized for both types Chebyshev poly- nomials, while z-score was employed for Taylor series and Fourier series. The performance of all four univariate func- tions was compared using the same configuration. As results presented in Figure 9, Fourier series consistently achieved the top two performance across all datasets. In contrast, Tay- lor series exhibited persistent bias due to non-zero function values most cases, hindering optimal model performance. The objective of both types of Chebyshev polynomials is to minimize the maximum error, which potentially con- flicts with anomaly detection methods that minimize squared prediction error, thus leading suboptimal perfor- mance.
4.6. Ablation on function deconstruction mechanism
To investigate the impact of the function deconstruction mechanism, we compared the model's detection capabilities under different univariate function combination strategies. For clarity, the specific definitions are provided in Equa- tion (5). As the results presented in Table 5, the model's detection performance exhibited a notable improvement with an increasing number of univariate functions. Both Fourier series and cosine waves outperformed the raw input data, likely due their smoother representations compared to the original signal, enabling higher detection accuracy. The combination of different features, particularly those involving Fourier series and cosine waves, resulted in signif- icant performance gains as the feature count increased. Ulti- mately, KAN-AD achieved optimal detection performance by integrating features. is worth noting that even the variant of KAN-AD utilizing only the raw time series X outperforms KAN, clearly demonstrating the advantage of Fourier series the of spline functions for optimizing univariate functions.
4.5. RQ4. Robustness to Anomalous Data
To evaluate KAN-AD's robustness anomalies in the train- ing set, we conducted additional experiments using synthetic datasets constructed in accordance with the TODS dataset generation methodology. We synthesized test datasets containing local peaks and drops anomalies, and progres- sively increased the proportion of these anomalies in the initially anomaly-free training set. As illustrated in Fig- ure 10, KAN-AD demonstrates stable performance across all anomaly ratios. Popular methods such LSTMAD, perform well at lower anomaly ratios but experience a sig- nificant decline as the ratio increases. Other approaches, like TranAD, fail to achieve optimal performance due to overfitting to fine-grained structures within the time series.
4.7. Performance Multivariate Time Series
To extend KAN-AD's application to the multivariate time series (MTS) scenario, we adopt a chamnel-independent approach. Specifically, an MTS input with the shape (batch_size, window,lengtin is reshaped into (batch.size * n.features, window length). Each of the features channels
n_features)
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
Table Best Fl and parameter counts for multivariate time series anomaly detection. Best and second hest results are in bold and underline.
Methods
SMD
MSL
SMAP
SWaT
PSM
Avg F1
Parameters MSL
Informer (Zhou et al., 2021)
0.8165
0.8406
0.6992
0.8143
0.7710
0.7883
504,174
Anomaly Transformer (Xu et al., 2022)
0.8549
0.8331
0.7118
0.8310
0.7940
0.8050
4,863,055
DLinear (Zeng et al., 2023)
0.7710
0.8488
0.6926
0.8752
0.9355
0.8246
20,200
Autoformer (Wu et al., 2021)
0.8511
0.7905
0.7112
0.9274
0.9329
0.8426
325,431
FEDformer (Zhou et al., 2022)
0.8508
0.7857
0.7076
0.9319
0.9723
0.8497
1,119,982
TimesNet (Wu et al., 2023)
0.8462
0,8180
0.6950
0,9300
0.9738
0.8526
75,223
UniTS (Gao et al., 2024)
0.8809
0.8346
0,8380
0.9326
0.9743
0.8921
8,066,376
KAN-AD (ours)
0.8429
0.8501
0.9450
0.9350
0,9650
0.9076
4,491
is thus treated as an independent univariate time series instance. KAN-AD is then applied to these individual series. This channel-independent strategy has proven effective (Nie et al., 2023). By adopting similar principle, KAN-AD can leverage its robust univariate modeling capabilities across all channels of an MTS dataset. The model is trained on the collection of these reshaped univariate instances, allowing it to learn generalized normal patterns.
construction capabilities: TranAD (Tuli et al., 2022) em- ploys adversarial learning for robust pattern capture, while OFA (Zhou et al., 2023) leverages GPT-2 (Radford et al., 2019) for modeling complex temporal dependencies.
These approaches identify anomalies through comparative analysis of current and historical patterns. Early methods, like SubLOF (Bre- unig et al., 2000) quantify pattern variations using window- based distance metrics. SAND employs temporal shape- based clustering to distinguish anomalous patterns. Recent advances, exemplified by TriAD (Sun al., 2024), leverage multi-domaii contrastive learning frameworks, demonstrat- ing superior performance on UCR datasets.
Pattern Change Detection Methods:
We implemented MTS versions of KAN-AD in popular time series library (THUML) and evaluated them on the com- mon SMD (Su et al., 2019), MSL (Hundman et al., 2018a), SMAP (Hundman et al., 201 8b), SWaT (Mathur & Tippen- hauer, 2016), and PSM (Abdulaal et al., 2021) datasets. Our evaluation metric uses the Best Fl score which is consistent with the baseline methods. We introduce these datasets and baseline methods in detail in the Appendix B. As detailed in Table 6, KAN-AD achieves the highest average Best Fl score of 0.9076, across all five benchmark datasets, outper- forming all listed SOTA methods. A significant advantage of KAN-AD is its exceptional parameter efficiency. With only 4,49/ trainable parameters (measured on MSL), KAN- AD utilizes substantially fewer parameters than all other compared methods.
6. Conclusion
Training time series anomaly detection models with datasets containing anomalies essential for deployment in pro- duction environments. Existing algorithms often rely on carefully selected features and complex architectures to achieve minor accuracy gains, neglecting robustness dur- ing training. This paper introduces KAN-AD, robust anomaly detection model rooted in the Kolmogorov-Aroold representation theorem. KAN-AD transforms the predic- tion of time points into the estimation of coefficients of Fourier series, achieving strong performance with few pa- rameters, significantly reducing costs while enhancing ro- bustness to outliers. KAN-AD includes constant term elimination module to address temporal trends and lever- ages frequency domain information for better performance. KAN-AD surpasses the SOTA model across four public datasets with a 15% improvement in average Event Fl score, simultancously achieving an 80% reduction in parameter count and 50% faster inference speed compared vanilla KAN. With KAN-AD, a promising research direction is to explore whether normal patterns in time series can be represented more efficiently by leveraging additional data.
5. Related Work
Time Series Forecasting Methods: These methods can be categorized into predcction-based and reconstruction-bart based methods, both aiming to identify deviations from normal patterns through temporal analysis. Prediction-based meth- ods, like FITS (Xu et al., 2024) achieves efficient detection through frequency domain analysis with minimal param- eters, while LSTMAD (Malhotra et al., 2015) leverages LSTM networks (Hochreiter & Schmidhuber, 1997) to cap- ture complex temporal dependencies. Reconstruction-bart approaches, like Donut (Xu et al., 2018) 8) focus on time se- ries denoising, while FCVAE (Wang et al., 2024) enhances the VAE (Kingma& & Welling, 2022) framework by incor- porating frequency domain information. Recent advances in Transformer architectures have further strengthened re-
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
Acknowledgments
Hendrycks. D. and Gimpel, K. Gaussian error linear units
(gelus). arXiv preprint arXiv: 2016.
This work was partially funded by the National Key Research and Development Program of China 21o.2022YFB20118000 the National Natural Science Foun- dation of China (62202445), the National Natural Science Grants Council (RGC) Joint Research Scheme (62321 166652), and the National Natural Science Foundation of China (Grant No. W2412136).
Hochreiter, S. and Schmidhuber, J. Long short-term memory.
Neural computation,
1997.
Foundation of
Hundman, K., Constantinou, V., Laporte, C., Colwell, I., and Soderstrom, T. Detecting spacecraft anomalies us- ing Istms and nonparametric dynamic thresholding. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 387-395, 2018a.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Hundman, K., Constantinou, V., Laporte, C., Colwell, I,, and Soderstrom, T. Detecting spacecraft anomalies us- ing Istms and nonparametric dynamic thresholding. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 387-395, 2018b.
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
A. Datasets and Baseliens on Univariate Time Series
A.1. Datasets
We selected four datasets from diverse domains, with samples originating from:
KPI (Competition, 2018): This dataset comprises service metrics collected from five major Internet companies: Sogou, eBay, Baidu, Tencent, and Alibaba. The data points primarily recorded every 1-2 minutes, with some sections exhibiting 5-minute interval.
TODS (Lai et al., 2021): TODS comprises artificially created time series, each designed to present specifie types of anomalies. Its excellent interpretability and carefully constructed data distributions make it suitable for in-depth case studies.
WSD (Zhang et al., 2022): This dataset consists of web server metrics collected from three companies providing large-scale web services: Baidu, Sogou, and eBay.
UCR (Wu & Keogh, 2021): This archive contains data from multiple domains with single anomalous segment on each time series. In addition to real anomalies, UCR also includes synthetic but highly plausible anomalies.
A.2. Baselines
We selected the following baseline approaches to further elaborate on the performance differences between KAN-AD and SOTA methods:
SubLOF (Breunig et al., 2000) represents traditional outlier detection techniques based on distance metrics.
SRCNN (Ren et al., 2019) is supervised approach reliant on high-quality labeled data.
LSTMAD (Malhotra et al., 2015) leverages Long Short-Term Memory (LSTM) networks (Hochreiter & Schmidhuber, 1997) for deep learningbbased anomaly detection.
FITS (Xu et al., 2024) achieves parameter-efficient anomaly detection by upsampling frequency domain information using low-pass filter and simple linear layers.
FCVAE (Wang al., 2024) is unsupervised reconstruction method based on Variational Autoencoder (VAE) (Kingma & Welling, 2022), designed to reconstruct normal patterns.
Anomaly Transformer (Xu et al., 2022) employs attention mechanism to computer the association discrepancy.
TranAD (Tuli et al., 2022) incorporates the principles of adversarial learning to develop training framework with two stages while integrating the strengths of self-attention encoders to capture the temporal dependency embedded in the time series.
SAND (Boniol et al., 2021) utilizes novel statistical approach based on curve shape clustering for anomaly detection in a streaming fashion.
TimesNet (Wu et al., 2023) leverages an Inception (Szegedy et al., 2015)-based computer vision backbone to enhance learning capabilities.
OFA (Zhou et al., 2023), with GPT-2 (Radford et al., 2019) as its backhone, improves its ability to capture point-to-point dependencies.
KAN (Liu et al., 2025) leverages Kolmogorov-Amold representation theory to decompose complex learning objectives into linear combinations of univariate functions.
These baseline methods encompass a variety of anomaly detection paradigms: shape- based SAND, subsequence distance- based SubLOF, Transformer-basee approaches like OFA, TranAD, and Anomaly Transformer for modeling sequence relationships, and frequency domain information enhanced methods FCVAE and FITS.
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KAN-AD: Time Series Anomaly Detection with Kolmogorov-Arnold Networks
B. Datasets and Baselines on Multivariate Time Series
B.1. Datasets
We evaluated KAN-AD on five widely-used public benchmark datasets for multivariate time series anomaly detection:
SMD (Su et al., 2019): A dataset collected from large internet company, containing data from many server machines several weeks.
MSL (Hundman et al., 2018a): A dataset from NASA containing telemetry data from the Mars Science Laboratory rover.
SMAP (Hundman al., 2018b): Another NASA dataset, containing telemetry data from the SMAP satellite.
SWaT (Mathur& & Tippenhauer, 2016): A dataset generated from scaled-down real-world water treatment testbed, including normal and attack scenarios.
PSM (Abdulaal et al., 2021): A dataset from eBay, consisting of aggregated metrics from multiple application servers.
B.2. Baselines
We selected the following baseline approaches to further evaluate KAN-AD and SOTA methods on multivariate time series datasets:
Informer (Zhou et al., 2021): A Transformer-base model designed for long sequence time-series forecasting, featuring ProbSparse self-attention mechanism to improve efficiency. For anomaly detection, it typically relies on reconstruction error or forecast error.
Anomaly Transformer (Xu et al., 2022): A Transformer architecture specifically tailored for time series anomaly detection, which aims to learn prior-associations and serics-associations to better distinguish anomalies.
DLinear (Zeng et al., 2023): A simple yet effective linear model that decomposes the time series into trend and remainder components, challenging the necessity of complex Transformer architectures for some forecasting tasks and adaptable for anomaly detection via reconstruction.
Autoformer (Wu et al., 2021): A Transformer model with a novel decomposition architecture and an Auto-Correlation mechanism, designed discover series-wise connections and improve long-term forecasting accuracy.
FEDformer (Zhou et al., 2022): A Transformer variant that enhances performance for long sequence forecasting by employing frequency-enhancee decomposition and a mixture of expert design in the frequency domain.
TimesNet (Wu et al., 2023): A model that transforms ID time series into of 2D tensors based on identified periods and applies 2D kernel (e.g., Inception block) to capture both intra-period and inter-period variations for general time series analysis.
UniTS (Gao et al., 2024): Aims provide unified framework for time series analysis, often leveraging large-scale pre-training diverse datasets build universal representation for both univariate and multivariate time series tasks.
14


Reviews:
[{"review_id": "12cf35QmEN", "reviewer": "Reviewer_GaXC", "summary": "This paper proposes a method for univariate time series anomaly detection. In particular, they aim to approximate the time series using smooth univariate functions. They build upon a method that uses Kolmogorov-Arnold Networks to approximate the time series, by replacing B-splines functions with Fourier series. They demonstrate that the new model is more robust to anomalies in the training set and therefore tends to perform better on the test set, especially, not surprisingly, where the training set has a larger percentage of anomalies. The models also are smaller compared to state-of-the-art anomaly detection algorithms, in terms of the number of parameters, and also have lower running times.\n\n## update after rebuttal\n\nI had minor questions, which the authors answered. For this reason, I will keep my review as it is.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to reviewer GaXC\n\nWe appreciate the constructive comments provided by the reviewer and will incorporate improvements in the subsequent version of the paper.\n\n**Comparative analysis of algorithm strengths**: Indeed, different methods are good at detecting specific types of anomalies. For example, frequency-domain-based methods (FCVAE [1], FITS [2], etc.) excel at detecting periodic anomalies, methods with differential modules [3] are more adept at handling spike-type anomalies, and methods with shape clustering modules (e.g., SAND [4]) are more effective at detecting shapelet anomalies.\n\n**Bias in the Taylor series**: The Taylor series can keep the value to be fitted around 0 through a constant term elimination mechanism. However, based on the mathematical properties of the Taylor series, when using a finite number of terms to fit curves with large fluctuations in absolute values, there will always be substantial residual errors (this is what we refer to as \"bias\" in the original paper) [5].\n\n\n[1] Z Wang, et al. Revisiting VAE for Unsupervised Time Series Anomaly Detection: A Frequency Perspective. WWW 2024.\n\n[2] Z Xu, et al. FITS: Modeling Time Series with 10k Parameters. ICLR 2024.\n\n[3] Wu R, et al. Current time series anomaly detection benchmarks are flawed and are creating the illusion of progress. TKDE 2021.\n\n[4] P Boniol, et al. Sand: streaming subsequence anomaly detection. VLDB 2021.\n\n[5] Rudin W. Principles of Mathematical Analysis. 2021.", "claims_and_evidence": "The algorithmic claims made by the paper are clearly true. The algorithm is clearly explained, and performance metrics (F1-based metrics) and running times and model sizes are given. The latter two are particularly notable, since Machine Learning papers rarely state their running times and model sizes, even when these are presented as advantages of the proposed methods. I'm glad the authors are bucking this disturbing tendency.\n\nThe number of datasets and types of datasets on which the algorithm is tested is insufficient. The percentage of anomalies in the training and test set range from zero to 7% is low, but I think that this is okay since the proposed method is aimed at addressing robustness to anomalies, so actually having a relatively low fraction of anomalies is more difficult for the method. However, the range of types of anomalies and the range of normal behavior seems to need broadening---amplitude, frequency, combinations of these.", "methods_and_evaluation_criteria": "While the range of benchmark datasets needs broadening, as I mentioned in the claims and evidence section, the metrics that they use for evaluation are quite appropriate.", "theoretical_claims": "There are no proofs provided, which is reasonable given the nature of this paper, which is more algorithmic and experimental.", "experimental_designs_or_analyses": "I did check the soundness of the experimental designs and analysis. The analysis provided is fairly thorough and appropriate. I have a few questions/comments on this:\n\n1. In figure 4, it appears that the 5-delay PA and Point-wise PA rows are swapped.\n2. What are the differences in the types of anomalies detected by the different methods?\n3. In 4.4.2, the authors state \"In contrast, Taylor series exhibited persistent bias due to non-zero function values in most cases, hindering optimal model performance.\" Is this true specifically in the context of this ablation study? I can't see this being an issue in general, since Taylor series, like the other methods, allows for constant term elimination.", "relation_to_broader_scientific_literature": "This paper yields an anomaly detection method that has greater robustness to anomalies in the training set, is smaller in terms of the number of patterns, and runs faster; relative to recent methods that have a large number of parameters.", "essential_references_not_discussed": "I cannot think of any essential references that were not discussed in this paper.", "other_strengths_and_weaknesses": "I have no other notable strengths and weaknesses to bring up beyond what I wrote in other sections.", "other_comments_or_suggestions": "1. In figure 2, the anomalies in the bottom row appear to be more difficult to detect than the anomalies in the top row, which seems to better explain the differences in performance than the claimed difference, which is the level of noise in the training sample.", "questions_for_authors": "1. The main question that I have is about differences in the types of anomalies detected by the different methods, going beyond simply performance measures. This likely will not shift my decision by a full level given that there are five choices, but this is nevertheless an important question to answer in general, and one that is only occasionally answered in Machine Learning papers.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Uh72x18t93", "reviewer": "Reviewer_HZbR", "summary": "This paper focuses on time series anomaly detection (TSAD), and proposes KAN-AD, which models \"normal\" behavior of time series using smooth functions. The paper addresses the shortcoming of existing methods that tend to overfit local variances in time series data. Proposed KAN-AD is a clever and novel approach -- introduces Fourier expansion within the KAN network -- which shows its effectiveness on benchmark time series datasets.\n\n## update after rebuttal\nI appreciate authors for addressing my comments. I think including the rebuttal responses makes the paper stronger. Achieving competitive/superior performance on the MTS task with only a fraction of parameters points to the superior architecture KAN-AD offers for the task.\n\nMy other comment was about testing the model's performance on real use-case dataset outside of the benchmarks that often contain synthetic anomalies. Nevertheless, I appreciate your efforts in providing detailed responses to my comments. I already liked this work, and will keep my score as is.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to reviewer HZbR\n\nWe appreciate the constructive suggestions provided by the reviewer and will incorporate improvements in the subsequent version of the paper.\n\n**Performance of KAN-AD on MTS**: We can reshape the original multivariate input `(batch, window, n_multivariate)` into `(batch * n_multivariate, window)`, which effectively supports multivariate time-series data. We implemented KAN-AD in a popular third-party Times-Series-Library [2] and trained it using the same seed (seed=2021) as the SOTA methods. We compared KAN-AD trained in this way with SOTA multivariate anomaly detection algorithms across several datasets, with the following results:\n\n| Methods             | SMD       | MSL       | SMAP      | SWaT      | PSM       | Avg       | Parameters@MSL |\n| ------------------- | --------- | --------- | --------- | --------- | --------- | --------- | -------------: |\n| Informer            | 81.65     | 84.06     | 69.92     | 81.43     | 77.10     | 78.83     |        504,174 |\n| Anomaly Transformer | *85.49*   | 83.31     | 71.18     | 83.10     | 79.40     | 80.50     |      4,863,055 |\n| DLinear             | 77.10     | *84.88*   | 69.26     | 87.52     | 93.55     | 82.46     |         20,200 |\n| Autoformer          | 85.11     | 79.05     | 71.12     | 92.74     | 93.29     | 84.26     |        325,431 |\n| FEDformer           | 85.08     | 78.57     | 70.76     | 93.19     | 97.23     | 84.97     |     1,119, 982 |\n| TimesNet            | 84.62     | 81.80     | 69.50     | 93.00     | *97.38*   | 85.26     |         75,223 |\n| UniTS               | **88.09** | 83.46     | *83.80*   | *93.26*   | **97.43** | *89.21*   |      8,066,376 |\n| KAN-AD              | 84.29     | **85.01** | **94.50** | **93.50** | 96.50     | **90.76** |      $\\mathbf{4,491}$ |\n\nAs seen from the results, KAN-AD outperforms the SOTA methods in average, while using only 0.05% of the parameters compared to UniTS. We have made the MTS version of KAN-AD available in an anonymous repository for further review. https://anonymous.4open.science/r/TSL-C6AC\n\n**Impact of data characteristics on performance**: In our paper (Column 2, Lines 292-295), we briefly discuss how different types of data across datasets influence the final performance. We will further elaborate on this in the next version.\n\n**Additional data**: Thanks to KAN-AD's parameter efficiency, we believe incorporating textual information (such as metric name) could enhance its performance. Furthermore, leveraging a Large Language Model to route multiple downstream KAN-AD models could further boost overall performance. Since KAN-AD is a very lightweight approach, this integration will be much easier.\n\n**Challenges with rapidly oscillating data**: For EPG-type time-series data in the UCR dataset, the detection accuracy of both KAN-AD and other baselines remains relatively low due to the highly volatile nature of these signals, even within very small time windows.", "claims_and_evidence": "I think the claims made — better detection, faster inference, robustness to noise — are supported through empirical results.", "methods_and_evaluation_criteria": "KAN-AD definitely makes sense. With respect to evaluation, the metrics Event F1 and Delay F1 align with real world use cases of detecting sustained as well as varied length anomalus segments in the time series.", "theoretical_claims": "The paper doesn’t introduce a new theory/theorems, instead relies on the Kolmogorov-Arnold representation theorem as the theoretical foundation. Though I haven’t verified it, I think the correctness of the Kolmogorov-Arnold theorem is well established in the literature.", "experimental_designs_or_analyses": "I’d answer yes. Experimental setup is clearly described, with details on datasets, training and testing setup, baselines and evaluation metrics as well as ablation studies (Section 4 as well as appendix C). I do not find issues with designs as such.", "relation_to_broader_scientific_literature": "The paper draws on KAN networks, and clearly points to overfitting phenomena in deep anomaly detection models. I actually like the paper, and find that it provides a new perspective on TSAD approach, overcomes the overfitting issues of existing methods, and cleverly uses Fourier transform to improve current KANs for time series anomaly detection, which is an extremely relevant problem to study.", "essential_references_not_discussed": "I think the paper references related literature.", "other_strengths_and_weaknesses": "++ The paper presents an original approach to time series anomaly detection by reformulating the problem and leveraging KANs. The modifications made to the original KAN architecture, including the use of Fourier series and the lightweight learning mechanism, are also original and well-motivated.\n   \n++ The paper studies an TSAD which has significant practical applications. Importantly, the proposed model is nimble with only about 300 parameters making it suitable for real-world environment with improvements in detection accuracy and inference speed.\n\n++ The paper is well-written, organized, and is easy to read. \n\n\n–  Not really a weakness, but something that I find worth mentioning. Most of the real world time series datasets, are naturally multivariate, while the paper focuses on univariate series. How hard would it be to extend the framework to multivariate setup? How will it affect the design choices in the proposed KAN?\n\n– The experiments cover a wide range of time series dataset, however, these are mostly well curated datasets, with many containing synthetic anomalies. I think the empirical study could be stronger if the method is applied on a real world dataset. It would be particularly interesting to see how well does it detect anomalies around 2008 recession, covid 19 etc.\n\n– Discussions around why on some datasets, KAN-AD is comparable to other methods while in some datasets it clearly outperforms (UCR vs WSD). What are the failure cases or scenarios of KAN-AD?  What dataset properties support or oppose KAN-AD?", "other_comments_or_suggestions": "Please double-check the references; I believe there are duplicates.", "questions_for_authors": "In the discussion of future work, the paper mentions exploring whether normal patterns in time series can be represented more efficiently by leveraging additional data. What types of additional data could be considered and could it be integrated into the KAN-AD framework? \n\n\nAs stated earlier, are there specific types of time series data or anomaly patterns where KAN-AD might be expected to struggle?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "F5ZJ33mvPd", "reviewer": "Reviewer_HB8y", "summary": "The paper discusses about the problem that most TSAD methods using forecasting models tend to overfit to local fluctuations, and reformulates time series modeling while approximating to smooth univariate functions. The paper adopted KAN backbone for TSAD, while replacing B-spine function with Fourier series for local smoothness. The proposed method KAN-AD is composed of mapping phase where raw time series is converted into multiple univariate functions, reducing phase where the coefficients of univariate functions are obtained, and projection phase where the coefficients are aggregated into the normal pattern. Experimental results show that KAN-AD achieves state-of-the-art performance on univariate time series datasets, showcasing its effectiveness.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to reviewer HB8y\n**Synthetic Method of the TODS Dataset**: We used the synthetic TODS dataset from [1], which includes all five anomaly types with diverse durations and the non-trivial characteristics introduced by TODS. This dataset is publicly available in their repository.\n\n**KAN-AD's Performance on MTS Data**: We can reshape the original multivariate input `(batch, window, n_multivariate)` into `(batch * n_multivariate, window)`, which effectively supports multivariate time-series data. We implemented KAN-AD in a popular third-party Times-Series-Library [2] and trained it using the same seed (seed=2021) as the SOTA methods. We compared KAN-AD trained in this way with SOTA multivariate anomaly detection algorithms across several datasets, with the following results:\n\n| Methods             | SMD       | MSL       | SMAP      | SWaT      | PSM       | Avg       | Parameters@MSL |\n| ------------------- | --------- | --------- | --------- | --------- | --------- | --------- | -------------: |\n| Informer            | 81.65     | 84.06     | 69.92     | 81.43     | 77.10     | 78.83     |        504,174 |\n| Anomaly Transformer | *85.49*   | 83.31     | 71.18     | 83.10     | 79.40     | 80.50     |      4,863,055 |\n| DLinear             | 77.10     | *84.88*   | 69.26     | 87.52     | 93.55     | 82.46     |         20,200 |\n| Autoformer          | 85.11     | 79.05     | 71.12     | 92.74     | 93.29     | 84.26     |        325,431 |\n| FEDformer           | 85.08     | 78.57     | 70.76     | 93.19     | 97.23     | 84.97     |     1,119, 982 |\n| TimesNet            | 84.62     | 81.80     | 69.50     | 93.00     | *97.38*   | 85.26     |         75,223 |\n| UniTS               | **88.09** | 83.46     | *83.80*   | *93.26*   | **97.43** | *89.21*   |      8,066,376 |\n| KAN-AD              | 84.29     | **85.01** | **94.50** | **93.50** | 96.50     | **90.76** |      $\\mathbf{4,491}$ |\n\nAs seen from the results, KAN-AD outperforms the SOTA methods in average, while using only 0.05% of the parameters compared to UniTS. We have made the MTS version of KAN-AD available in an anonymous repository for further review. https://anonymous.4open.science/r/TSL-C6AC\n\n**Impact of Parameter N on Model Performance**: Your suggestion is great, larger N allows for more precise modeling of the time-series data, but it also comes with the risk of \"overfitting\" to the anomaly patterns within the time window. When N becomes too large, the model creates an accurate prediction of the anomaly pattern, which leads to very small anomaly scores when the anomaly occurs. As a result, the model might fail to detect anomalies, leading to performance degradation.\n\n\n[1] Si H, et al. Timeseriesbench: An industrial-grade benchmark for time series anomaly detection models. ISSRE 2024.\n\n[2] https://github.com/thuml/Time-Series-Library Machine Learning Group of Tsinghua University.", "claims_and_evidence": "The claim that existing time series forecasting methods tend to overfit to local fluctuations is well illustrated with qualitative results.", "methods_and_evaluation_criteria": "The proposed method is designed for only univariate time series, and experimented only in univariate datasets, making it impossible to be used in multivariate time series, which is the case in many cases in real life. I assume that the proposed method can be extended to multivariate time series, and I wonder if there is any specific reason for KAN-AD not applied to multivariate time series in the paper.", "theoretical_claims": "The paper does not contain any theoretical claims.", "experimental_designs_or_analyses": "The paper does not clarify how TODS benchmark is synthesized, which can affect the performance highly. The authors should clarify detailed information about the benchmark for its validity.", "relation_to_broader_scientific_literature": "The key contribution of the paper that it adopts KAN backbone for efficient TSAD is related to recent advent of KAN network. The paper adequately adopted Kolmogorov-Arnold representation theorem that it decomposes multivariate continuous function into a finite sum of univariate functions to time series data.", "essential_references_not_discussed": "The paper cited the related works appropriately.", "other_strengths_and_weaknesses": "The attempt to integrate Kolmogorov-Arnold Network for TSAD is original, demonstrating further expandability of KAN into time series analysis in the future. In addition, the design of KAN-AD is clearly described to understand the key components of it. However, it was only employed to univariate time series, covering very limited scope of time series data.", "other_comments_or_suggestions": "There is no other comment.", "questions_for_authors": "1. I wonder if KAN-AD works with multivariate dataset, since the absence of results on multivariate dataset is a critical limitation of the paper. I am willing to change my evaluation of the paper if KAN-AD proves it to be effective in multivariate datasets.\n2. Why is using N=2 the best? I think using bigger N enables more accurate modeling of time window into fourier series, and is more helpful for the task. It would be helpful if the authors provide analysis on it.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gfcfRzablC", "reviewer": "Reviewer_UWPv", "summary": "The paper introduces KAN-AD, a novel approach to time series anomaly detection (TSAD) based on Kolmogorov–Arnold Networks (KANs). The motivation for this work stems from the limitations of existing TSAD methods, particularly those relying on forecasting models, which often overfit to local fluctuations and fail to generalize well in the presence of noise. The authors argue that effective TSAD should prioritize modeling \"normal\" behavior using smooth local patterns rather than attempting to capture every minor variation in the data.\n\nTo address this issue, the authors reformulate time series modeling by approximating the series with smooth univariate functions. They observe that KAN, in its original form, is susceptible to localized disturbances due to its reliance on B-spline functions. To overcome this limitation, they propose KAN-AD, which replaces B-splines with truncated Fourier expansions. This modification enhances robustness against local fluctuations while preserving the ability to capture global patterns. \n\nThe core methodological contributions include the reformulation of time series modeling using Fourier series for improved smoothness, the development of a novel coefficient-learning mechanism based on one-dimensional convolutional neural networks. The authors validate their approach through empirical analysis, demonstrating its effectiveness across various datasets, including those with noisy training data.\n\n## after rebuttal\nI changed my score from reject to weak accept. The authors did address my questions, however I'm still not certain about the authors' choice of evaluation metric. If you see from their tables in the second round, the real performances of the detectors are not as high as they claim with $F1_e$.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to reviewer UWPv\nWe thank the reviewer for the constructive suggestions and will further revise the manuscript accordingly. For **KAN-AD's Performance on MTS**, please refer to our response to reviewer HB8y.\n\n**Clarification on the Claim of Local Smoothness in Normal Curves**: The smoothness of normal patterns is a fact, independent of the type of model or training method used. The Anomaly Transformer [1] also leverages this observation to design its algorithm. Moreover, TSAD methods are not trained solely on data containing normal patterns but are trained on data that mixes both normal and anomalous data. In Table 1 (220-226, Column 2), we show that the training sets for KPI, TODS, and WSD datasets contain a certain proportion of anomalous data.\n\n**Data Splitting**: We have detailed data splitting in 4.1.2 (270-274 col 1).\n\n**Evaluation Metrics**: We are glad that you mention the healthcare example, what you mentioned illustrates that each evaluation metric has its appropriate use case. For the healthcare example, we should use F1 delay, as it allows us to select methods that detect anomalies as early as possible. However, for internet service systems, some anomalies last longer (as shown in Figure 6, where the anomaly duration can exceed 300 points). Using only point-wise PA or AUPRC in such cases may inflate the F1 score, since the algorithm only detects one point in the long anomaly segment will be considered as successfully detecting the entire segment. Event F1 is designed to overcome this issue by measuring whether the algorithm can detect more anomaly **events**, which is more appropriate and aligns with findings in [2].\n\n**Inference Time of SAND and SubLOF**: We tested the inference time under the same experimental conditions as in the paper.\n||CPU Time|UCR F1e|\n|-|-|-|\n|SAND|5637s|0.5108|\n|SubLOF|299s|0.4772|\n|KAN-AD|36s|0.5335|\n\n**Faster CPU Inference Time**: Your observation is correct. CPU inference time can be faster because the GPUs cannot fully leverage parallel processing for models with fewer than 1k parameters.\n\n**Reproducibility Environment**: As indicated in the first line of the list of incompatible dependencies you provided, the issue arose because the conda-forge channel was not configured on your local machine. We understand that no channel can guarantee 100% coverage, so to further facilitate reproducibility, we have provided a Dockerfile in the original repo to assist with this.\n\n**Yahoo and NAB Datasets**: These datasets were considered flawed in [3], and we have adopted their suggestions by not using these datasets in the main experiments. In fact, our algorithm performs even better on these two datasets, especially Yahoo. The AUPRC results for these datasets are shown below. Additionally, the KPI dataset is also known as the AIOps dataset, and we refer to it as AIOps in the code because the KPI dataset was originally used in the AIOps challenge.\n||NAB|Yahoo|\n|-|-|-|\n|SRCNN|0.8561|0.1459|\n|SAND|0.6595|0.5412|\n|Anomaly Transformer|0.9624|0.1109|\n|TranAD|0.9965|0.568|\n|SubLOF|0.9582|0.5222|\n|TimesNet|0.9842|0.4736|\n|FITS|0.9916|0.7803|\n|OFA|0.9947|0.7777|\n|FCVAE|0.9861|0.7049|\n|LSTMAD|0.9932|0.5655|\n|KAN|0.9976|0.7142|\n|KAN-AD|0.9918|0.9547|\n\n**Discussion on Hyperparameters**: We have discussed hyperparameter settings in 4.3 (342-344, col 2).\n\n**Reconstruction-based Baselines**: As outlined in Sec. 5 col 1, forecasting methods can be divided into reconstruction and prediction categories. For the reviewer’s interest in reconstruction methods, we have selected popular SOTA methods: Anomaly Transformer (ICLR 2022), TranAD (VLDB 2022), TimesNet (ICLR 2023), FITS (ICLR 2024), OFA (NeurIPS 2023), and FCVAE (WWW 2024) as our baseline.\n\n**AUPRC of Figure 9**: Due to space constraints, we will only show the AUPRC, and if necessary we will show the F1d in round 2.\n||KPI|TODS|WSD|UCR|\n|-|-|-|-|-|\n|Taylor|0.9411|0.9572|0.9757|0.6904|\n|CI|0.9585|0.9714|0.9832|0.7911|\n|CII|0.9616|0.9584|0.9819|0.7835|\n|Fourier|0.9693|0.9716|0.9868|0.8188|\n\n**Explanation of Figure 10**: The x-axis represents the anomaly ratio in the training set. Since real-world anomaly detection always involve some proportion of anomalies in training data, this figure aims to evaluate algorithm robustness across different anomaly ratios. Performance degrades as anomaly ratios rise, since models are increasingly influenced by anomalous data. This reduces normal pattern accuracy, causing more false positives in detection. The lower part methods show more sensitivity to training anomalies. Even at low ratios, they fail to maintain consistent normal-pattern accuracy, resulting in unpredictable performance curves.\n\n[1] Anomaly Transformer: Time Series Anomaly Detection with Association Discrepancy. ICLR 2022.\n\n[2] An Evaluation of Anomaly Detection and Diagnosis in Multivariate Time Series. TNNLS 2021.\n\n[3] Current Time Series Anomaly Detection Benchmarks Are Flawed and Are Creating the Illusion of Progress. TKDE 2021.", "claims_and_evidence": "The authors makes claims that raise some questions. Here, I'm reporting some of them:\n1. In the introduction the authors state that normal sequences exhibit greater local smoothness than abnormal ones. Isn't this claim just a collorary of reconstruction-based approaches? In other words, in reconstruction-based detectors, the anomalies have higher reconstruction error than normal ones, since the detectors (usually autoencoder-based) are trained on normal sequences only? How do the authors justify this?", "methods_and_evaluation_criteria": "The benchmarks are fine, however the evaluation metrics don't seem to make sense (see next section for my doubts, please).\nAlso, the authors fail to show whether they do any sort of cross-validation after they split the series in windows, or whether they use the time component to define the train, validation, and tests sets. In other words, are you using time, say first 10 windows for training, the 11th and 12th window for validation, and from the 13th window you test? Or are you mixing these windows, say 80% of all windows goes to training, 2% to validation, and the rest to test? If the former is the case, then how many $k$-cross validations are you doing?", "theoretical_claims": "The authors don't make any theoretical claims, which haven't been shown/proven before: i.e., using Fourier transformata instead of B-splines when wanting to capture periodic patterns, or having more local smoothness.", "experimental_designs_or_analyses": "I don't know why the authors are proposing different evaluation strategies. All of the F1 variants adopted seem to misrepresent the performances of KAN-AD. Sure, the delay penalizes KAN-AD, but why do you need to penalize it? Here's an example.\n\n|  | $t_1$ | $t_2$ | $t_3$ | $t_4$ | $t_5$ | $t_6$ | $t_7$ | $t_8$ | $t_9$ | $t_{10}$ |\n|---|---|---|---|---|---|---|---|---|---|---|\n| GT |  |  |  | $\\times$ | $\\times$ | $\\times$ | $\\times$ |  |  | $\\times$ |\n| Detection |  |  |  |  |  |  | T | T |  | T |\n| Timestep-based | TN | TN | TN | FN | FN | FN | TP | FP | TN | TP |\n| 3-Delay PA | TN | TN | TN | FN | FN | FN | FN | FP | TN | TP |\n| Point-wise PA | TN | TN | TN | TP | TP | TP | TP | FP | TN | TP |\n\n$\\times$ is true anomaly; T = detected anomaly; TP = True Positive; FN = False Negative; TN = True Negative; FP = False Positive. \n\nSo, my doubt is on the evaluation itself. Since the authors treat univariate series only, this is even clearer. Each $x_i$ at any timestamp $t_i$ can be either normal or anomalous. Therefore, the predictor can be evaluated for each $t_i$ as shown above: i.e., timestep-based. In these conditions, the authors report $F1_d = 0.2857, F1_e = 0.5714$ - since it doesn't consider the length of the detected anomalies nor false positives. Nevertheless, the real F1 score should be that for each time-step: i.e., $F1_t = 0.5$. The authors claim that they use the event-based calculation - i.e., $F1_e$ - which smears false negatives as being true positives (see the point-wise PA and Figure 4 in the paper). The authors were self-critical with the delay in the beginning and then decide to use the event-based one. It's very confusing, and the reason the authors provide is *\"For the sake of convenience [...] use Event F1 [...] as it is more alignment (**typo**) with the need for real-time anomaly detection in real-world situations.\"* Who says that event F1 is more aligned with real-time anomaly detection? Is there any other work that argues this? If not, this statement is unfounded, and actually harmful, since, again, event F1 obfuscates false negatives, a critical aspect in critical domains, especially healthcare: e.g., if you miss an anomaly in neurodegenerative patients, they might die.\n\n If we take the example in Figure 4 of the paper, we have $F1_d = 0.1429$, and $F1_e = 0.5714$, which is an overestimate of the true performance according to the timestep-based approach proposed above $F1_t = 0.2667$. Therefore, $F1_t$ emulates quite closely the \"self-harming\" $F1_d$. Again, $F1_e$ is an overestimate of the true performances. \n\nThe only sound metric here is AUPRC, which doesn't actually show that KAN-AD is way better than the second-best (i.e., KAN in KPI and WSD). So, in KPI you're comparable to KAN (difference of .029 which is statistically insignificant) and in WSD you underperofm by .013 (which again is statistically insignificant). Let's agree that in 2/4 datasets you're comparable to SoTA, and you win in 2 others. Therefore, across the board KAN-AD is not the best. Moreover, as per my comment in Sec. **Methods And Evaluation Criteria**, we can't even be sure that these metrics make sense since we don't know the number of folds you tested this on and what is the train-validation-test split you used.\n\nMoreover, the average $F1_e$ reported in Table 2 isn't indicative of KAN-AD's overall performance. There are statistical tests that showcase how much better is one method compared to others. For example the Friedmann test with a post-hoc Bonferroni-Dunn could actually show if KAN-AD is better than KAN in the previous two edge cases, which I doubt it would. However, these tests work if you have multiple runs/folds for each of the detectors compared.\n\n**In Table 3, why isn't SAND's resource consumptions reported? It has the second-best $F1_e$ score according to Table 2!** Even though you can execute it on CPU, the reader still should have these values reported. What does the justification you provided *\"SAND’s CPU-only\nexecution requirement and SubLOF’s limited multi-core utilization capabilities preclude fair comparison in modern hardware acceleration contexts\"* mean? You're using most of the other methods on GPU which actually accelerate the methods. What fair comparison are you doing then here by excluding SANDS or SubLOF? Also, how come KAN-AD has a lower execution time on CPU rather than on GPU? Are you doing a lot of I/O transfer operations? Actually this happens for all methods with 1k or less parameters.", "relation_to_broader_scientific_literature": "The scope of the paper seems shortsighted. Nowadays, especially foundational models [1], treat multivariate time-series [2]. KAN-AD treats only univariate ones. Real-world scenarios usually have multiple variables which makes detecting anomalies more complicated especially if one wants to produce an explanation/interpretation of why the anomaly happened. I'm sceptical about KAN-AD's usage in broader TSAD. The overall experiments (especially the evaluation) raises a lot of question about KAN-AD's utility (see **Experimental Designs Or Analyses**)\n\n[1] Gao et al. Units: A unified multi-task time series model. NeurIPS'25\n\n[2] Flaborea et al. Are we certain it's anomalous?. Workshops CVPR'23.", "essential_references_not_discussed": "**Why are the authors only considering forecasting approaches?** AD has also reconstruction-based approaches. This neglection hinders the generalizability of KAN-AD. Therefore, the authors have missed a lot of important competitors in TSAD. Here are a few that should be discussed and, perhaps, compared against. Most importantly, the comparison against UniTS [11] is paramount since they are SoTA in all time-series problems. I'm aware that some of these are >5 years old, however [1,9,11] should definitely be considered in the experiments.\n\n[1] Flaborea et al. Are we certain it's anomalous?. Workshops CVPR'23.\n\n[2] Audibert et al. Usad: Unsupervised anomaly detection on multivariate time series. KDD'20\n\n[3] Bieber et al. Low sampling rate for physical activity recognition. PETRA'14\n\n[4] Geiger et al. Tadgan: Time series anomaly detection using generative adversarial networks. Big Data'20\n\n[5] Hundman et al. Detecting spacecraft anomalies using lstms and nonparametric dynamic thresholding. KDD'18\n\n[6] Li et al: Mad-gan: Multivariate anomaly detection for time series data with generative adversarial networks. ICANN'19\n\n[7] Su et al. Robust anomaly detection for multivariate time series through stochastic recurrent neural network. KDD'19\n\n[8] Zhang et al. A deep neural network for unsupervised anomaly detection and diagnosis in multivariate time series data. AAAI'19\n\n[9] Zhang et al. Unsupervised deep anomaly detection for multi-sensor time-series signals. IEEE TKDE'21\n\n[10] Zhao et al. Multivariate time-series anomaly detection via graph attention network. ICDM'20\n\n[11] Gao et al. Units: A unified multi-task time series model. NeurIPS'25 (**however, the arxiv was published since February 2024, so this cannot be considered simultaneous and concurrent paper, and the authors should've included it in their discussion, especially since it can be used also in forecasting modality to detect anomalies**). I'm mostly interested to see KAN-AD compared against UniTS, the latter beign a foundational model for time-series problems. I suggest the authors look at [12] to see how to adapt it for anomaly detection.\n\n[12] Gabrielli et al. Seamless Monitoring of Stress Levels Leveraging a Universal Model for Time Sequences. arXiv preprint arXiv:2407.03821. 2024 Jul 4.", "other_strengths_and_weaknesses": "### Weakness\nI can't seem to find how the anomaly is detected. After the projection component of KAN-AD, you have a forecast on the time series. Then you have a ground truth (GT) series corresponding to the same window. How do you compare the forecasted and the GT? Nowhere in the paper this is written. Or are you predicting the classes $c_i$ for each input value at timestamp $t_i$? This is confusing.\n\n### Strength (but not so much)\nThe authors rely on the EasyTSAD (probably they forget the project on github) framework to implement their KAN-AD class. This framework provides a straightforward evaluation pipeline which contains the also $F1_d$ the authors show in the paper, among other things. However, although using an-already-built framework makes it easier to prototype and evaluate stuff, the authors refrain from implementing the works I enlisted in **Essential References Not Discussed**. All the reported SoTA methods were already implemented in EasyTSAD, which is ok, but doesn't justify the authors' choice of not discussing the rest, because \"probably it's a hassle to port them to EasyTSAD's design patterns\".", "other_comments_or_suggestions": "1. Why is the first figure on the paper labelled as figure 2? I can't seem to find figure 1.\n2. Figure 7 should rather be a 2d plot where the AUPRC axis can be transformed into a heatmap/colorbar. This improves readability of the plot. 3d-plots are often discouraged, but this is a minor suggestion.\n3. Why is Table 2 the first table in the paper. Please check your labelling system in latex.\n4. Figure 5 isn't much useful. I notice that you provide anomaly scores for each method, but the reader would much appreciate to see which are the detected anomalies. Here you might want to change the GT anomaly color to green, and the detected ones to red. Only in this way one can appreciate the detection capabilities of KAN-AD vs. rest. How the figure is now shown doesn't make much sense. Anomaly errors don't tell me anything.", "questions_for_authors": "I've put them in the other sections to stay consistent with my thought process. This layout of reviewing doesn't allow for cross-referencing. I'm sorry if I scrambled the questions above. Alas, this is my way of reviewing papers, as I feel it's more organic to put questions on each of the sections above.\n\nHere are some more due to character limitation in Sec. **Experimental Designs Or Analyses**.\n\n1. Figure 9 doesn't actually tell me, most of the time, that Fourier is the best. The most indicative example is in UCR, but the others don't show much of difference with the others. So why do you opt for Fourier? Also, what are the $F1_d$ and the AUPRC for these cases? I'd like to see AUPRC in the rebuttal, please.\n2. Why aren't the trends in Figure 10 non-increasing? This doesn't make sense, especially for SoTA forecasting methods. The more the training is tainted, the lower the performances are going to be. It's very wierd, especially for the lower portion of the figure.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Other expertise"], "ethical_review_concerns": "The authors missed the impact statement which is mandatory according to the submission guidelines at https://icml.cc/Conferences/2025/CallForPapers.\n\nIs this paper a desk reject?"}}]

Author rebuttal:

Meta-review: There is a clear consensus in the PC that this is a strong paper that should be accepted. Congratulations to the authors ! I am requesting author to carefully incorporate the suggestion provided by the review in camera ready version.
Decision: Accept (poster)