FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Authors: Xiwen Chen
Wenhui Zhu
Peijie Qiu
Hao Wang
Huayu Li
ZIHAN LI
Yalin Wang
Aristeidis Sotiras
Abolfazl Razi
Venue: ICML 2025 poster (2025)
Keywords: Time Series Classification
Primary area: applications->time_series
TL;DR: An efficient training framework for time series classification that leverages Fisher information as the constraint.

Abstract: Analyzing time series data is crucial to a wide spectrum of applications, including economics, online marketplaces, and human healthcare. In particular, time series classification plays an indispensable role in segmenting different phases in stock markets, predicting customer behavior, and classifying worker actions and engagement levels. These aspects contribute significantly to the advancement of automated decision-making and system optimization in real-world applications. However, there is a large consensus that time series data often suffers from domain shifts between training and test sets, which dramatically degrades the classification performance. Despite the success of (reversible) instance normalization in handling the domain shifts for time series regression tasks, its performance in classification is unsatisfactory. In this paper, we propose $\textit{FIC-TSC}$, a training framework for time series classification that leverages Fisher information as the constraint. We theoretically and empirically show this is an efficient and effective solution to guide the model converges toward flatter minima, which enhances its generalizability to distribution shifts. We rigorously evaluate our method on 30 UEA multivariate and 85 UCR univariate datasets. Our empirical results demonstrate the superiority of the proposed method over 14 recent state-of-the-art methods.

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Xiwen Chen
Wenhui Zhu?
Peijie Qiu Hao Wang' Huayu Li*
Zihan Li
Yalin Wang?
Aristeidis Sotiras
Abolfazl Razi 1
A bstrract
1. Introduction
Time series analysis is crucial in wide range of applica- tions, such as network traffic management (Ferreira et al., 2023), healthcare (Vrba & Robinson, 2001; Niknazar & Mednick, 2024), environmental science (Wu et al., 2023b), and economics (Sezer et al., 2020). This is due to the fact that much of the data they generate or collect naturally fol- lows time series format. Representative examples include stock prices in finance, electrocardiogram (ECG) signals in healthcare monitoring, and network traffic, activity logs, and social media data. In this work, we focus on the study of the time series classification (TSC) problem, one of the most important tasks in time series analysis (Bagnall et al., 2017; Chen et al., 2024; Ruiz et al., 2021; Wang et al., 2024c), which enables the categorization of these data sequences into distinct behaviors or outcomes.
Analyzing time series data is crucial to wide spectrum of applications, including economics, online marketplaces, and human healthcare. In particular, time series classification plays an in- dispensable role segmenting different phases in stock markets, predicting customer behavior, and classifying worker actions and engagement levels. These aspects contribute significantly to the advancement of automated decision-ne makinn and system optimization in real-world applica- tions. However, there is a large consensus that time series data often suffers from domain shifts between training and test sets, which dramatically degrades the classification performance. Despite the success of (reversible) instance normalization in handling the domain shifts for time series re- gression tasks, its performance in classification is unsatisfactory. In this paper, we propose FIC- TSC, a training framework for time series classi- fication that leverages Fisher information as the constraint. We theoretically and empirically show this is an efficient and effective solution to guide the model converge toward flatter minima, which enhances its generalizability to distribution shifts. We rigorously evaluate our method on 30 UEA multivariate and 85 UCR univariate datasets. Our empirical results demonstrate the superiority of the proposed method over 14 recent state-of-the- art methods.
There is broad consensus that time series data often ex- periences training/testing domain shifts, where the testing time series distribution differs significantly from that of the training data, resulting in a significant drop in perfor- mance (Kim et al., 2022b). This domain shift issue is largely attributed to several factors, including sensor variability, en- vironmental changes, differences in measurement or pre- processing methods, and data collection at different times. A common approaches to mitigate domain shifts involve designing normalization to reduce domain -omain-specific biases, enhance feature invariance, mitigate covariate shifts, and promote more generalizable representation learning (loffe & Szegedy, 2015; Ulyanov et al., 2016; Wu He, 2018 8; Li et al., 2018). One representative normalization method is Reversible Instance Normalization (RevIN, Kim al., 2022b; Liu et al., 2023), which has been widely used in time series forecasting (regression) tasks. However, ad- justing normalization may pose significant challenges to model training, e.g., gradient instability, increased sensi- tivity to hyperparameters, difficulty in optimizing the loss landscape, and potential overfitting to the training domain. Consequently, this leads to mismatch between training and inference loss (Dinh et al., 2017b; Zhou et al., 2021). More importantly, its effectiveness in time series classification remains unexplored in previous literature.
'Clemson University, USA. AArizona State University, USA. Washinntonn University in St. Louis, USA. "University of Arizona, USA 'University of Massachusetts Boston, USA Correspon- dence Xiwen Chen wewreneeg.clemoon edu>, Abolfazl Razi <arazi mazzi@ceemson.cdu...
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
To fill this gap, our study starts the investigation of domain shifts in time series classification task across several datasets. We confirm that the domain shift issue is also common in time series classification tasks. We further conduct an addi- tional analysis of RevIN IN in time series classification tasks, revealing that it is surprisingly ineffective this task. This motivates us to develop a method that is more tailored to domain shift issues in time series classification tasks. To this end, we propose novel learning method based on Fisher information, which measures the amount of information an observable random variable carries about an unknown parameter (Ly et al., 2017). In this context, we can utilize it as a measure of a neural network's 's sensitivity to small changes in input data. However, directly applying Fisher information as a regularization term in gradient-based op- timization suffers from a high computational and memory cost: (i) the Fisher Information Matrix (FIM), matrix that contains the Fisher information for all parameters, is pro- hibitively large due to its quadratic complexity w.r.t the number of parameters; (ii) it needs to propaproe twice (the first pass for computing the FIM and the second for updating the neural network), which introduces relatively substantial computation. To this end, we utilize diagonal approximation and propose gradient re- re-normalization based on FIM to tackle these challenges.
Testing Function
Bad Generalization
Lasss
Training
Good
Generaizztion
Flot Miniiuu
Figure I. A conceptual visualization of Flat and Sharp Minima. The Y-axis indicates loss, and the X-axis represents the variables (neural network parameters). Under train (blue) and test (purple) data domains, due potential distribution shift, the landscapes differ, i.c., with the same network parameter, the loss is often different. flat minimum can potentially lead to a low test error, while sharp minimum potentially leads to high test error.
ford, 1994; Seto et al., 2015), while further, the advent of deep learning has transformed TSC by enabling automated feature extraction and improving performance significantly. They employ or develop based on different architectures, such as CNN/LSTM hybrid (Karim et al., 2019; Zhang et al., 2020), purely CNN (Ismail Fawaz et al., 2020; Li al.,
We find our method is surprisingly effective and has a nice theoretical interpretation. First, it is easy to be integrated into the existing learning framework with automatic differ-
2021; Wu et al., 2023a), and Transformer (Zerveas et al.,
2021; Nie et al., 2023; Foumani et al., 2024; Eldele et al.,
entiation without an additional back-propagation It only requires a (n) memory complexity w.r.t. number of neu- ral network parameters. Second, our method can guide the neural network to converge to flat local minimum, poten- tially resulting in better generalizability in dealing with the issue of domain shifts (see e.g.. Keskar et al., 2016; Zhang & Xu, 2024). Finally, we show that despite the constraints, the theoretical convergence rate remains on par with that of standard neural network training.
2024). Instead of supervised learning, there are also recent methods that benefit from self-supervised pre-training, such as (Lin et al., 2023; Li et al., 2024). However, one significant issue is the non-convex nature of the optimization problem inherent in training deep neural networks. Due the com- plex structure of these models, the loss landscape is often filled with numerous local minima. This non-convexity poses challenges in finding global minimum or even suf- ficiently good local minimum. Hence, our proposed method can be one of the elegant solutions to mitigate this issue and guide the neural network to converge to a flat minimum with desirable generalizability (for details, see Sec. 4.2).
In summary, our contributions are two-fold: (i) We analyze the domain shift problem in time series classification and highlight the ineffectiveness of previous methods in address- ing this issue in classification tasks. (ii) To resolve this, we propose FIC-TSC, a time series classification model trained with a novel Fisher Information Constraint. We rigorously evaluate our approach on 30 UCR multivariate time series classification datasets (Bagnall al., 2018) and 85 UEA univariate time series classification datasets (Chen et al., 2015), demonstrating the superiority of our method over state-of-the-arn methods.
Sharpness and Model Generalizability. In the context of deep learning optimization, sharpness refers to the curva- ture of the loss landscape. Several works have illustrated that sharper minima often lead to poor generalization per- formance (see e.g., Keskar et al., 2016; Neyshabur et al., 2017: Zhang & Xu, 2024). This is because models that converge to sharp minima may overfit the training data, leading to a larger generalization gap. In contrast, flat minimum is less sensitive to the small perturbation of pa- rameters, and hence, is more robust to domain shifts (see Fig. 1 for illustration). Accordingly, some works (Foret et al., 2020; Andriushchenko & Flammarion, 2022; Kim et al., 2022a; Yun & Yang, 2024) have been proposed to ac-
2. Related Work
Time Series Classification. Traditional methods for TSC focus on similarity measurement techniques (Berndt& & Clif-
2

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
count for the sharpness of minima during training explicitly. They achieve this by modifying the traditional gradient- based optimization process: at each iteration, they compute the loss gradient w.r.t. parameters perturbed with small noise. This perturbation encourages the optimizer to move toward regions of the loss landscape where the loss remains low over a larger neighborhood (i.e., flatter minima). Re- cent work (Ilbert et al., 2024) designed for forecasting also falls into this category. However, the obvious issue is that they need to back-propagate twice at each training iteration, which introduces considerable computational and memory overheads. Consequently, their methods may have not been widely applied. In contrast, our method only requires a single propppppaaatiionn at each iteration, just like the stan- dard model training. More importantly, we demonstrate that despite the inclusion of the constraint, the convergence rate of our method is on par with standard training processes.
testing sets on several datasets. For convenience, following (Kim et al., 2022b), in each dataset, we illustrate the his- togram of the first channels of all samples from both training and test sets, offering a statistical perspective to interpret the distribution. As shown in Fig. 2, the observation is aligned with (Kim et al., 2022b), and it is evident that the distribution discrepancy is a common phenomenon across different datasets from two perspectives: (i) distribution between the entire training and testing sets and (ii) distribu- tions of the same class from the training set and test set. We further employ the Wasserstein-1 I distance to evaluate the distribution distance, which can be mathematically defined as below,
Definition 1. ((Peyré al., 2019)) The Wasserstein-1 distance between two probability distributions P and Q with cumulative distribution functions (CDFs) Fr(x) and (2( is defined as:
Fisher Information in Time Series Analysis. We notice some conventional approaches (i.e., not based on deep learn- ing) have utilized Fisher information as a tool for time se- ries analysis. For example, authors in (Dobos & Abonyi, 2013) use the Fisher Information to identify changes in the statistical properties of time-series data, facilitating the segmentation of the data into homogeneous intervals. Like- wise, authors in (Telesca & Lovallo, 2017; Wang & Shang, 2018; Contreras-Reyes & Kharazmi, 2023) employ Fisher information and Shannon Entropy to measure the temporal properties of time series in dynamic systems. In this con- text, the parameters often represent statistical properties of the series, like mean, variance, or even parameters defining windowed segments of the series for localization analysis. Therefore, our methods are essentially different from theirs, where we propose the Fisher information based constraint to guide the optimization of the neural networks.
W2(P,Q)= = inf B(x.)) (((z, v)l 7EI(P.QQ
(1)
where T(P,Q) is the set of all couplings (joint distribu- tions) 7(x V) with marginals P and Q and d(x, = Ix - yl is a absolute value distance function between points x and y.
Here, we used the discrete case Wasserstein-1 distance to calculate the distance between 1D) discrete distribu- tions (Ramdas et al., 2017). The dissimilarity matrices of these datasets are also shown in Fig. 2, where each element denotes class distribution from different sets. For example, the upper left element represents the distance between class from the training set and class from the test set. It is worth mentioning that we apply min-max normalization here for better visualization. We observe that the within- class distance (i.e., the same class distributions from differ- ent sets) can even be equal or greater than between-class distance (i.e., different classes from the same set), such as FaceDetect. ion and Sel fReguRlllll innCCP... These results reveal some potential reasons why the general clas- sification methods have poor performance time series data, and they motivate us to develop method that can be somewhat robust for the distribution discrepancy.
Domain adaptation. It addresses distribution shifts by enabling models trained on a source domain to general- ize related target domain. Common strategies include aligning feature distributions by minimizing statistical dis- tances (Chen et al., 2020) or using adversarial training to make features indistinguishable across source and target domains (Purushotham et al., 2017; Jin et al., 2022). In con- trast, our method is orthogonal to these approaches. does not require access to target domain data or labels during training, making it suitable when the target distribution is unknown or unavailable. Importantly, domain adaptation could be applied as post-training or downstream enhance- ment once target domain data becomes available. Hence, our method and these techniques can be complementary.
Effectiveness of RevIN. The invention of Reversible In- stance Normalization (Kim et al., 2022b) is designed to solve the train/test domain shift and has substantially pro- moted the performance of machine learning methods for souuence-to-sqquene regression tasks, particularly fore- casting tasks. This method applies instance normalization at the beginning to remove the non-stationary information (i.e., subtracting the mean and dividing by the standard de- viation. This leads the sequence to be a zero-mest and standard deviation of one) and perform denormalization at
3. Preliminary Analysis
Distribution Discrepancy. First, validate our conjec- ture of the distribution discrepancy between the training and
3

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
ArrowHead
FaceDetection
(class
(cless 1)
(class
0.096
0.92
(class 2)
1
Test (cless
1)
Thaining Set clos
Test (cless 1)
class Trinnin Set dless
Density
0.23
0.32
class
0.37
0.37
class
cless
Test Set
Test Set
-10
-3
10
PEMS-SP SF
SellRegulationSCPI
(class
Train
Triin (class 21
0.28
0.77
0.12
Train (cless 2)
0.25
0.20
(cless
Class Thaining Sox cluso
0.10
Icless
Density
Test (class 2)
Density
(class
0.1
0.00
0.52
0:3
02
61
Set
0.00
-50
Figure 2. Histograms representing sequences from the two selected classes of exemplary datasets with train/test distribution shift. The dissimilarity matrix illustrates the min-mat max normalized wasserstein-1 distance between class distribution from different sets. A lower value implies two distributions are more similar. is observed that distribution shifts exist between the entire training and testing sets and within the same classes across these sets.
the end to restore that information. In the classification task, only normalization at the beginning is appropriate since the output logits rather than a sequence. Hence, we just use the term in the following text. However, this technique has not illustrated any benefits in classification. This is reasonable since although IN reduces the distribu- tion shifts between the entire training and testing sets, it can also reduce the distances between different class dis- tributions within the same set. We show this fact in Fig. 3. We further empirically evaluate this technique on ten datasets: EC: EthamolComcentration) FD: FaceDetection, HW: Handwriting, HB: Hcartbeat, JV: Japanese JapaneseVVwells
Inin ICPSS
Trams t0
Boan toiss
Demsiiy
Denatty
IN-nomalized Figure 3. An illustration of the negative effect of Instance Nor- malization (IN). Left: The original input, and Right: Input after applying IN. It observed that IN reduce the difference of two class distributions. This may be disadvantagcous for classifi- cation.
Onginal Input
SCP1: SelfRegulationSSCP1, SCP2: SelfRegulationSCCP2
1.0
Baseline
SAD: SpokenArabicDiggin UW: UWaaeGessturlLiiree
0.8
+IN
and PS: PEMS-SF. According to the results shown in Fig. 4, IN does not positively affect most datasets' performance. In some datasets, IN even suppresses the performance. Ap- plying IN finally lowers the average accuracy, aligning with our conjecture. Therefore, IN is not desirable solution to solve the distribution discrepancy issue for the classification task. Thus, we seek to propose this issue, and we will discuss it in the next section.
Accuracy
0.6
0.4
0.2
0.0
6C F0
HW
H6
N SCP1 S0P2 SAD P5 Ave.
Figure 4. Comparison of classification accuracy between base- line and the model applying IN. Avg. indicates Average accuracy across all datasets. It is observed that IN does not have any positive effect on model for most datasets. statistical test is conducted in Appendix E.I.
method for addressing
4. Method
4.1. Problem Formulation
Time series data can be presented as ix..... ".,,,,,,,,,, where xx!" denotes time sequences containing T time points, with each time point a. E R being vector with a dimension of d. The goal of the task is to learn machine-learning model that directly maps feature space X' to target space y. The d implies this sequence is
univariate, while 1 is variate Our work can be generalized for both cases.
X: = {
4.2. Learning with the FIM constraint
As discussed Section 3, while normalization can miti- gate train-test distribution shifts, it may also reduce inter-

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
class distances, potentially hindering training effectiveness. Hence, we believe this is not feasible solution for the clas- sification task. In contrast, we expect to train more robust network to handle the distribution difference among training and test sets from a gradient optimization perspective. To this end, we propose an FoC-constrained strategy to guide the optimization of the network.
Additionally, simply applying FIM as regularization suf- fers double backward passes, where the first backward is used to compute the FIM and the second backward is used to impose regularization for updating the neural network. This again will introduce extra computation. Therefore, to avoid the computational cost of performing double back- ward passes, we constrain the optimization by introducing normalization strategy. Let be the pre-defined upper- bound and ||Fl]2 denote the entrywise I-norm' of the FIM. If ||FII 2 e, the gradient of each parameter 0, is normalized as follows:
FIM is used measure the amount of informa- tion/sensitivity about unknown parameters O carried by the data D, and lower Fisher information often represents the parameters that are less sensitive to small change of the data, which potentially improves its robustness. We first give its definition here:
Vel(e)
VIFIII Ve((e).
(5)
Definition 2. ((Kay, 1993)) Given a model parameter- ized by and an observable random variable DD the Fisher Information Matrix (FIM) is defined as,
It is worth mentioning that under the diagonal approxima- tion, [|F(()]lz = [|diag(F(()))l where diag(F(e))
is a vector.
F(0)
(2)
Considering these two aspects mentioned above, we sum- marize the proposed FlC-constrained optimization as,
=Be()))) (Ve log (V) lo [
min L(D;)) s.l. [[(O)]] Se. 9
(6)
where log (D () denotes the log- aigllikelihodd
In the context of the classification task, this log- likelihood can be interpreted as the negative cross-entropy (maximizing the log-likelihood is empirically equivalent to minimizing cross-entropy loss). To this end, we present the commonly used cross-entropy loss here.
Remark 1.
The complete algorithmic summary is provided in Algo- rithm 1, located in Appendix B.
Theoretical Justification. theoretical support of our method and illustrate our method has the potential to lead to better convergence with theoretically guaranteed rate. We first note that there is a strong relationship between FIM and the second derivative of the loss,
Now, we will delve into the
1 Σ log P(D, (),
C((D.)) ()
(3)
where D; denotes a training pair, n denotes the number of samples, log P(D: @) denotes the logarithm of the predicted probability of the true label.
Lemma 1. At a local minimum, the expected Hes- sian matrix of the negative log-like log-likeldood is asymp- totically equivalent to the Fisher Information Matrix, W.E. O which is presented as,
Remark 2. Directly computing FIM suffers from large com-- putation and memory cost (n²), where n denotes the num- ber of parameters.
Ep(D(e) [V (-loogp(D 0))] =
(7)
Remark 2 naturally motivates the simplification of the com- putation to facilitate the broader adoption of our method, particularly for users with limited computational resources. Hence, we apply the diagonal approximation, which ignores the off diigona elements and only interest diag in the FIM. Here, d-ag(() denotes the diagonal elements of matrix. This drastically reduces the computational burden and memory needs to (n). More importantly, it still has reasonable information, and we will employ this approxima- tion in our theoretical justification later. Empirically, given a set of data pairs {D,}² y 1. the FIM is estimated as,
Remark 3. See Appendix A for the proof. As previously dis- cussed, minimizing the negative log-likelihood is equivalent to minimizing the loss.
Accordingly, we realize that FIM can be bridged to the principle of sharpness. Sharpness measures the curvature around a local minimum in a neural network's loss land- scape. It is formally defined as follows,
Definition 3. (Keskar et al., 2016) For a non-negative valued loss function Le,, given B2(a, Q). a Euclidean ball with radius a centered at Q we can define the
(ia)(F(()) = Ve((e) o VeL(e),
(4)
where o denotes the clement-wise product and again the off-diagonal elements of F(O) is zero.
this norm will be utilized in the subsequent operations involv- ing matrices
5

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
a-sharpness as,
traffic management, human activity recognition, sensor data interpretation, healthcare, and complex system monitoring. A summary of the datasets is shown in Table 1. Please refer to Appendix C for more details about the datasets. The data pre-preesssng follows (Wu et al., 2023a; Foumani et al., 2024).
a-sharpness maxe (8) Ler - Le
(8)
1 Le
We observe this can be efficiently estimated at a local mini-
mum,
Corollary At a local minimum, the upper bound of shsharpnsss is able to be approximated via Taylor expansion as
Table 1. A summary of UEA and UCR datasets.
Dataset
Statistic of Variates
Length
Training Size
Text Size
of classs
UEA 30
max
minn
1345
2
8 17984
12 30000
20000
15
a²2 1111 2(1 + (()))
UCR 85
min 1
1
24 2709
16 8926
8236
20
2
a-sharpness
x
(9)
Baselines, We compare our method with multiple al- ternative methods, including recent advanced representa- tion learning- approaches: TsLaNet (Eldele et al., 2024), GPT4TS (Zhou et al., 2023), TimesNet (Wu et al., 2023a). PatchTST (Nie al., 2023), Crossformer (Zhang & Yan, 2023), TS-TCC (Eldele et al., 2021), and includ- ing classification-seeiffe approaches: ROCKET (Dempster et al., 2020), Inception Time (ITime) (Ismail Fawaz et al., 2020), ConvTran (Foumani et al., 2024) (MTSC only), MIL- LET (Early et al., 2024), TodyNet (Liu et al., 2024) (MTSC only), HC2 (Middlehurst et al., 2021) (UTSC only), and Hydra-MR (Dempster et al., 2023) (UTSC only). Finally, we involve the model with a single linear layer for compari- son.
Proof. Please refer to Appendix A.
is worth reiterating that a lower sharpness often implies higher generalizability (Keskar al., 2016; Zhang& Xu, 2024). Hence, according to Corollary 1, we can conclude:
Proposition 1. (Achievability.) With the same train- ing loss, appropriately constraining the Fisher in- formation can potentially converge to flatter minima and result in better generalizability.
Now, we are interested in the convergence of the proposed method. We offer the analysis under some mild and common assumptions.
Implementation. As a proof-of-concept, we implement our algorithm on network based on ITime (Ismail Fawaz et al., 2020), while we modify the original one linear classier to a two-layer MLP. We apply two hyperparameter strategies for our implementation: (i) Uni.: For the sake of the robustness and rigor of our method, we apply universal hyperparame- ters for all datasets in this model. We fixed the to 2 and mini-batch size to 64 for UEA datasets and 16 for UCR datasets, and (ii) Full: To fully explore the ability of our method, we perform a grid search for hyperparameters for each dataset. Specifically, we search the mini-batch size from {16, 32, 64, 128) and e from {2, 4, 10. 20). We place their performance aside from the main results. Please refer to Appendix D for more details about the implementation.
Theorem 4. Consider a L-Lipschitz objective func- tion L(0). defined such that for any two points O and 22 the inequality [[))),))
[( S LI[O1 02ll holds. By appropriately choosing the learning rate n and the FIC constraint e, the conver- gence rate of gradient descent can be expressed as O where T represents the number of iterations.
Proof. Please refer to Appendix A.
This suggests the empirical convergence time may be little slow due to the constraint, while asymptot- ically, the order of the theoretical convergence rate is not explicitly related to the constraint w.r.t T.
Remark 5.
Main Results. We report results on UEA multivariate datasets in Table 2 averaged over 5 runs. It is worth mention- ing that this result only includes 26 datasets since several methods (following TsLaNet (Eldele et al., 2024)) are not able to process the remaining datasets due memory or computational issues. However, our method does not have any obstacles learning on these datasets, and hence, we re- port the comprehensive results with additional metrics (i.e., balanced precision, Fl score, precision, recall) in Table 4 and Appendix F. It is known that adjusting hyperparameters is crucial for maintaining optimal performance across di- verse datasets, and we note that this strategy employed by
5. Experiment
Setup. To thoroughly evaluate our method, we conducted experiments on both MTSC and UTSC tasks. Specifically, we utilized the UEA multivariate datasets (30 datasets) (Bagnall et al., 2018) and the UCR univariate datasets (85 datasets) (Chen et al., 2015), which are among the most com- prehensive collections in the field. These datasets involve a wide range of applications, including but not limited to
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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Table Comparison of classification accuracy by our method with recent alternative methods for multivariate time series classification (26 UEA datasets). We highlight the best results in bold and the second- best results with underlining. Uni. indicates we set the consistent hyperparameters for all datasets, while Full indicates we perform dataset wise grid search for hyperparameters. To ensure the robustness
and rigor of our approach, we included Full at the end but did not participate in the comparison. The accuracy is averaged over 5 runs.
TodyNet Inf: Sci:24
ConvTran ECME/23
ROCKET BCML/20
GPT4TS NeurIPS
TimesNet ICLR'23
Crossoormer ICLR'23
Dutaset
(Uni.)
ICLRR124
DMKD 20
ICME/24
ICLR'23
LJCAP'2
(Full)
Anteulary Wortleogiionnno
99.3
99.0
98.3
99.3
98.5
99.0
91.3
95.2
98.0
97.7
98.0
93.1
99.8
AmalFFibillztton
56.7
43.3
66.7
40.0
20.0
44.0
40.0
33.3
33.3
46.7
51.3
33.3
46.7
60.0
RovicMetions
108.0
100.0
100.0
100.0
100.0
100.0
100.0
92.5
100.0
90.0
92.5
100.0
85.0
100.0
Cricket
108.0
100.0
100L0
Epilepay
98.6
98.6
76.1
60.3
106.0
Coneentiiinnnn
39.2
32.3
42.6
36.1
42.6
26.3
304
25,5
27.7
25,0
28,9
32.3
33.5
40.1
FaceDoitation
68.4
69.2
67.7
67.2
64.7
66.5
66.
65.6
67.5
66.2
60.0
63.1
67.
60.2
Finga Movemeats
65.0
64.0
62.5
56.0
61.
60.3
61.0
57.0
30
64.3
62.0
44.0
64.0
71.5
40.5
50,0
58.1
54.7
Handwating
48.5
22.5
Heartheat
81.0
76.8
79.0
78.5
69.8
13.8
77.6
36.6
74.5
76.5
76.6
72.1
73.2
R2.0
Winghet
71.1
72.0
63.0
21.3
41.8
70.9
10.0
10.0
10.0
10.0
10.0
10.0
10.0
72.3
Japanese Vawels
99.1
99.6
97.6
96.9
95.7
97.5
99.2
98.1
97.8
98.9
98.7
97.3
97.8
99.5
Libra
79.4
84.6
92.8
$3,9
68.1
92.8
77.8
73.3
91.7
Moneelmagery
64.0
56.0
53.0
60.2
62.0
51.0
61.0
61.0
61.0
68.5
NATOPS
98.9
98.3
95.6
94.4
83.3
96.8
95.6
91.7
81.8
58.3
96.7
96.1
93.9
99.2
PEMS-SP
79.2
63.0
96.1
82.8
75.1
60.4
83.8
N7.3
15.1
$2.1
K5.4
86.7
82.1
$7.9
PenDight
98.7
97.3
98.9
98.2
92.9
Spectra
2.1
RacketSpocts
84.5
86.2
96.2
90.8
H2.6
79.0
96.8
SellRegulalosSCPI
9011
38.1
90.4
91.8
34.5
$8.1
91.8
91.5
77.4
92.5
89.8
91.1
38.4
90.3
SeltRegulatinSC22
59.4
59.4
99.6
38.3
54,1
56.4
61.7
51.7
52.8
53.3
54,5
53.9
51.7
50.8
SpelenAmbiciiiis
99.9
99.5
99.2
99.7
99.9
99.4
95.4
96.4
99.8
96.7
100,0
Wall/up
UWWae ihouyy
98.8
86.3
89.1
94.4
$4.8
21.3
84.4
83.1
$1.6
81.0
99.0
Avg
74.8
74.
730
70.0
72.7
66.6
65.6
79.5
Table Comparison of classification accuracy by our method with recent alternative methods univariate time series classification (85 UCR datasets). We highlight the best results in bold and the second-best results with underlining. Please see the full results in Appendix
Method
Ours (Uni.)
MILLET
HC2
Hydra-MR
ITime
ROCKET
TsLaNet
GPT4TS
TimesNet
CrossFormer
PatchTST
TS-TCC
Linear
Ours (Full)
AVg.
86.2
85.6
86.0
85.7
85.6
83.1
83.2
61.6
65.3
73.4
71.8
75.1
69.7
87.3
Table 4. Classification results obtained using our method, evalu- ated across a comprehensive of metrics. Acc.: Accuracy, Bal. Acc.: Balanced Accuracy, Fl: Fl score, P: Precision, R: Recall.
Comparison
p- Value
A similar conclusion can be reached on UCR uni- variate datasets as the re- sults presented in Table
Uni. vs MILLET
0.039
Uni vs TodyNet
0,020
Dataset
Model
Acc.
Bal. Ace.
F1
P
R
Uni.
78.0
76.7
76.3
78.0
76.7
UEA 30
Full
80.9
79.7
79.3
80.7
79.7
3, where we achieve 0.2% gain over the previous SOTA method (HC2), Our full model can obtain I 1 further en- hancement over our method with universal hyperparameters. This is reasonably significant as we evaluate models across 85 datasets as optimal hyperparameters in different datasets can be various. We also realize that HC2 is an ensemble method of multiple different classifiers with expensive com- putational costs. Therefore, performance but also offers more efficient solution. In conclusion, all these results highlight the effectiveness of our method.
Uni.
86.2
83.4
83.3
85.2
83.4
UCR 85
Full
87.3
84.8
84.8
86.4
84.8
several models, such as TodyNet and TimesNet. However, the main observation is that even with universal hyperparam- eters, method can achieve the best overall performance and obtain a gain over previous SOTA methods (TodyNet and MILLET) with an average of 2.1% % accuracy improve- ment.
method not only improves
We also present the single-tailed Wilcoxon signed rank test to compare our methods against the most compet- itive candidates (MILLET and TodyNet) in the fol- lowing table, which confirms the statistical superiority. More surprisingly, our full model can achieve an aver- age accuracy of over 79.6%, with an additional 2.7% boosting of our model with universal hyperparameters.
6. Model Analysis
In this section, will delve into the analysis of model from several perspectives. We will use the ten datasets as the same as those used in Section since they are com- monly selected by previous works (Zerveas et al., 2021;

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Wu al., 2023a). We the model with the universal hyperparameters in the following investigations.
1.0
PatchTST
Ablation Analysis. We first investigate the effectiveness of our methods over baseline in Fig. 5. It is evident that train- ing with our proposed constraint can obtain accuracy gain in all datasets. Particularly, we observe that even though there are fluctuations on some datasets, setting 2 can obtain relatively considerable gain with 4% average improvement. We include statistical test in Appendix E.2.
Accuracy
0.6
0.4
EC
FD
HW
HB
IV
SCP1
SCF2
SAD
UW
FS
Avg.
Figure 7. Comparison of the classification accuracy between origi- nal PatchTST and it after applying proposed method.
0.05
Sharpness Reduction (x 100%)
0.04
0.03
0.6
0.02
0.4
0.01
0.2
EC
FD
HW
HB
0.25
FD
HW
IV
SCP1
SCP2
SAD
UW
PS
Ang.
Figure 8. The sharpness reduction by applying our methad.
0.20
0.15
landscape caused by our method. Specifically, compute the sharpness (Eq. 9) between our and the baseline models after well training. We reiterate here that lower sharpness often implies a better generalization. We show the results in Fig. 8, which confirms that using our method can obtain an average 40% reduction in the sharpness across all datasets. We illustrate this by visualizing the landscape in Fig. 9.
0.05
0.00
Aug.
Figure accuracy improvement our method over the base- line (i.e. standard training without using any constraint). We present average accuracy improvement the end.
44
1.0
TimesNet
62
FIC
Accuracy
-42
0.6
04
Figure 9. The landscape (error contour lines) of the well-trained network bascline (Left) and Ours (Right). The contour lines of our method are more spread out around the center compared to the baseline, indicating flatter region.
0.2
Aeg.
Figure Comparison of the classification accuracy between origi- nal TimesNet and after applying our proposed method.
Comparison with SAM. We also want to highlight the advantage of our method over previous sharpness-awe minimization (SAM) (Foret et al., 2020; Ilbert et al., 2024). For the sake the fair test, we fix the mini-batch size to 64 for both cases. We evaluate the accuracy and the runtime as shown in Table 5. This result demonstrates that our method improves accuracy by 2.6% and enhances efficiency, reducing the computation time by half.
Case Study on TimesNet and PStchTST. We conduct case study TimesNet (Nie al., 2023) and PatchTST (Wu et al., 2023a) to illustrate the proposed FIC can be generalized to other models. We implement it using the open source 2ode2 with default hyperparameters. The results shown Figs. and 7, where both models can obtain around 4% gain in accuracy. This confirms the effectiveness of our method.
Explicit Domain Shift Scenario. Here, we adopt a realistic evaluation scenario. In real-world healthcare applications, datasets are commonly partitioned by patient, such that in- dividuals included in the training set are distinct from those
Landscape Analysis. Here, we validate the change in the
httpps: /ttp:://iitii //te:////tttt..ttee cot.e...omtttttt com/thuml/ Ti Time- Seriee Library
8

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
approach is both theoretically sound and computationally efficient, with empirical results that strongly support the- oretical insights. Specifically, our method not only achieves superior performance on both univariate and multivariate time series datasets compared to several state-of-the-a methods, but also leads to a notable reduction in sharpness. These observations confirm that constraining Fisher infor- mation encourages flatter minima, ultimately enhancing generalization.
Table Comparison of classification accuracy and runtime per iteration by SAM and our method. We fixed both bati- sizes to 64 for a fair comparison. statistical test conducted in Appendix E.3.
Metries
Accuracy(1)
Runtime (s)()
Dataset
SAM
Ours
SAM
Ours
EC
36.1
39.2
0,103
0.046
FD
61.0
68.4
0.057
0.036
HW
58.6
61.6
0.468
0.182
HIB
79.0
81.0
0.050
0.032
JV 98.7
99.1
0.045
0.028
Acknowledgment
SCPI
88.1
90.1
0.061
0.032
SCP2
60.6
59.4
0.072
0.036
This material is based upon the work supported by the National Science Foundation under Grant Number CNS- 2204721. It is also supported by our collaborative project with MIT Lincoln Lab under Grant Numbers 2015887 and 7000612880 Additionally, this work was partially supported by National Institutes of Health, United States (R01EY032125 and R21 RG06559492 and the State of Ari- zona via the Arizona Alzheimer Consortium.
SAD
99.9
100.0
0.203
0.091
UW
89.7
90.2
0.048
0.028
PS
69.9
79.2
0.472
0.185
Avg.
74.2
76.8
0.158
0.070
Table Comparison classification accuracy on healthcare datasets with explicit domain shift. We include Medformer (Wang et al., 2024a), a recent SOTA method, for reference.
Impact Statement
Dataset
Method
Ace.
Bal. Acc.
R
Baseline
93.0
93.0
93.0
93.4
93.0
This paper proposes FIC-TSC, novel training framework for time series classification that leverages Fisher informa- tion as a constraint. Our approach effectively mitigates the domain shift problem and enhances model generalization by guiding optimization toward flatter minima.
TDBrain
+FIC
96.2
96.2
96.2
96.3
96,2
Medformer
89.6
89.6
89.7
89.6
Baseline
46.0
43.9
44.0
44.1
43.9
ADFTD
+FIC
52.8
49.9
48.3
52.3
49.9
Medformer
53.3
50.7
51.0
50.7
Baseline
73.9
59.2
60.0
65.4
59.2
(i) Theoretical View. We establish a Fisher Information
PTB-XL
+FIC
75.1
61.4
63.0
68.6
61.4
Constraint (FIC) to regulate training, theoretically linking to sharpness reduction and generalization under distribution shifts. Our framework maintains competitive convergence rates while improving robustness.
Medformer
72.9
62.0
64.1
60.6
Baseline
85.1
74.3
74.8
76.5
74.3
SleepEDF
+FIC
86.7
75.5
75.5
78.3
75.5
Medformer
82.8
71.1
71.4
74.7
(ii) Empirical Validation. Our empirical analysis strongly aligns with our theoretical findings, confirming that con- straining Fisher information effectively reduces sharpness and enhances generalization.
in the testing set. This setup introduces an explicit domain shift, as highlighted by Wang et al. (2024b), due to inter- patient differences in physiological characteristics, signal noise, and device-specific variations. We select four popular publicly available datasets: TDBrain (Van Dijk et al., 2022), ADFTD (Miltiadous et al., 2023b;a), PTB-XL (Wagner et al., 2020), and SleepEDF (Kemp al., 2000). As shown in Table 6, our method can consistently obtain improvement over the baseline, and can outperform Medformer (Wang et al., 2024a), a recent SOTA method, in most cases.
(iii) Applicability. FIC-TSC is designed to be broadly ap- plicable across diverse time series classification tasks. We validate its effectiveness on 30 UEA multivariate and 85 UCR univariate datasets, which span multiple domains, in- cluding healthcare, finance, human activity recognition, and industrial monitoring. By outperforming 14 state-of-the-an methods, FIC-TSC demonstrates superior robustness do- main shifts crucial capability for real-world deployment.
Discussion. We have included additional discussion on model analysis. please refer to Appendix G.
In summary, this work sets a foundation for integrating Fisher information constraints into time series learning, with broad implications for both theory and practical ap- plications.
7. Conclusion
In this work, we propose FIC-TS, a novel learning strategy that leverages Fisher information as a constraint to address train/test domain shifts in time series classification. Our

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
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13

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FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
A. Proofs
Lemma 1. At local minimum, the expected Hessian matrix of the negative log-likelihood is asymptotically equivalent to the Fisher Information Matrix, w.r. θ which is presented as,
Ex(D)))
V (-loog po [8)]] =
(10)
Proof. We first obtain,
E Bn(ee) (2 P( [))]
(11)
BBp (ole)
V2 (D)
0))
Now, we focus on the term p(D (),
Vloog((D| ())
(12)
= -V (Vlogp(( ()) = 4 Vp(D ())
p(D ())
V'r(D) (D )) - (p(D () (ee
P (D
P'p(D (D) 9 Vp(D ()Vp(D 16)
P(D (D) ())
p(D
O)p(D ())
V'p(D 9) Vp(D (
Vp(D (9) T
p(D ()) p(D
p(D ())
At a local minimum, Vp(D ()) 0. Therefore,
Epprle <Plog +)]
(13)
P'p(D +)
P +) p(D O)dz
(D(e) Voggp(D (ogg [)
v2 P(D O) F
F.
Substitute Eq. 13 to Eq. 11, the proof is completed.
Corollary 1. At local minimum, the upper bound of a-sharpness is able to be approximated via Taylor expansion
a² a|1
a-sharpness 2(1 + (()))
(14)
Proof. We first applying the Taylor expansion for L(O'),
C(e') = +(9)+ VC
(15)
+ 1 (0' -0)
2
When è is a local minimum, VL()) = 0, accordingly,
L(0') L()) = - -)
(16)
14

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
In this quadratic form, we that:
ma (L'()) (()))
<2]]V> 2(+ ((+)
(17)
1 1+ ((+)
where [00)l22 denotes the spectral norm of the matrix, which is equal its largest eigenvalue Amar- We also note, at local minimum, V2L(() is positive semidefinite, hence,
[[d]ag((=<(0)]]]> = Σ> 2 Amaz
(18)
therefore,
Ba (L())) - (()))
a c())l2 2(1 2(0))
(19)
+ (())
2(1 + L()))
Under the diagonal approximation of FIM, after applying Lemma above, the proof is completed.
Theorem 1. Consider a L-Lipschitz objective function [(O), defined such that for any two points O₁ and O2 the inequality [VL())) (l())) & L||0, - G holds. By appropriately choosing the learning rate n and the FIC constraint the convergence rate of gradient descent can be expressed as (++), where T represents the number of iterations.
Proof. We first reiterate the update rule at iteration via gradient descent,
+ L+1 0, -
(20)
where denotes the learning rate, and V ((e)) denotes the gradient constrained by FIC. The Liptinitzzconiinuityy assumption can bound the change the loss function:
((0++1) S VZ()) - θ.) L 11 0,112.
(21)
Substitute Eq. 20 to above equation, and after simplifying and rearranging,
Len2
-(0+1))-(eA)) -
(22)
Since VL(O,))TL(()) 2 0 always holds, when selecting a suitable and v, we can ensure the loss is sufficiently decreasing. Subsequently, when the model at Tth iteration from a initial point O1,
((OT)- - C(O,) < (VVC(()) + Len
(23)
Suppose a stationary point exists during these iterations, which is denoted as Or, The following inequalities hold,
C( _(e,) < ((er) L(O1),
(24)
and
|V[0')] min |L(0,) [E[T]
(25)
'At this stage, for simplicity, we assume is minimum.
15

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
where [T] (1, T. Substitute Eq. 24 to Eq. 23 and rearrange.
1
VL(O.) Vi(OY) -
LEn
+
(26)
T
nT
2
We also have the below inequality,
1 T
min V S DV
T VC(O,)
(27)
EE(T]
We can select suitable (i.e. not too small) such that
min E
[V
min VL(O.) (C(..))T((()))) tE[T]
(28)
Consequently, according to Eqs. 25, 26, 27, and 28, we can immediately obtain convergence rate of OL/TT), which demonstrates a sublinear rate of convergence.
B. Algorithm Detail
Please refer to Algorithm 1.
Algorithm FIC-TS (Training Phase)
Input: Initial Neural network O, Optimizer, number of iterations T, loss function L. dataset S
Output: Trained network +
1: for t = 1 to T do
2: Sample a mini-batch of data (Bx, By) E S
Forward pass By: (Bxx
4: Compute loss L(Br,
5: Backward pass VoL(By,By,
6: Compute ||FI
7: if [| 2 then
VL(O)) - / | VL(
9: end if
Update parameters Optimizer.
11: end for
C. Dataset Description
UEA 30 Datasets.The detail of all 30 datasets is provided in Table S7. It should be noteworthy that the datasets JapaneseVowell i and SpokenArabi cDi piiiite ts used in the Group 2 Experiment originally have varied lengths of sequences. We pre-process data following (Wu et al., 2023a), where we pad them to 29 and 93, respectively.
UCR dataset. We present the full list of UCR datasets in Fig. $10.
D. Implementation Detail
We use AdamW optimizer (Loshchilov & Hutter, 2017) with learning rate of 5e-3 and weight decay of lc-4. We implemented all experiments on cluster node with NVIDIA A100 (40 GB). We use Pytorch Library (Paszke et al., 2019) with version of 1.13. we implement our algorithm on a network based on (Ismail Fawaz et al., 2020). The architecture of the network can simply presented as:
X RdXT Jeear X m 128xT mean pool
X
IR 128
fue
Y C
(29)
16

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Table S7.
Dataset Summary
Dataset
Training
Test Size
Dimensions
Length
Classes
Articulary WordRecognition
275
300
144
25
AtrialFibrillation
15
2
640
BusicMotions
40
40
6
100
4
Character/rajectorii
1422
1436
3
182
20
Cricket
108
72
6
1197
12
DuckDuckGeese
40
1345
270
5
Eigen Worms
128
131
6
17984
Epilepsy
137
138
206
EihunolConcentration
263
3
1751
4
ERing
30
30
4
65
6
FaceDettection
5890
3524
144
62
2
FingerMovements
316
100
28
50
2
HandMovemenntDireee
320
147
10
400
4
Handwriting
3
152
26
Heartbeat
204
405
JapaneseVowels
270
370
12
29
Libras
180
180
2
45
15
LSST
2459
2466
6
36
14
InsgcrWingbeat
30000
20000
200
78
10
Motorlmagery
278
100
64
3000
2
NATOPS
180
180
24
S1
6
PenDigits
7494
3498
10
PEMS-SF
267
173
963
144
Phoneme
3315
3353
11
217
39
RacketSports
151
152
6
30
4
SellRegulationCCII
268
293
6
896
2
SellRegulationSCCCP
200
180
7
1152
2
SpokenArabicDigits
6599
2199
13
93 (max)
10
StandWalkJump
2500
UWaveGesuurLibbrry
120
320
3
315
8
Adiac, ArrowHead, Beef, BeetleFly, BirdChicken, Car, CBF, ChlorineConoentratiion CinCECGTorso, Coffee, Computers, Cricketx, Crickety, Cricketz, Diatomsi DistalPhallmxor11neesgeCCoo
DistalPaalaxx0011IeeGrree
DistalPhalanxIW,
Earthquakes, ECG200, ECG5000, ECGFiveDays, ElectricDevices, FaceAll, FaceFour, FacesUCR, FiftyWords, Fish, FordA, FordB, GunPoint, Ham, HandOutlines, Haptics, Herring, Inlineskate, InsettWingbattSonn, ItalyPowerDemand, Largekitchennkpllannee. Lightning2, Lightning7, Mallat. Meat, MedicalImages, anggle Middee MoteStrain, NontiwaasiveeertaaSSSttoooa NoniwasiveePttaaECChhooaz... (1teeaa 0live0il, oSULeaf, Phalaaggsoottlneeccrreett Phoneme, Plane, wwwp://naaPaaaanotttteeeggggooor ProximalPhallanxTM, ScreenType, Shwpe ShapesAl1, 3mal1kiitchhnnppl1aaccc,s SonyATRORobotSurracee
Middle
Proxdime  PhaaaaanoooLnneeerrrr
RefrigeraationDericcs,
$onyAlBoRobbotSurfacee. StarLightCurves, Strawberry, SwedishLeaf. Symbols.
SyntheticControl, ToeSegmentation1. ToeSegmentation2, Trace, TwoLeadECG,
TwoPatterns, UNavaGesturetiibraayylll uNaveGestureuibraryy UWwveessture
UfaveGeestureL.bbaayyyy Wafer, Wine, WordSynonyms, Norms, WormsTwoClass, Yoga.
Figure S10. Full list UCR datasets.
where freat denotes the backbone feature extractor, following the specifications detailed by (Ismail Fawaz et al., 2020). The MLP-based classifier, denoted as fNTPP, comprises two sequential layers: the first layer features 128 x 128 neurons with ReLu activation function, and the second layer, designed to output class probabilities, includes 128 x neurons, where C represents the number of classes.
17

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Table S8. A summary of the Wilcoxon signed-rank test Effectiveness of RevIN.
Comparison
p Value
w/ RevIN w/o RevIN
0.889
Table $9. A summary of the Wilcoxon signed-rank test Ablation Analysis.
Comparison
p- p-Value
Itime Itime+FIC
0.001
E. Additional Statistical Test
As suggested by (Demšar, 2006), we can conduct the Wilcoxon signed-rank test to compare the performance of two classifiers across different datasets.
E.1. Statistical Test on Effectiveness of RevIN
Please refer to Table S8, which indicates that there no significant difference classification performance between using RevIN and using RevIN. This suggests that RevIN not helpful for classification.
E.2. Statistical Test on Ablation Analysis
Please refer to Table $9, Due to the p-value being tiny and much smaller than 0.05, we have the confidence to conclude that our method is statistically superior to the baseline.
E.3. Statistical Test on SAM and FIC
Please refer Table $10.
F. Full Results
F.1. Multivariate Time Series Classification
Please refer to Table S11 below.
F22. Univariate Time Series Classification
Please refer Table $12 below.
Table S10. summary of the Wilcoxon signed-rank the comparison between SAM and FIC.
Comparison
p- Value
Accuracy
0.006
Runtime
0.001
18

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Table S11. The full results on 30 UEA datascts. We reported multiple motrics, including Accuracy, balanced Accuracy, F1, Precision (P), Recall (R).
Uni.
Full
Dataset
Accuracy
Bal. Accuracy
Fl marco
P marco
R marco
Accuracy
Bal. Accuracy
Fl marco
P marco
R marco
Articulary WordRecognition
0.993
0.993
0.993
0.994
0.993
0.998
0.998
0.998
0.998
0.998
AtrialFibellation
0.567
0.567
0.505
0.486
0.567
0.600
0.600
0.522
0.477
0.600
BasicMotions
1.000
1,000
1,000
1,000
1.000
1,000
1,000
1,000
CharacterTrnjoctoriee
0.997
0.997
0.997
0.998
0.998
Cricket
1,000
1.000
1,000
1.000
1,000
1.000
1.000
1,000
1.000
1,000
DuekDuckGeese
0.650
0.650
0,637
0.694
0.650
0.720
0.720
0,718
0.769
0,720
EigenWorms
0.855
0.797
0.804
0.850
0.797
0.924
0.896
0.893
0.892
0.896
ERing
0,919
0.919
0.919
0.924
0.919
0.954
0,954
0.954
0.955
0.954
Epilepsy
0.990
1.000
ElhansiConncemtration
0.392
0.392
0,386
0.402
0.392
0.401
0.401
0.395
0.418
0.401
FaceDeteccton
0.684
0.684
0.684
0.685
0.684
0.692
0.692
0.691
0.692
0.692
FingerMovements
0.650
0.647
0.639
0.666
0.647
0.715
0,716
0.714
0,718
0.716
HandMooemeenDireectinn
0.493
0.437
0,443
0.506
0.437
0.547
0,514
0,517
0.584
0,514
Handwriting
0.613
0.647
0.613
0.725
0.714
0.725
Heartbeat
0.810
0.715
0.735
0.780
0.715
0.820
0.746
0.760
0.784
0.746
InseceWingbeat
0,711
0.711
0.711
0.713
0.711
0.723
0.723
0,721
0.722
0,723
JapaneseVowet
0
0.989
0.990
0.990
0.989
0.996
0.996
0.996
0.995
0.996
Libras
0.794
0.794
0.790
0.811
0.794
0.917
0,917
0.916
0.923
0.917
0.443
Motorlmagery
0.650
0.650
0.649
0.651
0.650
0.685
0.685
0.683
0.690
0.685
NATOPS
0,989
0.989
0.989
0.989
0 989
0.992
0.992
0.992
0.992
0.992
PEMS-SF
0,792
0.797
0.788
0.796
0.797
0.879
0.878
0.875
0,883
0.878
PenDigits
0.976
0.976
0.976
0.978
0.976
0.985
0.985
0.985
0.986
0.985
PhonemeSpectra
0.323
0.314
0.323
RacketSports
0.898
0.906
0.904
0.905
0.906
0.908
0.915
0,914
0.915
0.915
SellRegulationSCPI
0
0.901
0.901
0.901
0.901
0.903
0.903
0.903
0.904
0,903
ScllRegulationSCP2
0.594
0.594
0.589
0.599
0.594
0.608
0.608
0.608
0.609
0.608
Stand WalkJump
0.633
0.620
0.640
0.633
0.667
0.631
0.715
0.667
SpokenArabicDiiits
1.000
1,000
1.000
1,000
1,000
UWWaeGesturrlLibmgg
0.902
0.902
0.900
0.906
0.902
0.930
0.930
0.929
0.933
0.930
Avg.
0.780
0.767
0.763
0.780
0.767
0.809
0.797
0.793
0.807
0.797
19

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
Table S12. The full results on 85 UCR datasets. We reported multiple metrics, including Accuracy, balanced Accuracy. F1, Precision (P). and Recall (R).
Detaset
Accurssy
Bul. Accuracy
Usi.
Fl murrcn
P asurco
R Rmsrco
Accarue
Bal. Accursc
Full
Fl marco
P Fimarco
R =marro
Adiac Acroiiiedd
0.778
0:760
0.760
0.292
0.780
0.766
0.791
0.774
0.312
0.791
0.909
0.900
0.906
0.906
0.508
0.911
0.505
0.907
0.913
0.905
Beat
0.808)
0.3201
0.708
01875
0.300
0.831
0.851
0.874
0.856
0.851
FuzilFFly
1
1.000
1.200
1.330
1.500
1.000
1.000
1.000
1200
1.000
BindChider
1.000
1.000
1.000
1,300
1.000
1.000
1,000
1.000
1.00
1.000
Car
0,95
0.11
0521
0.11
0.93
0.924
0.551
0.953
0.924
C3P
1.200
1.200
1.200
1,000
1.000
1.000
1.000
1.000
1.000
1,000
Chirried Cowortatio
0.444
0.806
0314
01875
0.800
0.547
0.614
0.821
0.84
0.81
CinC COCCBO
0.765
0.765
0.782
11.9%
0.785
0.511
0,511
0.813
0,319
0.511
Coffee
1.000
1.000
1.000
1,000
1.000
1,000
1,000
1,000
1.000
1,500
Componees
0.554
03344
03544
113334
0.854
0.590
0.500
0.500
0.80
0.590
CriceeX
0.541
0.343
0.40
0.850
0.843
0.554
0.557
0.853
0
0,557
Criceet
0.825
0.827
0836
0.8344
0.827
0.544
0.545
0.843
0.348
0,545
Cricktt
0.550
0,842
0.41
01.49
0.842
0.522
0.554
0.855
0
0354
Deaum Succeaaaiin
0.979
0.961
0.68
0.978
0.961
0.995
0.962
0.987
0.992
0.982
0.788
0.769
0.7%
0.789
0.506
0.779
0.505
0.850
0.77
0.12
0,700
0.798
01823
0.700
0.19
0.509
0.812
0,318
0
Eanbyrakes
DasulPtalasTTW
0.791
0.745
0,63
0,577
10034
1930
0,577
0655
0500
0741
0.864
0.88
0.596
0.84
0.767
0.558
0.564
0.60
Focelbuu
FaxsAll
EOG200
EDG3000
0.954
0.930
0.955
0.340
0.945
1.000
0.958
0.915
0,949
0.60
0,554
1.000
039
1060
19113
1050
1867
1666
1011
0915
0.554
1200
0000 0000
0184
0085
0900
1440
0.963
0.921
0,961
17009
0.64
0950
1805
0.711
0929
0616
0.722
0.945
0.55
0.744
10000
0.721
0,961
0.951
0.925
1,000
0,509
UUoeUUU
0.957
0.90
0350
0500
0.944
0.90
0.94
PetrWonk Words
0.401
0.67
0.67
3311
0,681
0.676
0.30
0.881
FondA
0.966
0.991
0000
0000
1066
0.90
0366
0597
0961
0.991 1999
1367
0.997
0.67
0.997
0.967
0.995
PordB
0
02000
0364
0.90
0.809
0,800
0,89
GusPuint
1,000
1000
1,000
1000
1,000
1,000
1,000
HandCuttnes
0.958
0.824
0884
1065
0.47
0.13
0990
0.43
0.844
0.968
0.843
0.345
0.971
0,544
Hapties
0,528
0.54
0,53
0.742
0660
0790
0.72
Inlineskue
0.356
0.584
0.303
0.395
0.627
0618
0.41
0.54
0.641
0.64
0.914
0074
0000
4000
0.9%
0.917
0017
0:17
0000
0,980
0.30
Lichning2
0.18
0000
0218
0,19
0904
0.006
0,926
0.936
0.925
Lighning?
0200
12800
0910
9818
0,921
0,915
0954
0.91
0000
0870
0.900
0990
0.99
0802
0920
0940
0993
0.984
0.90
0.944
Medicallinges
0.508
0760
0751
0818
0.7M2
0.00
0.66
0461
9674
0111
0.534
0770
0511
0574
0.773
0.875
0.77
0873
0.627
0.35
0110
0-120
0435
0633
04463
0.450
0,470
0463
MoreStais
0.925
0925
0824
02224
0925
0928
0.924
0,927
0.227
0.22
0.27
0.26
0823
0926
0996
0.25
0.934
0937
0923
Nonlin www.eeeeeeeeeeiinee
0.366
0.34
0932
0934
9964
0.90)
0,938
0.443
0,90
OliveOi
0.800
0.719
0672
0774
0719
0.33
0.764
0.744
0,58
07664
OStLead
0.936
0.922
0927
0222
0922
3949
0.900
0.999
0.55
0.900
MallagesOutiieeeeeeee
0.857
0846
0357
0330
3560
0.540
0.44
0,622
0550
Phoramm:
0.352
0.186
0.168
0228
0.186
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20

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
G. Discussion
G.I. Why Choose ITTmme, PatchTST, and Timesnet as baselines?
The selection is based on their impressive impact on the TS community.
G.2. Why choose 10 datasets in the Model Analysis Section?
These are widely chosen in different works, such as TimesNet (Wu et al., 2023a).
G.3. Why is Our Method Potentially Better than SAM in TSC?
As SAM is not central to our work, we defer its detailed theoretical analysis to future work while providing insights for this superiority. We conjecture the following two reasons as following:
Training sample size. Most SAM papers focus on image datasets (Foret et al., 2020), which typically have large training sizes (60k 14M). In contrast, TSC datasets often have very limited training sizes, e-g., the UW and SCP2 datasets have only 120 and 200 training samples, respectively. SAM's effectiveness in such cases remains unexplored.
Sensitive to peeee-rrraeters. As studied in (Andriushchenko & Flammarion, 2022) (e.g., Fig.16), SAM's performance is highly sensitive to its hyper-paramente (e.g., dataset-dependent batch size and perturbation radius). Poor choices can easily lead to worse performance than standard training. Given diverse TSC datasets, identifying universal hyper-parameters for SAM performing well on most datasets is challenging.
G.4. Is Diagonal Approximation Necessary?
Yes. DL models typically have huge parameters (e.g., ITime has 600k parameters on SCPI, so Transformer- based models can have even more parameters). Therefore, computing and storing the full FIM on a typical GPU is extremely inefficient or even not feasible. We have also tested on an 100 GPU, and the results support this claim. We also found related works (Kirkpatrick et al., 2017: Lee et al., 2017; Jhunjhunwala et al., 2024) that consistently apply diagonal approximation tackle similar computational issue, and they mention that the diagonal elements contain sufficiently important information.
It is worth mentioning that EWC (Lee et al., 2017) preserves prior knowledge by penalizing changes to important weights, using Gaussian posterior centered at previous weights with precision from the observed Fisher information (Laplace approximation). Notably, EWC uses a diagonal approximation, aligning with and supporting the efficiency goals of our work.
Another relevant work, K-FAC (Martens & Grosse, 2015), addresses the high computational cost of the FIM by approximating large blocks of it, corresponding entire layers, the Kronecker product of two much smaller matrices. We consider this promising direction for future work to achieve more accurate and efficient FIM approximations.
G.S. Is Analysis of Non-Minimum Points Needed?
No. Our theoretical analysis aims to deliver the achievability of a better convergence. Since the optimizers [by simply adjusting dyper-parametery, in general, can easily reach local minima after convergence in the TSC task, it is sufficient to evaluate sharpness at local minima and their neighbors. In Proposition 1, we only claim that an appropriate FIC could potentially lead to a convergence to flatter minima. Hence, non-minimum points do not affect our conclusion regarding achievability.
This focus on achievability is analogous to the approach commonly used in Coding Theory (Cover, 1999), where initial results often emphasize achievable rates demonstrate feasibility before refining practical implementation further. Similarly, work lays the groundwork for future exploration of theoretical optimality.
Moreover, this claim is strongly supported by empirical evidence, where method achieves 40% reduction in sharpness and ~ 4% gain in accuracy as presented in Figs. 5, 8, and 9, These results validate the practical implications of our theoretical analysis and demonstrate the effectiveness of our proposed approach.
21

---

FIC-TSC: Learning Time Series Classification with Fisher Information Constraint
G.6. Can FIC Compete Related Methods Related to Domain Shift Problem?
Yes. We compare two related methods that target solve domain-shift problem:
RevIN. Our method outperforms RevIN, which is the most common approach for addressing domain shift in time series. The main motivation of our work is that RevIN's effectiveness in time series classification remains unexplored. Accordingly, we conducted an empirical investigation that demonstrates RevIN's ineffectiveness (see Sec. 3).
SAM. Our method outperforms SAM in both accuracy and efficiency in TSC, as presented in Table
G.7. Sharpness and Generalization
While the link between sharpness and generalization is out of our focus, here we want to include more discussion about them. We fully acknowledge that the relationship between flat minima and generalization remains an open and nuanced research question. Rather than taking definitive stance in this ongoing debate, our work aims to contribute to this conversation by demonstrating that a regularization strategy informed by Fisher information and sharpness can lead to improved robustness and generalization in real-world time series tasks. Importantly, we have taken care to avoid overclaims in the paper, using qualified language such "potential" and "achievable" to reflect the limitations inherent in this area. While authors in (Dinh et al., 7a; Petzka et al., 2021) raise concerns about its limitation, these results are derived under specific assumptions (e.g., fully connected ReLU networks and carefully constructed reparameterizations) Their applicability to general architectures and practical training setups remains limited.
Moreover, recent empirical studies (Jiang al., 2019; Andriushchenko & Flammarion, 2022) suggest that in practical settings, where such reparameterizations are not applied, sharpness commonly measured) can still correlate meaningfully with generalization. These observations support the idea that sharpness-base -base metrics, while theoretically imperfect, can still provide practical value. In addition, discussed in our related work section, several recent papers (Neyshabur et al., 2017; Zhang & Xu, 2024; Foret et al., 2020; Andriushchenko & Flammarion, 2022; Kim et al., 2022a; Yun & Yang, 2024) supported the utility of sharpness-related methods and successfully leveraged them to improve learning outcomes.
Therefore, we believe our results add to this growing body of evidence, particularly in the underexplored domain time series data, and we remain cautious yet optimistic about the promise of these methods.
H. Graphic Summary See Fig. $11.
Sharpness (+b)
Accuracy (1)
Runtime (+)
Baseline
FIC-TSC
Baseline
FIC-TSC
Baseline
FIC-TSC
Baseline: min L(D;))
θ
FIC-TSC: min L(D s.t. || S E
Figure $11. Comparison the baseline method (standard training) and our proposed FIC-TSC approach. Training with FIC-TSC leads to convergence flatter minimum, potentially enhancing performance. The additional runtime incurred is insignificant and considered negligible.
22


Reviews:
[{"review_id": "hrMqGBUait", "reviewer": "Reviewer_kJBN", "summary": "This paper addresses the failure of RevIN in out-of-distribution (OOD) scenarios for time series classification and proposes a constraint method based on the Fisher Information Matrix to enable smoother model optimization, thereby improving the generalization ability of the classification model. Furthermore, considering the quadratic complexity of computing the Fisher Information Matrix and the need for two rounds of backpropagation, this paper introduces a diagonal approximation information matrix and a Fisher information constraint method to reduce the computational cost. The effectiveness of the proposed method is validated through performance evaluation experiments on the UEA 30 and UCR 85 datasets, case studies, sharpness evaluation, and landscape visualization.\n\n## update after rebuttal\n\nThe authors' responses addressed my concerns, and I chose to maintain my original score.", "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": "We sincerely thank the reviewer for the thoughtful and insightful comments, especially for recognizing our contributions and empirical validation. We addressed the main concerns as follows.\n\n---\n\n### **Q1. Data and Sharpness.** \n\nTime series datasets are often small in size (e.g., UW has 120 samples, SCP2 has 200 samples), which increases the risk of overfitting. In such low-data regimes, models often exhibit low training error (low bias) but high variance, making them prone to overconfidence and instability. This instability is reflected in the geometry of the loss landscape: models may converge to sharp minima, where small perturbations in input or parameters cause large changes in loss. Sharpness thus captures this sensitivity, and sharp minima are empirically linked to poor generalization.\n\nAdditionally, time series tasks frequently involve domain drift, a shift between the distribution of training and test data, due to factors like temporal changes or varying sensor conditions. Even when the dataset is moderately sized, this distributional shift can exacerbate the generalization gap. Encouraging the model to find flatter minima, i.e. correspond to solutions that are less sensitive to such shifts, can improve robustness and performance on unseen data.\n\n---\n\n### **Q2. Analysis of Epsilon.**  \n\nThe sensitivity analysis of $\\epsilon$ is presented in Section 6 and Fig. 5. While some datasets exhibit performance fluctuations, setting $\\epsilon = 2$ provides a generally reasonable balance. To be more comprehensive, we have extended the analysis to include all 26 datasets (consistent with Table 1).\n\n|$\\epsilon$|0.02|0.5|2|4|20|100|\n|-|-|-|-|-|-|-|\n|Acc.|69.7|75.2|76.9|76.2|75.5|72.6|\n\nWe observe that when $\\epsilon$ is set too high, the gain is marginal. This is due to, in this case, the fisher information of the network is very likely below  $\\epsilon$, and accordingly, we do activate the renormalization mechanism. Conversely, if $\\epsilon$ is too small, model updates become challenging because we always renormalize its gradient. This may lead to performance degradation.\n\nThe optimal choice of $\\epsilon$ depends on both the dataset and model architecture. While $\\epsilon = 2$ serves as a strong default, we also report the results obtained using the best-matched $\\epsilon$ for each individual dataset.\n\n|Uniform $\\epsilon$| Best-matched  $\\epsilon$|\n|-|-|\n|76.2|**77.5**|\n\n\nFor a new dataset, $\\epsilon$ should be treated as a hyperparameter. However, **its search space can be efficiently narrowed by considering the initial range suggested by the Fisher information estimated during the first few iterations.**\n\n---\n\n### **Q3. More Comparison.** \nWe compare our method with NuTime (29 datasets selected by NuTime) as follows. SAN is a variant of RevIN that may have similar issues to RevIN, while NuTime is a self-supervised method. Our method demonstrates better overall performance than both of them.\n**Full comparison is available at https://github.com/AnonymousUserss/ICML2025-4119-Response.**\n\n||Ours (UNI.)|NuTime (self-supervised)|ITIME+SAN|\n|-|:-:|:-:|:-:|\n|Avg.|**78.3**|77.8|74.4|\n\nWe will include it in our final version.\n\n---\n\n### **Q4. Rationality of the diagonal approximation.** \nWe argue that this is a necessary trade-off, i.e., sacrificing the precision of FIM to achieve a feasible computational cost. As mentioned in G.4, we have tested computing the full FIM with a mini-batch size of 64 in **one iteration** on InceptionTime ($\\sim$ 0.6M parameters) on an A100 40GB GPU, and it has been running for more than 15 minutes and is still ongoing. We realized that even in such a small-scale network, the computation required for one iteration is prohibitively heavy, and we would need thousands of iterations for full training; hence, applying full FIM is infeasible in broader scenarios with larger networks and/or larger-scale datasets.  \n- We also found related works that consistently apply diagonal approximation to tackle a similar computational issue, and they mention that the diagonal elements contain sufficient important information.\n\nWe will include these discussions in our final version, but indeed, this is a potential limitation of our work, and we plan to investigate it in the future.\n\n[1] Overcoming catastrophic forgetting in neural networks\n\n[2] Overcoming Catastrophic Forgetting by Incremental Moment Matching\n\n[3] Fedfisher: Leveraging fisher information for one-shot federated learning\n\n---\n\n### **Q5. How FIC works.**\n\nIn FIC-TSC, whenever the Fisher Information exceeds some threshold $\\epsilon$, the parameter gradients are downscaled accordingly. By “capping” how large Fisher information can grow, the algorithm actively steers the model away from sharp or overly sensitive parameter regions and potentially reduces the overall Fisher information.\n\n---\n\n### **Q6. Grid Search.** \n\nFollowing TsLaNet and ConvTran, we split 20\\% of data from the training set as a validation set.", "claims_and_evidence": "Yes, the claims are convincing.", "methods_and_evaluation_criteria": "Yes, the evaluation criteria and benchmark datasets are appropriate for the problem and application at hand.", "theoretical_claims": "Yes, I have checked.", "experimental_designs_or_analyses": "Yes, I have checked.", "relation_to_broader_scientific_literature": "No.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tThis paper clearly articulates the limitations of RevIN in out-of-distribution (OOD) scenarios for time series classification and innovatively leverages Fisher information constraints to mitigate OOD issues in time series classification.\n\n2.\tTo address computational cost concerns, the paper proposes a diagonal approximation of the Fisher Information Matrix and a Fisher information constraint method.\n\n3.\tThe experiments are comprehensive, including performance evaluation on UEA30 and UCR128 datasets, case studies, sharpness evaluation, and landscape visualization, which thoroughly validate the effectiveness of the proposed method.\n\nWeaknesses:\n\n1.\tAlthough the authors suggest that using Fisher information can smooth the model's optimization space, the paper lacks a thorough analysis of whether and why time series data itself or the model inherently causes sharpness phenomena in time series classification.\n\n2.\tThere is insufficient justification for the rationality of the diagonal approximation of the Fisher Information Matrix and the Fisher information constraint method.", "other_comments_or_suggestions": "No other comments or suggestions.", "questions_for_authors": "1.\tIt would be better to clarify why time series data lead to sharpness, improving the rationale and persuasiveness of the proposed method.\n\n2.\tConsidering that Epsilon is a crucial hyperparameter, please conduct a sensitivity analysis and discussion of Epsilon.\n\n3.\tCompare with other advanced time series normalization methods, such as SAN [R1], Numerically Multi-scaled Embedding in NuTime [R2].\n\n4.\tExplain the rationality of the diagonal approximation of the Fisher Information Matrix under the premise of significant parameter correlation or strong model non-linearity (which is common in deep learning).\n\n5.\tProvide a reasonable explanation of how the Fisher information constraint reduces the Fisher Information Matrix.\n\n6.\tThe authors state in the experimental section: \"To fully explore the ability of our method, we perform a grid search for hyperparameters for each dataset.\" However, for the UCR 85 archive and the UEA 30 archive, each time series dataset only contains a training set and a test set, without a validation set. In the absence of a validation set, how did the authors conduct grid search to select hyperparameters?\n\n[R1] Liu Z, Cheng M, Li Z, et al. Adaptive normalization for non-stationary time series forecasting: A temporal slice perspective. Advances in Neural Information Processing Systems, 2023, 36: 14273-14292.\n\n[R2] Lin, Chenguo, et al. \"NuTime: Numerically Multi-Scaled Embedding for Large-Scale Time-Series Pretraining.\" Transactions on Machine Learning Research, 2024.", "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": "Z4VwlD6OA9", "reviewer": "Reviewer_QtfR", "summary": "FIC-TSC introduces a novel training framework for time series classification by enforcing a Fisher information constraint to guide the optimizer toward flatter minima, aiming to improve robustness against domain shift. The method leverages two key approximations—a diagonalized Fisher information matrix and a gradient normalization strategy that requires only one backward pass—to achieve computational efficiency. Experiments on standard UEA and UCR datasets demonstrate that FIC-TSC outperforms several state-of-the-art methods in terms of classification accuracy and runtime, while also reducing the sharpness of the loss landscape.  \n\n## Update After Rebuttal\nI raised my score as a result of a discussion with the authors (see below).", "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": "Thanks for these insightful suggestions, and we are very glad to hear that you enjoyed the reading. \n\n---\n\n### **Unjustified motivation/Questionable assumption/Sharpness in data domain.** \n- Our primary motivation is that time series data often suffers from domain shift between train and test set, and we propose to constrain FIM during learning to alleviate the issue, which potentially results in a flatter minimal and improves the generalization. \n- Regarding the concern about citing prior work for the claim that \"a flat minimum is less sensitive ...\", we have referenced studies earlier in the sentence (Keskar et al., 2016; Neyshabur et al., 2017; Zhang & Xu, 2024). Notably, in Keskar et al. (2016), the authors explicitly define sharpness as a measure of sensitivity (see p.5, Section 2.2.2). We will cite them in a more proper place in the final version.\n- **Fig. 1** is a conceptual illustration of landscapes with domain shifts. **We do not have the assumption that the shapes of the landscapes on test and train data remain the same.**\n- **The landscape is related to both data domain $\\mathcal{D}$ and the weight domain $\\Theta$, i.e., calculating loss $\\mathcal{L}(\\mathcal{D};\\Theta)$ needs dataset and weights of a network**, so when training data and test data have domain shift, the landscapes with same weights can be different, i.e. $\\mathcal{L}(D_{train};\\Theta)$ and $\\mathcal{L}(\\mathcal{D}_{test};\\Theta)$. \n**In Fig. 1, we use different colors to denote different data domains $\\mathcal{D}$**. We improve it at https://github.com/AnonymousUserss/ICML2025-4119-Response.\n- **Fig. 9 is a visualization of the landscapes to better illustrate the concept and show the relative flatness reduction.** The **quantitative results** are presented in **Fig. 8** (not Fig. 9) and **Fig.5**, as the key evidence to support our claim: **Compared with baseline, using our method can obtain an average 40% reduction in the sharpness (see (i) and (ii)) across all datasets and finally translate to ~4% accuracy gain (iii).** Please refer to our response to **R#YXQa** for explicit domain-shift experiments.\n\n---\n### **Flatness and Generalization.**\nWe thank the reviewer for highlighting these important works. We fully acknowledge that the relationship between flat minima and generalization remains an open and nuanced research question. **Rather than taking a definitive stance in this ongoing debate, our work aims to contribute to this conversation by demonstrating that a regularization strategy informed by Fisher information and sharpness can lead to improved robustness and generalization in real-world time series tasks.** \n\n**Importantly, we have taken care to avoid overclaims in the paper, using qualified language such as \"potential\" and \"achievable\" to reflect the limitations inherent in this area.** \n\nWhile Dinh et al. (2017) and Petzka et al. (2021) raise concerns about its limitation, **these results are derived under specific assumptions** (e.g., fully connected ReLU networks and carefully constructed reparameterizations). Their applicability to general architectures and practical training setups remains limited.\n\nMoreover, recent empirical studies [1-2] suggest that in practical settings, where such reparameterizations are not applied, **sharpness (as commonly measured) can still correlate meaningfully with generalization**. These observations support the idea that sharpness-based metrics, while theoretically imperfect, can still provide **practical value**.\n\nIn addition, as discussed in our related work section, several recent papers (Zhang & Xu, 2024; Foret et al., 2020; Andriushchenko & Flammarion, 2022; Kim et al., 2022a; Yun & Yang, 2024; Ilbert et al., 2024) have \n- supported the utility of sharpness-related method; \n- successfully leveraged it to improve learning outcomes.\n\n**We believe our results add to this growing body of evidence, particularly in the underexplored domain of time series data, and we remain cautious yet optimistic about the promise of these methods.**\n\n[1] Fantastic generalization measures and where to find them\n\n[2] Towards Understanding Sharpness-Aware Minimization, ICML2022\n\n---\n\n### **Reproducibility.**\n**We have made the code and the trained weights at https://github.com/AnonymousUserss/ICML2025-4119-Response.**\n\n- **standard deviation (%)** on two benchmarks.\n||Acc.|Bal. Acc|F1|P|R|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|UEA 30|1.1|1.2|1.4|1.5|1.2|\n|UCR 85 |0.5|0.8|0.9|1.4|0.8|\n\n---\n\n### **Technical Simplifications.** \n- Please see our response to **Reviewer 3cHV Q4** regarding diagonal approximation.\n- Double-backward vs single-backward with FIC is presented as follows. **Our method is on par with double-backward in accuracy but substantially reduces the runtime. Full comparison is at the same repo**.\n\n|Metrics|Accuracy(%)| |Runtime(s)| |\n|:-:|:-:|:-:|:-:|:-:|\n|10 datasets|Double|Ours|Double|Ours|\n|**Avg.**|77.4|76.8|0.147|0.070|", "claims_and_evidence": "### Unjustified motivation.\nThe primary motivation of the paper is the assumption that \"a flat minimum is less sensitive to small perturbations of parameters, and hence, is more robust to domain shifts\" (p.2), as conceptually illustrated in Fig. 1. However, this assumption is not supported by rigorous theoretical or experimental evidence. The authors do not cite prior work that empirically or theoretically validates this assumption. Moreover, while Fig. 9 compares the loss landscapes of the baseline and the proposed method, both appear fairly flat, and there is no clear demonstration of how domain shifts would transform these landscapes.\n\n### Questionable assumption.\nThe assumption above tells that if domain shift occurs, the loss landscape plotted in 2-D plane, where x- and y- axes are neural network weight parameters and loss value respectively, \"slides\" without changing its shape (Fig. 1). However, it is unclear why distribution shift in *data domain* leads to the slide in *weight domain*. As I mentioned above, this nontrivial hypothesis is neither verified theoretically or experimentally.  \n\n### Sharpness in data domain is not considered.\nThe paper’s analysis and visualizations (e.g., Fig. 1) are confined to the loss landscape in the weight parameter space. It implicitly assumes that under a domain shift, the loss landscape \"slides\" without changing its shape—a nontrivial claim that connects a shift in the data distribution to a simple translation in the weight domain. This assumption is neither theoretically justified nor empirically verified. Moreover, since domain shifts occur in the data domain, for the hypothesis that flat minima yield better cross-domain generalization to hold, the authors need to demonstrate that:\n(i) existing approaches produce sharp minima when visualized in the data domain,\n(ii) FIC-TSC results in flat minima in the data domain, and\n(iii) FIC-TSC outperforms baselines in cross-domain experiments. Without such evidence, the connection between parameter-space flatness and robustness to domain shifts remains unproven.\n\n### Questionable link between flatness and generalization.\nThe relationship between flat minima and generalization is still a matter of debate. Works such as Dinh et al. (2017) (\"Sharp Minima Can Generalize For Deep Nets\") have shown that conventional flatness or sharpness measures are not invariant under reparameterizations—meaning that a flat minimum can be transformed into an arbitrarily sharp one without changing the network’s function. Similarly, Petzka et al. (2021) (\"Relative Flatness and Generalization\") argue that generalization is more closely tied to feature robustness than to absolute flatness in parameter space. The failure to discuss these critical findings raises concerns about the validity of the paper’s central premise.\n\n### Concern on reproducibility.　　\nThe experimental results are reported as scalar values without error bars or confidence intervals, making it challenging to evaluate the stability and robustness of the proposed approach. This lack of statistical rigor raises doubts about the significance of the observed performance gains. Furthermore, without a public release of the code, it is even more difficult for reviewers and future researchers to assess the reproducibility and practical limitations of the method.\n\n### Technical simplifications are not adequately evaluated.\nFIC-TSC introduces two technical simplifications for computational efficiency: (1) the diagonal approximation of the Fisher information matrix, and (2) enforcing the Fisher constraint using a single backward pass rather than a costly double back-propagation. While these tricks reduce computational load, the paper does not provide a rigorous ablation study comparing FIC-TSC with a variant that uses the full Fisher matrix and/or double back-propagation—even on simple artificial datasets or small networks. Without such controlled experiments, it is difficult to assess whether these approximations negatively impact the method’s robustness or generalization performance.", "methods_and_evaluation_criteria": "Please see the above Claims And Evidence section.", "theoretical_claims": "I have reviewed the high-level theoretical claims but there remains a possibility that oversights exist.", "experimental_designs_or_analyses": "Please see the above Claims And Evidence section.", "relation_to_broader_scientific_literature": "The paper builds on a long-standing debate in the literature regarding the relationship between loss landscape flatness and generalization. Early works (e.g., Hochreiter & Schmidhuber, 1997; Keskar et al., 2017) linked flat minima to improved generalization, but subsequent studies (e.g., Dinh et al., 2017; Petzka et al., 2021) have shown that conventional flatness measures can be manipulated through reparameterizations, challenging their direct connection to generalization. FIC-TSC contributes by proposing a computationally efficient Fisher information constraint to enforce flatness, aiming to enhance robustness to domain shifts in time series classification, and thereby adds to the ongoing discussion by attempting to operationalize flatness in a practical setting.", "essential_references_not_discussed": "Please see the above Claims And Evidence section.", "other_strengths_and_weaknesses": "FIC-TSC's main strength lies in its computational efficiency—it leverages a diagonal Fisher information approximation and a single backward pass to enforce its constraint, making it potentially scalable and applicable to real-world time series classification scenarios.", "other_comments_or_suggestions": "The paper was an enjoyable read. Please note that my comments represent my initial impressions and may include misunderstandings. I welcome further discussion on these points and am open to revising my score once my questions and concerns are adequately addressed.\n\n(Thank you for further clarification during the discussion period, I raised my score accordingly.)", "questions_for_authors": "I have incorporated my questions into Claims And Evidence section.", "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": "eP3W4Dijo4", "reviewer": "Reviewer_YXQa", "summary": "The paper  introduces a novel framework for time series classification (TSC) that addresses domain shift issues by leveraging Fisher information as a constraint.\n\nMain Contributions:\nDomain Shift Problem in TSC: The paper highlights the challenge of domain shifts in time series classification, where the test set distribution deviates from the training set, leading to reduced classification accuracy. It examines the limitations of existing normalization-based solutions, such as Reversible Instance Normalization (RevIN), which were effective in regression tasks but ineffective in classification.\n\nFisher Information Constraint (FIC-TSC):\nThe proposed method incorporates Fisher Information Constraint (FIC) into training, which guides neural networks toward flatter minima, enhancing generalization under distribution shifts.\nDirect computation of Fisher Information Matrix (FIM) is computationally expensive, so the authors use a diagonal approximation and gradient re-normalization to efficiently impose the Fisher information constraint.\n\nTheoretical Insights:\nThe method is justified mathematically, linking Fisher information to sharpness-aware minimization and showing that networks trained with FIC achieve flatter minima. Theoretical convergence guarantees are maintained while improving generalization.\n\nEmpirical Validation:\nThe method is evaluated on 30 UEA multivariate and 85 UCR univariate time series classification datasets.\nResults show superior performance over 14 state-of-the-art models, including TsLaNet, GPT4TS, ROCKET, and TimesNet.\nThe method is computationally efficient, achieving better generalization without requiring an additional backward pass per iteration.\n\nMain Findings:\nFIC-TSC outperforms prior approaches in handling domain shifts and achieves higher classification accuracy across diverse time series datasets.\nThe proposed Fisher Information Constraint reduces sharpness in loss landscapes, leading to better robustness and generalization.\nThe approach is efficient, requiring only a single backward pass per iteration, unlike prior sharpness-aware methods such as SAM.\n\nKey Takeaways:\nFIC-TSC introduces a novel constraint-based optimization approach that significantly improves time series classification under domain shifts.\nIt is both theoretically grounded and practically effective, making it a promising advancement in time series analysis.", "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": "We deeply appreciate the reviewer's effort in carefully reviewing our paper and giving very constructive suggestions. We also thank the reviewer for recognizing our novelty, significance, and theoretical and empirical analysis. \n\n---\n\n### **Explicit Domain Shift Scenario.**\n\n We consider the following two scenarios.\n- **Healthcare dataset**: Splitting data by patient introduces explicit domain shift due to physiological variability. We use four popular public datasets (anonymized in accordance with ethical standards): TDBrain, ADFTD, PTB-XL, and SleepEDF. \n\n- **Online Handwriting Recognition (OnHW-Chars)**: The training set contains characters from **right-handed** writers, while the test set features **left-handed** writers, who often differ in stroke direction, pressure, slant, and orientation. Additionally, the two datasets were released at different times, introducing potential temporal shifts. \n\n**Our method outperforms the baseline with accuracy gains of 1.6%–6.8% and F1 improvements of 0.7%–4.3%.**\n\n**View full results at https://github.com/AnonymousUserss/ICML2025-4119-Response.**\n\n\n|Dataset|Method|Acc.|\n|-|-|-|\n|TDBrain|Baseline|93.0|\n||**+FIC**|**96.2**|\n|ADFTD|Baseline|46.0|\n||**+FIC**|**52.8**|\n|PTB-XL|Baseline|73.9|\n||**+FIC**|**75.1**|\n|SleepEDF|Baseline|85.1|\n||**+FIC**|**86.7**|\n|OnHW-Char (R→L)|Baseline|44.1|\n||**+FIC**|**46.8**|\n\n\n[1] Wang, Yihe, et al. \"How to evaluate your medical time series classification?.\".\n\n---\n\n### **Systematic Results on $\\epsilon$.** \n\nPlease refer to our response to **Reviewer kJBN Q2**.\n\n---\n\n### **Diagonal Approximation Trade-Off Question.**\n\nYes, we did. Computing the full Fisher Information Matrix is theoretically and practically expensive, substantially growing the time complexity from $O(n)$ to $O(n^2)$. \nAs noted in Appendix G.4, our empirical analysis confirms that diagonal approximation is essential for practical use. **We tested the computation of the full FIM using a mini-batch size of 64 in a single iteration of InceptionTime (about 0.6M parameters) on an NVIDIA A100 40GB GPU. This operation took over 15 minutes per iteration, rendering full FIM computation infeasible for real-world training.** In contrast, with the diagonal approximation, the same computation can be completed in approximately 0.1 seconds (see Table 5), demonstrating a significant improvement in efficiency.\n\n---\n\n### **Domain Adaptation/Transfer Learning.**\n\nDomain adaptation and transfer learning address distribution shifts by enabling models trained on a source domain to generalize to a related target domain. Common strategies include aligning feature distributions by minimizing statistical distances [2] or using adversarial training to make features indistinguishable across domains [3,4].\n\nIn contrast, our method is orthogonal to these approaches. It does not require access to target domain data or labels during training, making it suitable when the target distribution is unknown or unavailable. \n**Importantly, domain adaptation/transfer learning could be applied as a post-training or downstream enhancement once target domain data becomes available.** In this sense, our method and these techniques can be complementary.\n\n[2] HoMM: Higher-order moment matching for unsupervised domain adaptation AAAI 2020\n\n[3] Purushotham, Sanjay, et al. \"Variational recurrent adversarial deep domain adaptation.\" ICLR 2017.\n\n[4] Jin, Xiaoyong, et al. \"Domain adaptation for time series forecasting via attention sharing.\" International Conference on Machine Learning. PMLR, 2022.\n\n\n---\n\n### **Runtime and Memory Scaling Question.**\n\nWe scale PatchTST by setting different numbers of blocks. The test is conducted on InsectWingbeat (30k training samples). **Our method reduces memory usage by 5–8% and cuts runtime by more than 50%.**\n |Model|#Params|Method|Memory(MB)|Time/Epoch(s)|\n|:-:|:-:|:-:|:-:|:-:|\n|PatchTST-1|4.5M|Ours|**706**|**1.8**|\n|||SAM|768|4.3|\n|PatchTST-5|17.1M|Ours|**1022**|**5.2**|\n|||SAM|1106|12.3|\n|PatchTST-20|64.4M|Ours|**2254**|**18.4**|\n|||SAM|2370|43.3|\n\n---\n\n### **Essential References to Be Included.**\n\nEWC preserves prior knowledge by penalizing changes to important weights, using a Gaussian posterior centered at previous weights with precision from the observed Fisher information (Laplace approximation). Notably, **EWC uses a diagonal approximation, aligning with and supporting the efficiency goals of our work.** \n\nK-FAC addresses the high computational cost of the FIM by approximating large blocks of it, corresponding to entire layers, as the Kronecker product of two much smaller matrices. We consider this a promising direction for future work to achieve more accurate and efficient FIM approximations.\n\n[5] Overcoming catastrophic forgetting in neural networks\n\n[6] Optimizing Neural Networks with Kronecker-Factored Approximate Curvature\n\n---\n\n**We will include the above discussion in our final version.**", "claims_and_evidence": "1. Claim: “Domain shifts in TSC degrade classification performance.”\nEvidence & Analysis:\nThe authors illustrate distribution discrepancies in the UEA datasets by plotting histograms for selected train/test sets and computing Wasserstein-1 distances.\nThey also show Reversible Instance Normalization (RevIN) helps reduce shift in regression but not in classification, motivating the need for an alternative method.\nVerdict: These examples, while drawn from a subset of datasets, convincingly demonstrate that train–test divergence is common enough to undermine classification accuracy.\n2. Claim: “Constraining Fisher information leads to flatter minima and improved generalization.”\nEvidence & Analysis:\nTheoretical Foundation: The authors use the known relationship between Fisher Information Matrix (FIM) and the Hessian at a local optimum, arguing that bounding Fisher information fosters lower sharpness.\nDiagonal Approximation & Gradient Renormalization: To keep computation feasible, they approximate the FIM by its diagonal and rescale gradients if the FIM norm exceeds a threshold. They prove the approach retains an 𝑂(1/𝑇)O(1/T) convergence rate.\nEmpirical Sharpness Reduction: Post-training, they compare the sharpness of baseline vs. Fisher-constrained models, showing the latter achieves consistently flatter minima.\nVerdict: Although off-diagonal elements are ignored, the approximation appears effective. Empirical results suggest that the constraint indeed decreases sharpness.\n3. Claim: “FIC-TSC outperforms 14 state-of-the-art methods on 30 UEA and 85 UCR datasets.”\nEvidence & Analysis:\nThe authors compare their approach (with both universal and dataset-specific hyperparameters) to a broad slate of methods, including ROCKET, InceptionTime, PatchTST, and TimesNet, showing average accuracy gains in the 1–3% range.\nThey incorporate Wilcoxon signed-rank tests to highlight the statistical significance of improvements.\nVerdict: The large-scale experiments are thorough. Since these are standard splits, additional “real shift” setups (e.g., temporal splits) could further confirm the model’s shift-handling capability. However, the reported gains over many baselines look convincing.\n4. Claim: “Our Fisher-based method is more efficient than other sharpness-aware solutions.”\nEvidence & Analysis:\nFIC-TSC uses only one backward pass per mini-batch, while methods like SAM require two, roughly doubling computation.\nThe authors compare iteration-level runtimes on selected datasets to demonstrate FIC-TSC’s speed advantage.\nVerdict: Given standard deep-learning frameworks, a single-pass approach should indeed be faster. This is plausible and supported by iteration-time measurements.\nPotential Weaknesses:\nDiagonal Approximation\nWhile standard in large-scale second-order methods, ignoring off-diagonal terms could limit accuracy of curvature estimates. An ablation or discussion on how robust performance is under this approximation might enhance confidence.\nRealistic Domain Shift Splits\nMost experiments still rely on standard train/test divisions from UCR/UEA. Inducing controlled or temporal shifts in the data could help illustrate exactly how robust FIC-TSC can be in true real-world drift scenarios.\nHyperparameter Sensitivity\nThe paper briefly explores 𝜖 settings (the Fisher constraint threshold) but does not deeply map how different ϵ values influence final accuracy or model stability across varied tasks.\nFinal Verdict\nOverall, the paper’s key contributions—(i) identifying domain shift issues in TSC, (ii) introducing Fisher-based constraints to improve generalization via flatter minima, and (iii) empirically surpassing a range of strong baselines—are supported by both theoretical arguments and extensive benchmark results. While points like diagonal approximation, real-world shift tests, and hyperparameter exploration merit deeper discussion, the evidence is generally strong enough to validate the authors’ central claims.", "methods_and_evaluation_criteria": "1. Proposed Methods\n1.1 Fisher Information Constraint (FIC)\nCore Idea:\nThe authors introduce a Fisher Information Constraint to encourage flatter minima. They approximate the Fisher Information Matrix (FIM) by its diagonal and rescale gradients if the overall Fisher norm surpasses a threshold 𝜖\nImplementation:\nUnlike double-backward techniques (e.g., SAM), FIC-TSC needs only a single backward pass each iteration.\nThis keeps overhead low while preserving benefits of second-order information.\nWhy It Makes Sense:\nTime series data often exhibit domain shifts, where distributions differ between training and testing. FIC encourages the model to be less sensitive to small input perturbations, boosting robustness.\nThe diagonal approximation is a practical trade-off, capturing enough curvature to reduce sharpness without excessive computation.\n1.2 Addressing Domain Shifts\nNon-Stationarity:\nMany real-world TSC tasks—sensor data, medical signals—can shift over time. A sharper minimum can lead to overfitting.\nBy bounding the Fisher norm, the paper aims to reduce such overfitting, achieving better generalization under unseen conditions.\n2. Evaluation Criteria\n2.1 Datasets\nUEA (Multivariate) and UCR (Univariate)\nCover 30 and 85 datasets respectively, spanning a wide array of TSC problems.\nThey are standard archives in the TSC community.\nWhile standard splits may not always replicate “hard” domain shifts, they do exhibit real differences between train/test distributions.\n2.2 Metrics and Comparisons\nAccuracy:\nThe paper primarily uses classification accuracy, supplemented by additional metrics like F1 scores on some datasets.\nStatistical Tests:\nWilcoxon signed-rank tests assess significance of accuracy improvements.\nBaselines:\nComparison with 14 methods, including ROCKET, InceptionTime, TimesNet, and PatchTST.\nThis broad coverage clarifies where FIC-TSC stands versus both older and newer TSC models.\n2.3 Suitability\nThe authors’ approach aligns with typical TSC evaluation practices—large-scale experimentation on UCR/UEA is widely recognized as a benchmark standard.\nReporting both mean accuracy and statistical tests is methodologically sound, ensuring that improvements are not mere artifacts of specific datasets or chance.\n3. Overall Alignment with the Problem\nFocus on Distribution Shifts\nTSC often suffers from training–testing mismatches, so a method that systematically curbs parameter sensitivity is well-justified.\nEfficiency\nFIC-TSC’s single backward pass makes it more practical than other sharpness-aware methods, especially for larger networks or frequent online training updates.\nComprehensive Benchmarking\nEvaluating on 115 total datasets (30 UEA + 85 UCR) ensures results are not tied to one domain.\nThe authors also highlight relevant statistics (precision, recall, F1) and run significance tests, reinforcing the robustness of their comparisons.\nConclusion\nFIC-TSC directly targets a core challenge in time series classification: the impact of domain shifts on model performance. By constraining the Fisher Information via a diagonal approximation, the method encourages learning a flatter solution that is less sensitive to distributional changes. The authors’ evaluation strategy—using well-known UCR/UEA benchmarks, reporting standard metrics, and conducting statistical significance tests—effectively demonstrates the method’s advantages over existing TSC models. Hence, the proposed methods and evaluation align well with the problem, showing both conceptual appropriateness (mitigating domain shifts) and empirical thoroughness (large-scale comparisons on standard archives).", "theoretical_claims": "1. Overview of Theoretical Claims\nEquivalence of Fisher Information and Hessian\n\nThe paper states that the Fisher Information Matrix (FIM) is asymptotically equivalent to the Hessian of the negative log-likelihood at a local optimum.\nThis is a well-established result in information geometry. The authors’ statement aligns with known derivations (e.g., under regularity conditions, the expected Hessian and FIM coincide).\nSharpness Reduction via Fisher Constraint\n\nThey link Fisher-based constraints to reduced sharpness, arguing that smaller FIM norms imply flatter minima.\nThe proof involves approximating sharpness around a local minimum using a Taylor expansion and relating the Hessian to the FIM. The steps appear consistent with prior sharpness-aware minimization work.\nConvergence Rate\n\nThe paper claims that imposing the Fisher constraint preserves an O(1/T) convergence rate under standard smoothness assumptions.\nThe authors outline a gradient-based convergence analysis, treating the constraint as a re-normalization step. While not exhaustive, the argument follows typical first-order proof templates and does not introduce obvious contradictions.\n2. Checked Details and Issues\nFIM–Hessian Equivalence:\nThe authors’ derivation is standard, relying on well-known results in maximum likelihood theory. No major oversights were found.\nFlatness and Taylor Expansion:\nThe paper’s link between smaller diagonal FIM and flatter minima uses the logic that the Hessian’s diagonal dominates local curvature in a diagonal approximation. Though it omits cross-terms, the rationale is mathematically sound for large-scale deep networks that rely on diagonal approximations.\nOne potential limitation is that ignoring off-diagonal terms may underestimate curvature in certain directions, but this is disclosed as a simplifying assumption rather than a full second-order analysis.\nConvergence Proof:\nThe authors provide a high-level sketch showing that re-normalizing gradients when the FIM norm exceeds a threshold does not break standard Lipschitz-based convergence arguments.\nA full, line-by-line formal proof with all constants and step sizes is not fully detailed, but the outline is in line with existing gradient-descent proofs.\n3. Conclusion\nOverall, the theoretical claims appear largely correct and adhere to established principles in optimization and information geometry:\nFIM–Hessian Relationship: Properly stated and well-known in literature.\nSharpness Reduction: Reasonably extended from the FIM–Hessian link and standard Taylor expansions.\nConvergence Rate: Based on recognized gradient-descent proofs, with a plausible argument that enforcing an upper bound on Fisher norm does not slow the asymptotic 𝑂(1/𝑇) rate.\nWhile the diagonal approximation and partial discussion of cross-terms mean the proofs do not capture every nuance of a full second-order method, they are consistent with standard practices and do not exhibit fundamental errors.", "experimental_designs_or_analyses": "1. Scope of Datasets and Baselines\nUEA (30 datasets) and UCR (85 datasets):\nThese are highly regarded open-source repositories covering both univariate and multivariate TSC, offering a broad distribution of domains (e.g., sensor signals, ECG data, image outlines).\nValidity: Using these archives is widely accepted for benchmarking TSC models, thus the selection is appropriate and representative.\nBaseline Comparisons:\nThe paper benchmarks against 14 diverse methods, including both classic (e.g., ROCKET, InceptionTime) and modern (e.g., TimesNet, PatchTST) approaches.\nValidity: This large set of baselines, covering a variety of algorithmic designs, strengthens the credibility of the results.\nStatistical Tests:\nThey employ Wilcoxon signed-rank tests to compare model accuracies across multiple datasets.\nValidity: The Wilcoxon test is a standard, non-parametric choice in multi-dataset settings. It bolsters confidence that reported gains are not merely coincidental or dataset-specific.\n2. Experimental Protocols\nTrain/Test Splits:\nThey follow the standard UCR/UEA data partitions rather than generating alternative splits.\nPotential Issue: These standard splits do not always strictly reflect real-world distribution shifts (e.g., chronological shifts). However, the authors do point out that some train–test differences already exist (shown via histograms/Wasserstein distances). More explicit or controlled shift scenarios might have provided stronger evidence of domain-robustness.\nMetrics:\nAccuracy is the primary metric, supplemented by F1, precision, and recall for selected experiments.\nValidity: Accuracy is typical for classification; adding more metrics helps address class imbalance or multi-class comparisons. This is consistent with established TSC practices.\nHyperparameter Selection:\nThe authors discuss two strategies: a “universal” (same hyperparameters for all datasets) and a “full” (dataset-wise tuning).\nValidity: Showing both strategies clarifies how stable the method is when hyperparameters are not exhaustively tuned, and how much improvement is possible with targeted tuning. This split is a fair approach and demonstrates practical applicability.\nComputational Efficiency Reporting:\nThe paper compares runtime per iteration with Sharpness-Aware Minimization (SAM), illustrating that the proposed Fisher-based approach is lighter due to its single-pass design.\nValidity: Presenting runtime differences is important for real-world feasibility. They show consistent speed gains over SAM, which seems credible given SAM’s two backward passes.\n3. Analysis of Results\nAverage Accuracies:\nThe authors list comprehensive tables of accuracy scores for all 115 datasets, often showing 1–3% gains over competitive methods.\nThey use statistical significance tests (Wilcoxon) to confirm whether these gains are systematic.\nValidity: Reporting means plus significance metrics is standard in large-scale TSC evaluations. The improvement margins appear reasonably consistent across many datasets.\nSharpness Measures:\nThey measure and visualize sharpness or curvature in the parameter space to show how the proposed method yields flatter minima.\nValidity: These additional visual/quantitative analyses back up the theoretical claims. While each sharpness metric is an approximation, it does support their notion that the method reduces sensitivity to domain shifts.\nAblation Studies:\nThey vary 𝜖 (the Fisher norm threshold) to see how it affects performance.\nPotential Issue: Some readers might want deeper exploration of the hyperparameter’s sensitivity across more datasets. The paper gives partial insights but could further detail the trade-offs between different 𝜖 values in a more systematic manner.\n4. Overall Soundness\nPositives:\nLarge benchmark coverage (115 datasets), multiple baselines, and recognized statistical tests.\nClear demonstration of both universal vs full hyperparameter settings, indicating robustness.\nReported run-time comparisons for efficiency claims.\nMinor Caveats:\nStandard train/test splits in UCR/UEA do not precisely mirror real-world domain shifts, although the authors do show partial evidence of distribution differences.\nThe hyperparameter search could be more comprehensively documented to fully assure reproducibility, though the current studies do give enough for a fair comparison.\nConclusion:\nThe paper’s experimental designs and analyses generally adhere to standard TSC practices and provide robust support for the proposed method. Although more targeted shift experiments or extended hyperparameter sensitivity tests could strengthen the domain robustness argument, the large-scale evaluations, statistical comparisons, and runtime analyses all appear fundamentally sound and valid.", "relation_to_broader_scientific_literature": "1. Fisher Information and Sharpness-Aware Optimization\nPrior Work:\nSharpness-aware methods like SAM (Sharpness-Aware Minimization) introduce a second backward pass to encourage flatter minima. In parallel, Fisher Information has long been connected to curvature in information geometry, relating to the Hessian of log-likelihoods.\nPaper’s Advancement:\nThis paper combines these strands by using Fisher information as a direct constraint, making the optimization process single-pass rather than double.\nIt thus situates itself alongside second-order and sharpness-focused approaches but does so with a diagonal FIM approximation, keeping computational overhead minimal.\n2. Time Series Classification under Domain Shifts\nPrior Work:\nTSC literature often centers on specialized architectures (InceptionTime, ROCKET, various Transformers) or similarity-based methods (DTW). However, handling non-stationarity or train–test distribution shifts in TSC has received comparatively less attention beyond methods like Reversible Instance Normalization (RevIN), which primarily aids forecasting/regression tasks.\nPaper’s Advancement:\nBy directly addressing distribution shifts within TSC (rather than purely focusing on new architectures), the paper provides a relatively novel perspective: it treats domain shifts as a fundamental optimization/robustness issue, not merely a data-augmentation or normalization challenge.\nThis complements prior TSC work on normalization techniques by proposing that regularizing curvature (via Fisher Information) can outperform, or at least fill the gap left by, normalization-based solutions in classification settings.\n3. Empirical Benchmarks and Method Comparisons\nPrior Work:\nMany TSC papers rely on UCR/UEA archives for evaluation, but most emphasize raw accuracy improvements with domain-specific innovations (CNN designs, ensemble methods, or transform-based approaches). Ensemble methods like HC2 or shapelet-based frameworks have historically been robust but often with higher computational cost.\nPaper’s Advancement:\nThe authors show that a general optimization-level technique—FIC—can match or surpass specialized TSC architectures on the same standard benchmarks.\nThis demonstrates that certain general-purpose enhancements (aimed at stability or curvature control) can be as crucial as sophisticated network designs in pushing performance bounds.\n4. Connection to Information Geometry and Generalization\nPrior Work:\nFisher Information has been used to study generalization, but mostly in contexts such as Bayesian inference, elastic weight consolidation, or advanced network compressions. These approaches often require extra memory or partial second-order updates.\nPaper’s Advancement:\nBy framing Fisher Information as a constraint on gradient norms (rather than storing or inverting any matrix blocks), the paper contributes a simpler, more scalable variant of these second-order ideas.\nIt thereby extends the principle that controlling curvature (through the FIM) can guard against overfitting, especially in domains—like TSC—where distribution shifts are common.\n5. Overall Positioning\nBringing Second-Order Insights to TSC:\n\nThe paper merges second-order optimization insights with TSC challenges, emphasizing domain shift resilience rather than purely augmenting classification architectures.\nBridging Normalization Gaps:\n\nTechniques like batch or instance normalization have been central in other time series tasks (especially forecasting). The paper positions itself as a complementary or alternative approach for classification scenarios where direct normalization sometimes obscures class differences.\nPotential for Future Extensions:\nThis approach could be extended or integrated with other robust classification methods (e.g., adversarial training, domain adaptation), indicating synergy with broader machine learning methods beyond TSC.\n\nConclusion:\nIn summary, the paper’s key contributions—a Fisher-constraint–based approach for flatter minima, tailored specifically to TSC’s domain shift challenge—fit well into existing second-order/sharpness-aware frameworks while tackling a recognized but less-explored problem in the TSC literature. It bridges general-purpose optimization insights (FIM-based regularization) with domain-specific concerns (non-stationarity in time series), making it a noteworthy addition to both communities.", "essential_references_not_discussed": "1. Prior Work on Fisher Information / Second-Order Methods\nKronecker-Factored Approximate Curvature (K-FAC)\nReference: Martens & Grosse, ICML 2015, “Optimizing Neural Networks with Kronecker-Factored Approximate Curvature.”\nRelevance: Demonstrates a practical way to approximate second-order information using Kronecker-factored matrices, thus improving training efficiency without resorting strictly to diagonal approximations. While K-FAC is mainly used for faster convergence, it also relates directly to the Fisher Information Matrix. The current paper’s diagonal-Fisher approach could be contrasted with or informed by this more elaborate approximation.\nElastic Weight Consolidation (EWC)\nReference: Kirkpatrick et al.  PNAS 2017, “Overcoming catastrophic forgetting in neural networks.”\nRelevance: Uses Fisher Information to preserve previously learned knowledge when new tasks arrive. Even though EWC focuses on continual learning, its reliance on Fisher-based constraints parallels this paper’s method for controlling sharpness. Citing EWC would acknowledge foundational uses of Fisher Information in deep learning optimization and illustrate that the notion of stabilizing parameter updates via FIM is an active line of research.\n2. Domain Adaptation / Distribution Shift in Time Series\nDeep Domain Adaptation for Time Series\nExample Reference: Purushotham et al., KDD 2017, “", "other_strengths_and_weaknesses": "Strengths\nNovel Combination of Existing Ideas\n\nThe paper blends known second-order optimization insights (i.e., Fisher Information, Hessian relationships) with a specific target of distribution shifts in time series classification. While Fisher-based methods are not new, applying them as a one-pass constraint to flatten minima for TSC is an original twist.\nSignificance and Broad Applicability\nDomain shift problems are pervasive in real-world time series tasks (e.g., evolving sensor conditions, non-stationary signals). Demonstrating a general solution that can integrate with both convolution- and transformer-based TSC models enhances the paper’s practical value.\nClarity in Writing and Structure\nThe paper is generally well-structured. The motivation (domain shift in TSC), main idea (constrain Fisher norm), and experimental evaluation (comprehensive benchmarking on UCR/UEA) are laid out in a logical progression. Key points—like the contrast to Reversible Instance Normalization—are relatively clear and straightforward to follow.\nExtensive Experiments and Statistical Significance\nThe authors present results on 115 datasets, with multiple baselines, and use Wilcoxon signed-rank tests. This thorough coverage supports the claim that the method is widely applicable and not narrowly tuned to a handful of problems.\nWeaknesses\nLimited Exploration of Realistic Shift Scenarios\nAlthough the paper shows distribution differences in standard train/test splits, it does not deeply investigate scenarios like chronologically separated training and testing. A controlled shift experiment (e.g., training on earlier time frames, testing on later frames) could further validate real-world domain-robustness.\nDiagonal Approximation Restricts Full Second-Order Information\nWhile efficient, ignoring off-diagonal terms may limit capturing interactions among parameters. An ablation comparing diagonal vs. block-diagonal approximations (or referencing Kronecker-factored approaches) would strengthen the paper’s second-order argument.\nHyperparameter Sensitivity Analyses\nThe paper briefly discusses setting the threshold ϵ but does not provide a thorough grid-based sensitivity exploration across multiple datasets. More systematic experimentation could clarify how much \nϵ-tuning is required for robust performance.\nSomewhat Limited Theoretical Details\nWhile the authors offer a high-level sketch of convergence arguments, more detailed proofs or bridging steps could help. In particular, it would be helpful to see how quickly the algorithm converges in practice under typical TSC conditions (e.g., smaller sample sizes, higher dimensional signals).\nOverall Assessment\nOriginality: The idea of enforcing Fisher constraints in a single-pass framework, specifically tailored to time series classification, stands out for handling domain shifts—an underserved area in TSC compared to the often-solved architecture innovations.\nSignificance: Given that domain shifts are a prevalent real-world challenge, the approach has practical and theoretical importance.\nClarity: The paper is mostly coherent and well-motivated, though a deeper discussion of certain hyperparameters or more elaborate shift scenarios might enhance understanding.\nDespite some noted weaknesses, the paper’s strengths—innovative re-interpretation of Fisher-based constraints for TSC, extensive comparative experiments, and strong performance—indicate a valuable contribution to time series research.", "other_comments_or_suggestions": "Below are additional suggestions and minor observations:\n\nHyperparameter Tuning Explanation\nIt would be helpful to have a more detailed discussion in the text on how you searched or selected the threshold 𝜖 for the Fisher Information Constraint. Although the paper mentions a small grid search, providing explicit rationale or heuristics would benefit readers trying to replicate or extend the approach.\nResults Section Organization\nYou might consider splitting the main performance tables into (a) univariate (UCR) and (b) multivariate (UEA) subtables for clarity. The current combined presentation is still understandable but separating them could help readers compare methods more intuitively by domain type.\nHighlighting Real-World Case Studies\nIf any of the UCR/UEA datasets map closely to real industrial or medical shift scenarios, explicitly mentioning them might further strengthen the practical motivation. A short  example could illustrate why flatter minima matter in that setting.", "questions_for_authors": "Below are several questions that address points where additional clarification or detail could potentially change the evaluation of the paper:\n\nExplicit Domain Shift Scenario\n\nQuestion: Have you conducted any experiments using a chronologically split setup or another explicit shift scenario (e.g., training on earlier data, testing on later data) to confirm the method’s robustness beyond standard UCR/UEA splits?\nWhy It Matters: If FIC-TSC demonstrates notable improvements under controlled real-world shifts, that would further validate the claims about domain-shift resilience. Conversely, if such tests haven’t been done, it’s possible that the standard splits do not fully capture how well the method handles non-stationary data.\nHyperparameter Sensitivity\n\nQuestion: Please provide more systematic results on the impact of different ϵ values across multiple datasets, and explain how a practitioner might choose ϵ for new tasks?\nWhy It Matters: The threshold ϵ appears central to the Fisher constraint’s effectiveness. Detailed guidance, or an ablation over a broader range of datasets, would clarify how sensitive the method is to this parameter and whether it needs per-dataset tuning.\nDiagonal Approximation Trade-Off\nQuestion: Have you tested any partial or block-based FIM approximations to see if ignoring off-diagonal terms significantly affects performance or memory demands?\nWhy It Matters: This would reveal whether the diagonal approximation is sufficiently capturing curvature or if a more nuanced approximation could yield further improvements (albeit at higher computational cost).\nComparison to Domain Adaptation Methods\nQuestion: How does FIC-TSC compare with existing domain adaptation or transfer learning approaches for time series, especially methods that explicitly align distributions (e.g., adversarial alignment techniques)?\nWhy It Matters: Although your focus is on robust optimization, the domain-adaptation community also tackles shifts in time series. Clarifying these potential synergies or differences may strengthen the positioning of FIC-TSC in the broader literature.\nRuntime and Memory Scaling\nQuestion: Beyond single-batch iteration timing, did the authors benchmark FIC-TSC’s training speed and memory use on large-scale datasets (e.g. tens of thousands of training samples) or with very deep networks?\nWhy It Matters: Demonstrating consistent scaling properties can solidify the paper’s claim that FIC-TSC is more efficient than other sharpness-aware approaches (SAM) in practical, large-scale scenarios.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper addresses the problem of time-series classification under distribution shift. Its main contribution is the proposed FIC-TCS method, which leverages a Fisher information constraint to guide the model toward flatter minima, thereby improving robustness. During the rebuttal phase, reviewers raised concerns about the motivation related to sharpness in time-series data and the assumption that flatter minima lead to better generalization. These concerns were adequately addressed by the authors. Following the rebuttal, all reviewers gave positive evaluations. The authors are encouraged to incorporate the reviewers’ constructive feedback into the revised manuscript.
Decision: Accept (poster)