Time Series Representations with Hard-Coded Invariances

We sincerely appreciate the reviewer's time and thoughtful feedback. In the following, we thoroughly address each identified weakness and question.

[Experimental Design]

• Performance on UCR normalized data: We address your concern in Q1 below. To clarify, synthetic deformations are used only in test sets for the robustness study of convolutions (Section 4.1, Table 1). In contrast, in all other experiments (Sections 4.2–4.4 for classification and Appendix A.5.1 for anomaly detection), performance is evaluated on default data without added deformations.

• Significance of results: We invite the reviewer to see the critical diagram difference for UEA in the rebuttal for reviewer gvG9.

[Other]

The presentation of the paper could be improved [...].

To enhance readability, we will move Figure 2 to an earlier section of the manuscript to better standard deformations and add a brief intuitive introduction in the method section. See our response to reviewer Bfty in section [Clarity] for details.

Replies to Questions:

Q1. InvConvNet does not exhibit optimal performance on normalized data [...]. What is the practical significance of artificially adding deformations?

In the robustness experiment ("I. Robustness to Deformations", section 4.1), the performance of invariant convolutions compared to normal ones on the plain (i.e., z-normalized input without deformations) is better for 2 out of 5 plain UCR datasets. It is still close for the rest of the cases (average -4.7% drop for the 3 data cases). Contrary when synthetic deformations are added, the performance drops for normal convolutions are around -50% (on average for the 4 considered deformations), whereas for the trend invariant convolution, this drop is just around -2% (on average for the 4 considered deformations) showing the competitiveness and robustness of the proposed layer in comparison to its standard counterparts.

The motivation for the robustness study on UCR datasets is as follows: Most UCR datasets, including those used in this study, are already z-normalized. Their univariate nature, limited class diversity, and relative trend stability make classification easier for normal convolutions. To systematically assess the impact of synthetic deformations, we use these datasets in a controlled setting, progressively increasing deformation complexity (from offset shifts to linear trends and smooth random walks) to evaluate how different convolutional components, including those invariant to such distortions, respond.

Q2. How can you ensure that the added deformations are justifiable for the specific dataset?

Offset and trend invariances are common in real-world applications like PPG monitoring and ECG analysis (see reply [Other] to reviewer gvG9). They are tackled by baseline wander removal and offset correction techniques that preserve physiological signals while eliminating low-frequency noise. The closest easy-to-generate deformation, a smooth random walk, is included in the robustness experiment (see Table 1).

Q3. Can invariant convolution enhance the performance of time series forecasting?

Our invariant layers, combined with the example decoder used for anomaly detection, naturally extend to forecasting by employing the same lightweight decoder based on linear layers applied to the learned coefficients. We next provide some preliminary results (in MSE) for the ETT-small datasets (for horizon length h=96), where we use a fixed seed and compare our approach against 8 baselines (most derived from the package Time Series Library).

These results, achieved without hyperparameter tuning (in less than a week), are close to SOTA performance, highlighting their potential. Future improvements include testing hybrid architectures for the decoder (e.g., transformers) to enhance finer granularity, capture longer dependencies, and unsupervised pretraining for better generalization.

We sincerely thank the reviewer for their time and thoughtful evaluation of our work. In the following section, we address any concerns raised.

[Claims and Evidence]

"However, the basic assumptions about the deformation phenomenon seem not strong enough[...] While in forecasting tasks, such decomposition methods (moving average, convolutions) are widely used [...]"

For time series forecasting, the suggested papers [1,2,3,4] assume a seasonal-trend decomposition of time series to derive suited ANN architectures. Most methods employ a (learnable) moving average kernel to infer the trend and propose different ways to deal with residual time series. The average moving kernel corresponds to a specific deformation in the proposed framework: the offset shift (see Figure 2). Assuming such offset-invariant kernels, the proposed layer offers different views of the residual time series. When focusing on trend deformations, the trend approximation could be better inferred by assuming a higher-order Taylor expansion (linear, quadratic, etc.). In the anomaly detection (Appendix A.5.1) and forecasting experiments (see rebuttal reply Q3 to reviewer MxbV), we also leverage trend-residual decomposition in lightweight architectures by feed-forwarding the projections coefficients to the reconstruction layer. In classification, PerimidFormer (Classification Table in rebuttal response to reviewer Bfty) and Dlinear (Table 2 of the manuscript) are also based on a trend-residual decomposition, yet are significantly outcompetes by the proposed invariant convolutions. For instance, on the UEA datasets, our example architecture InvConvNet, performs better than PerimidFormer and Dlinear (acc: 71.81 vs (59.71 and 61.51 resp.)), suggesting that deformation-based trend invariance is better suited than standard trend-residual decomposition for classification with neural networks. Overall, the proposed layer seems more versatile and robust compared to existing architectures on several tasks while guaranteeing state-of-the-art performances.

"The authors seem only consider some naive linear deformations [...] are ANNs really hard to handle these?"

Standard ANNs often struggle with generalization on noisy, non-stationary time series, as illustrated in Figure 1, where a standard CNN fails to capture relevant ECG features. To test whether CNNs can learn invariance, we conduct a robustness experiment (Table 1), showing that classification performance declines as datasets are progressively deformed. Standard convolutions prove sensitive to spatiotemporal distortions (avg. acc. drop: -50%), whereas hard-coded invariant convolutions offer better generalization (avg. acc. drop: -2%) than contrastive learning (avg. acc. drop: -23%). Figure 8 (Appendix) further visualizes this, revealing that standard CNN feature maps become distorted, while invariant convolutions retain structured representations (see Figure 7 for corresponding FordB dataset deformations). These findings align with (Kvinge et al., 2022) for image data. Finally, in transfer learning (Table 4), invariant convolutions surpass standard architectures and contrastive frameworks by at least 4%.

• Kvinge, H., Emerson, T., Jorgenson, G., Vasquez, S., Doster, T., & Lew, J. (2022). In what ways are deep neural networks invariant and how should we measure this?. NeurIPS, 35, 32816-32829.

The paper proposes a representation method [...] tasks such as forecasting, anomaly detection?

We conduct extensive experiments across diverse real-world applications to validate our theoretical framework of time series invariant convolutions, leveraging common offset and shift invariances in shallow (Figures 5 and 6) and lightweight example architectures (Figure 3). We have demonstrated robustness against offset and trend deformations (Section 4.1), strong performance in multivariate classification (Section 4.2), and transfer learning for classification (Section 4.3). Additionally, we extend our analysis to anomaly detection in Appendix A.5.1. Rather than proposing a general-purpose architecture, targeted to specific time series tasks, our goal was to develop versatile layers that integrate seamlessly with existing methods to enhance robustness. To illustrate this, we employ lightweight architectures—such as linear decoders for regression—that, despite their simplicity, match or surpass complex baselines. While our current focus is classification and anomaly detection, our approach naturally extends and holds promise for forecasting, as noted in our response and experiments provided to reviewer MxbV (rebuttal reply Q3).

[Experimental Designs Or Analyses] We show improved performance against the suggested baselines UP2ME [5] and PerimidFormer [6]. Please refer to the Tables in the rebuttal reply to reviewer Bfty. We will also include a discussion about the additional baselines in the revised version of the manuscript (not added here due to the word limit).

We sincerely appreciate the reviewer's time and effort in evaluating our work. Below, we provide our responses to the main suggestions.

[Claims And Evidence]

"authors claim computational efficiency benefits [...], additional explicit runtime comparisons [...]".

We have demonstrated the computational efficiency of the proposed invariant layers over baselines, with memory complexity and training time comparisons shown on the UEA Heartbeat dataset (with 61 channels and 405 timestamps) in Figure 3. Similar time comparisons on larger datasets will be included in the revised manuscript. More details on the fast computation of our convolutional layers via FFT can be found in Section 3.2 (pages 4-5).

[Methods and Evaluation Criteria]

• Additional Experiments. We next present comparisons for classification on the 26 UEA datasets and the 5 anomaly detection (AD) datasets, evaluating the additional methods: TimeMixer, iTransformer, Peri-midFormer, and UP2ME

Additional Results Classification Acc.(%)

Additional Results Anomaly Detection

Our method consistently retains its advantage in both classification and anomaly detection tasks. We were unable to obtain results for the SWaT dataset using UP2ME (marked with out-of-memory), even after reducing the hyperparameters to manage memory usage.

Implementation Details: Based on their official GitHub implementations, only iTransformer has been adapted for classification and anomaly detection. Optimal hyperparameters are provided for iTransformer on 10 UEA datasets and all AD datasets, while for TimeMixer, we used default values for non-forecasting tasks. Hyperparameters for Peri-midFormer are sourced from its GitHub for classification (on 10 UEA datasets) and anomaly detection tasks. Additionally, we test the backbone of UP2ME. We adapted TimeMixer and UP2ME for classification by reshaping the encoder outputs into a one-dimensional vector and passing them through fully connected layers for predictions.

• Visualizations of Capturing Invariances. We will definitely include additional visualizations of the invariances captured by our layers in the revised manuscript. For now, we have included visualizations of offset and trend deformations on UCR datasets, along with feature maps for normal and invariant kernels, in Figures 7 and 8 of the Appendix.

[Experimental Designs Or Analyses]

We will incorporate the reviewer's suggestions in the revised manuscript. We already provide extensive details on hyperparameter tuning for the proposed method and the baselines in Section A.4 Implementation Details. While results are currently marked based on mean performance and std overlaps, we will consider paired statistical t-tests for all datasets and best methods (not provided due to time constraints). Please also refer to the critical diagram difference for UEA in the rebuttal for reviewer gvG9.

[Clarity]

"The mathematics framework is hard [...] intuitive explanations."

To enhance readability, we will relocate Figure 2 to an earlier section of the manuscript to clearly showcase standard deformation types. The methods section will also add a short, intuitive introduction: "The proposed mathematical framework consists of two main components. The first is a group action that formalizes how certain deformations transform time series, resulting in their deformed counterparts. Only deformed time series are observable in practice, such as those influenced by noise or trends. However, embeddings that remain invariant to these deformations are essential for many applications. To address this, the second component is a mapping function that constructs embeddings of deformed time series while remaining invariant to specific deformations. Finally, these embedding maps can be efficiently integrated into convolutional operations to extract robust local features to non-informative deformations.

We would like to thank the reviewer for the time spent to evaluate our manuscript. We next reply to their key suggestions and comments on our work.

[Evaluation Criteria]

"The authors could have produced a synthetic diagram of the model's performance [...]."

We appreciate the reviewer's suggestion and will add a critical difference diagram, along with clarifying Rocket's first-count performance in the main section. As shown, InvConvNet achieves the best mean rank across 26 UEA datasets and 9 baselines. Using the aeon library [1], we compute mean ranks based on accuracy, confirming significance via a Friedman test (critical difference value of 1.6784 at $\alpha = 0.1$). The Nemenyi post-hoc test identifies three statistically distinct groups: 'InvConvNet' and 'Rocket' in the first, followed by 'TSLANet', 'ResNet', 'TimesNet', 'Cnn', 'Crossformer', and 'PatchTST' in the second.

Mean Rank for UEA Classification Acc.

[1] https://www.aeon-toolkit.org/en/latest/examples/benchmarking/published_results.html#References

[Broader Scientific Relations]

"Their proposed idea goes somewhat against the current trend [...] few real-world cases were invariances are so explicitly defined in my opinion."

Recent deep learning trends for time series rely on implicit invariances from augmentations using contrastive learning, often lacking generalization guarantees (see Table 1) and incurring high computational costs. Our approach directly encodes common time series invariances within the network, ensuring both efficiency and generalization (see Figure 3, Tables 1 & 3). Our convolutional design is rooted in group-theoretic modeling of time series deformations, which is a novel approach in this domain. Similar formalisms have introduced robust architectures for computer vision and graph neural networks. Our layers, coupled with lightweight modules (Figures 5, 6), can achieve robust and competitive performance, even on smaller datasets.

[Other]

"It is unfortunate to limit the study to trend and offset [...] discussed other types of invariants."

Baseline removal is common in applications for physiological signals, such as EEGs and PPGs, where baseline wander can mask critical variations, and removing trends can enable more accurate feature extraction [1,2]. The closest easy-to-generate deformation is a smooth random walk, which we incorporate in the robustness experiment as the most complex deformation following simple offset shift and linear trend (see feature maps visualizations in the last row of Figure 8 for smooth random walk deformation of the FordB dataset in Figure 7). Offset shifts and linear trends often arise due to sensor drift, calibration errors, or gradual measurement degradation, leading to misleading variations in the data. While z-normalization removes global offsets, it fails to address local distortions. Our model learns local invariance to offset shifts and trends, making it effective for real-world data. This justifies our choice of offset shift and linear trend as key deformations to experimentally validate the broader theoretical framework of time series invariant layers under group actions.

In the introduction (paragraphs 4–5, pages 1–2), we discuss time series invariances, including time-shift, time-rescaling, and contrastive learning-based transformations. Commonly addressed deformations include amplitude scaling, offset shifts, and trends, tackled via z-normalization or contrastive frameworks (e.g., TS-TCC). We introduce a formal mathematical framework for time series invariance to spatiotemporal deformations, providing exact formulations of invariances rather than relying on approximations.

We restrict invariant convolutions to simpler but common affine deformations, allowing invariance to trend when it can be assumed linear at the scale of the convolution kernel size. However, we plan to explore approximating the trend with non-linear functions, such as splines or higher-degree polynomials, and incorporating seasonal components with low-frequency cosine bases in future work.

[1] Kaur, M., Singh, B., & Seema. (2011). Comparing approaches for baseline wander removal in ECG signals. Proc. Int. Conf. & Workshop on Emerging Trends in Technology, 1290-1294.

[2] Awodeyi, A. E., Alty, S. R., & Ghavami, M. (2014). On filtering photoplethysmography signals. IEEE Int. Conf. Bioinformatics & Bioengineering, 175-178.