Shifting the Paradigm: A Diffeomorphism Between Time Series Data Manifolds for Achieving Shift-Invariancy in Deep Learning
Authors: Berken Utku Demirel
Christian Holz
Venue: ICLR 2025 Poster (2025)
Keywords: Time series analysis
invariance in neural networks
Primary area: learning on time series and dynamical systems

Abstract: Deep learning models lack shift invariance, making them sensitive to input shifts that cause changes in output. While recent techniques seek to address this for images, our findings show that these approaches fail to provide shift-invariance in time series, where the data generation mechanism is more challenging due to the interaction of low and high frequencies. Worse, they also decrease performance across several tasks. In this paper, we propose a novel differentiable bijective function that maps samples from their high-dimensional data manifold to another manifold of the same dimension, without any dimensional reduction. Our approach guarantees that samples---when subjected to random shifts---are mapped to a unique point in the manifold while preserving all task-relevant information without loss. We theoretically and empirically demonstrate that the proposed transformation guarantees shift-invariance in deep learning models without imposing any limits to the shift. Our experiments on six time series tasks with state-of-the-art methods show that our approach consistently improves the performance while enabling models to achieve complete shift-invariance without modifying or imposing restrictions on the model's topology. The source code is available on GitHub.

Published as a conference paper at ICLR 2025
SHIFTING THE PARADIGM: A DIFFEOMORPHISM BE- TWEEN TIME SERIES DATA MANIFOLDS FOR ACHIEV- ING SHIFT-INVARIACCY IN DEEP LEARNING
Berken Utku Demirel
Christian Holz
Department of Computer Science
Department of Computer Science ETH Zurich
ETH Zurich
ABSTRACT
Deep learning models lack shift invariance, making them sensitive to input shifts that cause changes in output. While recent techniques seek address this for im- ages, our findings show that these approaches fail to provide shift-invariance in time series, where the data generation mechanism more challenging due the inter- action of low and high frequencies. Worse, they also decrease performance across several tasks. In this paper, we propose novel differentiable bijective function that maps samples from their high-dimensional data manifold to another manifold of the same dimension, without any dimensional reduction. Our approach guarantees that samples-wher subjected to random shifts are mapped unique point in the manifold while preserving all task-relevant information without loss. We theoretically and empirically demonstrate that the proposed transformation guaran- tees thift-invariance shift. Our experiments on time series tasks with state-of-the-art methods show that our approach consistently improves the performance while enabling models to achieve complete shift-invariance without modifying or imposing restrictions on the model's 's topology. The source code is available on GitHub
deep learning models without imposing any limits to the
INTRODUCTION
Inference time series is essential for several important applications, such heart rate (HR) estimation (Koshy et al. 2018), activity recognition (Saint-Maurice et al. (2020), and cardiovascular health monitoring (Hannun al. 2019), which are generally performed using signals that are encoded as a sequence discrete values over time. Most of these signals contain features that characterize
the signal independently of their position in time (Waibel et al. 1989 Demirel & Holz 2023). In other words, the information content of signals generally remains unchanged under the action of finite groups such as translations (Mallat) 2012). Therefore, ensuring the ability accurately capture these inherent patterns is crucial for the reliability of the deep learning models in such critical human-involved health-related tasks (Akbar et al.) (utillo al. 2020).
Deep learning networks perform downsampling by using strided-convolutionn and pooling (He et al. 2015 Krizhevsky al. 2012), which cause loss of information due high-frequency components of the input alias into lower frequencies, i.e., aliasing (Oppenheim et al. 1996). Previous works have proposed to employ a low-pass filter to prevent the aliasing and mitigate information loss during downsampling (Zhang. 2019 Mairal et al. 2014). While this additional filtering improved the robustness, the effect of employed low-pass filters is quite poor compared to the ideal implementation (see Figure a and b), which still causes high-frequency components to alias into lower ones.
Despite the potential benefits of emphasizing low-frequency components for image recognition, as aligns with human perception (Subramanian al. 2023), the imperfect preservation of frequency components with each subsampling layer contributes to information loss, leads to performance degradation, especially in tasks where the significance lies in both low and fighfrrequency components with their interactions. A more recent approach to achieve shift-invariant neural networks involves the of adaptive subsampling grids (Chaman & Dokmanic 2021). However, these methods still

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(a)
(c)
(e)
Magnitude (dB 10#)
Amplitude
Amplitude
wwmm In w
xie)
Binomial Ideal
Allased
frequency band
www.
mrad/samoli
Time (s) 8
Time
0.5
0.8
10
(b)
Amplitude
Heart Rate (HR)
Probabiltty
AFIB
Binomial
140
M MMMMAM
sin/xra)
Time (s)
Shift
Shift e
-4
5
Figure 1: (a) The magnitude response of the ideal low-pass filter and binomial filter that is employed in (Zhang] 2019) for preventing aliasing. (b) Time domain representations of the ideal and binomial filters with interpolation for smoother waveforms.
(c) An 8-second signal for blood volume changes and its t' shifted version, obtained through photoplethysmograne
widely utilized signal for heart rate monitoring (Perez et al. 2019). (d) The heart rate prediction of a trained ResNet with binomial filters to prevent aliasing. Different amounts of shifts (t' [-4, 4]) change the trained model output drastically from 140 60 beats per minute (bpm). (e) A 10-second electrocardiogram (ECG) signal from patient with atrial fibrillation (AFIB). (f) The model misclassifies the abnormal AFIB pattern as a healthy sinus rhythm (SR), with shifts causing a complete change in output probability.
fail to guarantee ihit-iinvariancy due to the change in content at the boundary (Rojas Meaaa- et al. 2022] and impose constraints on the shift range maintain invariance.
Consequently, the evaluation of these methods is confined to a limited range of shifts while covering small subset of the space. Additionally, their reliance on grid scheme introduces a dependence on sampling rates, resulting in performance gaps across the entire shift space (Michaeli et al. 2023).
In this work, we propose a differentiable bijective function that maps samples from their high- dimensional data manifold to another manifold of the same dimension, without any dimensional reduction. Our method ensures that randomly shifted comples--repreentting variations of the same signal-are mapped to the same point in the space, preserving all task-relevant information.
Since our method modifies the data space, can be integrated into any deep learning architecture, offering adaptable and complementary solution for achieving shift- init--invarianyy in time series. Summarizing our contributions in this paper:
We introduce novel diffeomorphism to ensure caitt-invariancy in neural networks. Addi- tionally, we incorporate the proposed diffcomorphism into the network architecture using a novel, tailored loss term to further enhance performance while ensuring invariance.
We demonstrate both theoretically and empirically that the proposed transformation guaran- tees shift-invariancy in models without imposing any limits the range of shifts or changing model topology, which enable previous methods to be used in conjuction.
We conduct extensive experiments on six time series tasks with nine datasets. Our ex- periments show that the proposed approach consistently improves the performance while decreasing the variance and enabling models to achieve complete shift-invariance.
METHOD
2.1 NOTATIONS
We use bold lowercase symbols (x) for time series. The parametric mappings are represented (.). where 0 is the parameter. The discrete Fourier transformation of time series is denoted as F(x). yielding a complex variable [X (e²") (er((w) which contains magnitude and phase information each harmonic (sinusoidal). ((w)) and T₁ represent the phase angle and period of the k-th harmonic with frequency Wμ. We mainly used the textbook notations (Oppenheim et al., 1996) throughout the script, providing comprehensive list of notations and detailed definitions in Appendix A.
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2.2 OBJECTIVE
Given a dataset D - (x)()) where each x E X consists of uniformly sampled real-valued values and each y E Y represents the corresponding labels, the objective to have consistent and accurate outputs for all variants of a sample that are subjected to shifts such that when a parametric model fe X + Y is evaluated on the set Dust {(x(t), )= the output will be the same and true y, for all t' to be shift-invariant, i..., y; = fo(x - t)):), , Vt' E R.
We propose diffeomorphism that maps randomly shifted time series samples to the same point in data space, preserving all relevant information to ensure ihift-invariance. The motivation and theoretical derivation of our method are presented in the following steps.
Proposition 2.1 (Time shift a Group Operation). Shift operation in time domain defines an Abelian Group of phase angles in the frequency domain for each harmonic with frequency wk-
($x, + mod 2T), where & = {ol Φ = (Ø(w)) + wit') mod 2x, E R}
(I)
Proof. Using F(a(t t7) IX (e(())) pdut' and the multiplication of complex numbers
31' E IR, Ve E (-T, m] (Ø(wx) + wal') mod 2T
See Appendix A for detailed proof with group axioms.
Proposition 2.1 states that the shift variants of sequence define group of phase angles, known as circle group (Fuchs 1960) T. An impor- tant observation from Equation 2 is that different shift values (t') can map to the same phase angle (d) due to modulo operation with 2T.
(a)
(b) Amplitude
Im
t
kinlag)) stafuss
n
Ad
Re
We 2
Au
To
# T//
However, a closer look reveals that this mapping can be defined uniquely for specific harmonics using the cir- cular shift. Specifically, we can rep- resent every point in the shift space uniquely with the phase angle of harmonic whose period is equal to or longer than the length of sample, i.e., To t. In the remainder of this sec- tion, we explain how this observation is framed as novel diffeomorphism. We denote the frequency, period, and phase of this specific harmonic wh. To. and ((w)), respectively.
2
x/2
Time
21.) (x(e)")eed)))
x(1)
x(1) sin (wof) + +....
x(t F(., (e
x(1 (') (sond + Ad) +
Figure 2: (a) Frequency domain representation of a harmonic frequency wo with different phase angles in unit circle. (b) Time domain representation of signal x(t) and its shifted version x(t t'). The phase angle of the harmonic can cover all (i.e., surjective T(x, Φ)) potential shifts. Moreover, shifts in the time domain correspond
unique (i.e., injec- tive T(x, Φ)) angle rotations in the frequency domain for the sinusoidal with periodicity To. Therefore, the proposed transformation function T(x, Φ) is bijective.
The proposed transformation function, T(x, Φ), takes a sample and an angle applies a linear phase shift to each harmonic, mapping the time series to a new variant where the phase angle of the harmonic with frequency we matches the desired angle Φ. The proposed transformation. which converts a time series to another shifted variant, is defined as in Equations and
(- r). It then
x(1)
T(x,0)
F ((X (eeeoo))))) -iw where
(0
i > n and 0 (Ø(w)) Φ % 2n
A6
2n
d=To
else
2tr
Mainly, the transformation first decomposes signal to its harmonics, then it calculates the phase difference, denoted as AØ, between the harmonic with frequency wi and the desired angle d. Finally,
We represent time shift (t') for sample x as x(t +'), similar to Oppenheim et al. (1966). All the time shifts throughout the paper imply circular shift, i.c., (t t²) = (t fl) where % the modulus.
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It returns to the time domain by taking the inverse Fourier transform, (.), while applying linear phase shift to all harmonics to preserve the waveform morphology. In the end, the transfor- mation matches the phase angle of the harmonic at frequency wp with the desired angle ø. We first demonstrate that the proposed transformation is a bijective function, as shown in Theorem 2.2
Theorem 2.2 (Covering the Entire Time Space Injectively). Given a sample x, the defined function T(x, 6) 4 x Ha + Ra x A is bijective such that all shift variants of sample can be covered with the unique phase angle of a harmonic whose period longer or equal to the length xf
voa @b Φ, T(x, 0a) T(x, 06) = Qu 4=== x6
Vt' E, E Φ, T(x, (x(1)) Ao),
where the first and second equations represent the injection and surjection, respectively.
We provide an intuitive demonstration in Figure 2 with a detailed mathematical proof in Appendix A Since each point in the shift space can be uniquely defined by the phase angle of a harmonic with period To, we use the angle of this harmonic to define manifolds D .MP on which the samples lie. Specifically, we apply the proposed transformation T(x, の) for each sample to map it to manifold defined by the angle, i.e., T(x, @a) E MM*, T(x, db)) E MM and 2 M @ (See Appendices 12. and A.2 for detailed definition of manifolds and notations). We, therefore, can map sample and its randomly shifted variants to the same point in the space, which is sufficient for providing shift-invariancy as demonstrated Theorem 2.3 with a detailed proof in Appendix A
Theorem 2.3 (Guarantees for Shift- hhift-Invariancy).. Gäven x and randomly shifted variant of it x(1 a(t-t'), t'), (x, 6) is applied to both samples with the same angle &a. the resulting samples will be the same.
T(x(t), 6.) = (x(t), Adx()), T(x(t - t'), = (x(t), Adx(e-e))
Proof.
To
xx(1) = ((w); Px((-1()= =bww) wt', Adxit-1) Adx(t) -w
(5)
2T
To Wo w tt TT(x)))) = (T(x
(T(x(+-11).ba) (((x))))
(6)
Therefore, the output time series samples will be the same after applying the transformation.
The proof concludes by demonstrating that the harmonics retain the same phase and magnitude after transformation, despite an unknown shift applied the sample. Moreover, since the transformation only contains exponentials with Fourier transform, it is fully differentiable, allowing optimization with
x//2
do
(c)
x/2
La = Co + VarlJa.
T(x
(x,
A
or
Co fca (T(x0)))
0
④0
Re
Th
40
2m
8 - 00 0
To =
x())
T(x,
T
((
dv
(x(())
or
fcs(.)
fc,(.)
x(e
Figure 3: (a) An input signal in the time domain and complex plane representation of its decomposed 2x with the phase angle Φn- (b) Guiding the diffeomorphism to map samples between manifolds. (c) The obtained waveform with phase shift applied to all frequencies linearly, calculated by the angle difference, as in Equation 4 without altering the waveform. (d) The loss functions for optimizing networks with the cross-entropy and the variance of possible manifolds.
sinusoidal of frequency wp 2%
The manifold is defined d-dimensional Fuclidean space, matching the data's 's dimension, to hetter explain the abstract transformation. There manifold learning of low-dimensional space in transformation.
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guidance network fc. R + Φ with shift-invariant input, absolute Fourier transform of samples, to generate an angle in radians for mapping. Simultancously, the main classifier fc, X + Y maps the transformed samples to the label space. Both networks are optimized with cross-entropy loss. The optimizer for the guidance network (La) has an additional loss term reduce variations in batch (B) of angles, given in Equation 7
neural networks. Therefore,
N
Lc c
Σ Nig log fc, (T(x,, (x)) LG = Lo + / V (fae ((F(x)))
(7)
The guidance network, optimized by the proposed loss, works an adaptive linear constraint that limits the regions in the original data space where samples can be found. In other words, if we conceptualize the data space of samples as expanding with shift variants, illustrated in Figure 3
the model learns to reduce the potential points where samples can be found in the data space.
Additionally, for real-world samples where optimal phase shift values are unavailable, applying trivial phase shifting may lead to suboptimal data space representations. To address this, we transform the data space using the proposed diffeomorphism for the downstream tasks using the guidance network (see Appendix F for detailed analysis of the guidance network). Moreover, unlike traditional manifold learning methods (Lin & Zha 2008 Waan et al.] 2004), which project data into lower- dimensional spaces, our approach operates directly within the original data space. In our ablation studies, we thoroughly examine the impact of loss terms on the performance and present the findings.
3 EXPERIMENTS
3.1 DATASETS
We conducted experiments on nine datasets across six tasks, including heart rate (HR) estimation from photoplethysmoogrphh (PPG), step counting and activity recognition using inertial measurements (IMUs), cardiovascular disease classification from electrocardiogram (ECG), sleep stage classification from electreeneephalography (EEG) and lung sound classification from audio. We provide short descriptions of each dataset below, and further details can be found in Appendix
Heart rate We used the IEEE Signal Processing Cup in 2015 (IEEE SPC) (Zhang et al. 2015). and
DaLia (Reiss et al. 2019) for PPG-based heart rate prediction. We used the leave-one-session-vl (LOSO) volss-aali,ation, which evaluates models subjects/sessions that were not used for training.
Activity recognition We used UCIHAR (Anguita et al. 2012), and HHAR (Stisen et al. 2015) for activity recognition from inertial measurement units from smartphones. We evaluate the cross-person generalization performance of the models, i.e., the model is evaluated on previously unseen subjects.
Cardiovascular disease (CVD) classification We used Chapman University, Shaoxing People's Hospital ECG (Zheng et al. 2020) and PhysioNet 2017 (Clifford et al. 2017 Goldberger et al. 2000) datasets. We selected the same four leads for the Chapman as in (Alday et al. 2020). We split the datasets into training, validation, and test sets according to the patient ID (each patient's 's recordings appear in only one set) using 60, 20, 20 ratio in Demirel & Holz 2023) ; Zheng et al. (2020).
Step counting We used the Clemson dataset (Mattfeld et al. 2017), which released for pedometer evaluation. We conducted experiments using wrist IMUs where labels are available through videos.
Sleep stage classification We used the Sleep-EDF dataset, from PhysioBank (Goldberger et al.) 2000,, which includes whole-night PSG sleep recordings, where used single EEG channel (i.e., Fzz-Cz) with a sampling rate of 100 Hz, following the same setup as in Eldele et al. (22T))
Lung sound classification We used the Respiratory @ TR, which contains lung sounds recorded with two digital stethoscopes (Altan et al.) (017)). Two pulmonologists validated and labeled the recordings based on X-rays, pulmonary function tests (PFTs), and auscultation. The labels correspond to five COPD severity levels (COPD0-COPD4) as described in prior work (Zhang al. 2024).
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3.2 BASELINES
We compared our method and existing approaches including low-pass filtering (LPF) Zhang 2019). and adaptive subsampling grids (APS) (Chaman & Dokmanic 2021). In addition techniques, we evaluated our method against shift-equivariant Wavelet Networks (Romero et al. 2024) and canonical representation learning techniques for equivariance (Kaba et al. 2023 Mondal et al. ).23). Moreover, since our method can be integrated with any existing approaches, we investigate the performance of previous techniques for shift-invariancy when combined with our algorithm.
3.3 IMPLEMENTATION
We follow similar implementation setup previous work on shift-invariancy (hang 2019) in supervised learning, making architectural adjustments for time series. Specifically, we employed ResNet (He et al. 2015) with eight blocks designed for time series (Hong al.) (020), excluding signals from inertial measurement units with a single dimension. For the latter, we observed a better performance with fully connected networks (FCN). Therefore,
used a three-layer FCN for the single dimensional IMU- task, i.e., step counting. Similarly, for guiding the transformation function, we used an FCN with single output, which is the angle for the chosen sinusoidal. For each dataset, we set the Fourier transform length equal to the signal length, as the Fourier transformation of the same size inherently includes sinusoids with periods equal to or longer than the signal length. We use categorical cross-entropy loss, which is optimized using Adam (Kingma & Bal 2015). The learning rate is determined through grid search for each dataset and set to the same value for all baselines given in the Appendix. During training, it was halved when the validation loss stops improving for 15 consecutive epochs. The training is terminated when 90 successive epochs show no validation performance improvements. The best model is chosen the lowest loss the validation set. Detailed hyperparameters and architecture specifications can be found in the Appendix C.4
3.4 EVALUATION
We evaluate the performance of the models using the common evaluation metrics, i.e., accuracy, F1, for each task. For shift- chfttiinvariany,, we used the shift consistency (S-Cons.) metric which measures how often the network outputs the same classification, given the same time series with two different shifts, similar to (Zhang 2019) as in Equation [8] We applied shifts across the entire space in contrast
to previous approaches where the range of shift is heavily limited (Rojas-Gomez Gomez et al. 2022).
Ex 1 [fo(x((- (1)) fo(x(1-()))
(8)
where fc represents the classifier's output following the arg max operation. 4,2 are uniformly sampled integers from the interval [1, t), with & denoting the length the sample.
RESULTS AND DISCUSSION
We present the main results of our approach compared state-of-the-art methods across the six time series tasks on nine datasets. Overall, our method has demonstrated a substantial performance improvement, reaching up to 10-15% in some tasks, while increasing the shift consistency up to 50-60% compared to previous techniques.
The experimental results from all the time series tasks are given in Tables I 2 3
and These tables demonstrate that the previous techniques fail to provide shift-invariant models when applied to time series without limiting shifts. Additionally, the models exhibit extremely low consistency (as low as 32%) in HR prediction. More importantly, applying state-of-the-art methods enhance shift consistency in deep learning models for predicting the heart rate results in performance degradation.
We believe the main reason for the small improvements in the consistency of previous techniques is that the research to date has tended to focus on limited shifts rather than considering the whole shift space literature is mostly concemed about images. While restricting shifts can be valid assumption in computer vision, where the main reasoning is that the object being classified should not be near the boundary. This assumption does not apply time series, where the whole signal

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Table Performance comparison of our method and other techniques for HR estimation
IEEE SPC22
DaLiA
Method
S Cons (8)
RMSE
MAE. +
(
S Cons (%)
RMSE
MAE.
p (S) 1
Baseline
61.99=1.19
18. 2.999
10.28+1441
62.64:15.74
32.08=0.222
9.86:2023
4,40=0.03
86.01
Aug.
76.48=1.77
18.73=1.15
10.42+0.40
64.06-+3.70
52.77=0.39
9.85+0.21
4,47-0.66
85.99=0.49
76.88-0.73
20.20-11554
13.44=0082
65.40++1.92
38.67+0.30
4.67-0.12
85.68++051
APS
73.99-1.00 1.06
19.42-0.60
12.98:+0229
65.27++1.32
44.33++0.16
10.45=0.40
5.01-0.17
84.69+0.85
WaveletNet
$1.71=1995
21.56:11001
14.61=0.34
60.74-24.37
36.71=3.04
15.46=0064
7.67-0.23
76. 73:11:11.86
Canonicalize
63.52=11.20
19.02=00.2
10.40:009
61.27:+1.07
32.01=0.33
9.77+0.12
4.39-0.05
86.02:+0.50
70.12+22.0
9.75++015
86.06++0.19
Lie
100=00.00
20. 20.34-11.62
65.60+-2.31 12.32:20.44
13.77:0084
100+00000 67.01=3.79
10.72-0.11
5. 5.30-0.03
84. 12:00.3
Ours+APS
100=00.00
18.81=1.59
100:00.00
10.47=0.09
5.10=0003
84, 34.62:2031
Table 2: Performance comparison of ours and other techniques in ECG datasets for CVD classification
Method
Chapman
PhysioNer 2017
S Cons (%)
Acc 1
F1 T
AUC (R))
Cons (%)
Acc 1
7
AUC
Baseline
98.53=0.17
91.32:+::3
91 2::22:0022
98.34=0.16
98. 98.37-0.15
83.22 83.22:1.772
73 1:950:11:99
93 212110.300
Aug.
99.00-0.16 0.16
91.96:0.19
91.89:0.22
98.45=0.18
98 9.10..1
82.28 1.18
72.32:-220
93.20=0.42
LPF
92.0111:23
91.9410.58
98.50=0.24
98.94-0.39
84.4010.16
75.6810.76
93.80=00322
APS
08.60-0.17
90.6910.89
89.44:1.00
08.31=0.24
WaveletNer
91 11.02-1.14
90.87+1002
90.02:1.00
07.94-0.11
65.03-0.71
76.0620064
63.35==340
87.02=0.29
Canonicalize
08.80=0.24
91.93:0.13
90.8720.18
98.42-0.15
98.26-0031
83.3410.46
73.9720.67
93.68
Ours
1.000.00
92.10=0.25
91.93=0085
98.47-0.15
100-0.00
13.15=0055
14.12:11.80
93. 93.28-0.31
OursaI.P
92.05-+0.52
100-0.00
77.50=1.21
94.20-0019
OuS
100-0.00
91.61+1.111
91.10=0.56
98.36-0.20
carries the information (Demirel & Holz, 2023) additional to local waveform features, and as such, there is no explicit boundary condition imput consider for limiting the range of shifts.
The empirical results support our motivation for proposing diffcrentiable bijective function that maps samples with different shifts the same point the data manifold, avoiding the limited shift assumption. Additionally, applying low-pass filtering to prevent aliasing can degrade performance for certain tasks, where the interaction between frequencies plays critical role (Canolty et al. 2006).
Time delay adversary? An interesting finding from our experiments is the notable decline in model consistency as the number of output classes increases. This behavior in the models is similar previous findings adversarial examples, indicating that the robustness decreases with higher number of classes (Fawzi et al.) 2018). During our experiments, we observed the same phenomenon where the small shifts of the input change the output to another class, particularly when the task complexity increased with a higher number of classes. For example, in the case of HR estimation (Table D even short shifts (as low as 10-100 ms) can lead change in the prediction by over 80 bpm, despite no alteration in the periodicity of the signal, which is the main feature for this task.
Normally, is expected that models learn the periodicities in these signals and infer the heart rate. However, our results indicate that the models learn something else or in different way, because as the signal undergoes a slight shift, the model prediction jumps more than 100%, even though the periodicity of the waveform remains unchanged with the shift operation.
We believe these drastic output changes arise from the model's sensitivity to (shortcut) fea- tures (Geirhos et al. 2020 Zhang et al. 202T), resulting in performance decrease when evaluated on samples different from those encountered during training. Since our proposed transformation
Table Performance comparison of our method with other techniques on an EEG dataset for sleep stage classification and an audio dataset for lung sound classification in respiratory health assessment
Method
Sleep-EDF
Respiratory
S-Cons (%)
Acc
W-FI
x
S-Cons (%)
Acc
F1
W-FI 1
Baseline
95. 95.06=20.61
75.41:2.01
74.87+1.92
67. 12.49.6
99. 10:2043
25.21
21.21 57.01+3.62
Aug.
99 17
74.99+1.11
74.03++146
65.89+181 81
09.68-0042
20.32+5.18
45.81433.51
15.31++607
LPF
92.43-+1122 1.24
73.56++2991
7..01
65.68-13.46
99.50+0042
19.4749.78
46.534-3.04
11.89:44.98
WaveletNet
84.40-15.90
73.54
72.7413.45
64.66.14.12
91.38
28.57+10.81
44.23.1.12
17.1011.81
Canonicalize
93.95:10.51
77.1242.21
70.14412.25
69.8112766
98.28-10.4
22.68:110.52
45.331.5775
15.30.15.33
Ours
100-:0.00
7..90+11.22
76.77+2.58
70.01
100:-0.00
33.10+5.12
60..13+4.667
28.33++6.55
O.P
100+:0.00
1.89
75.34++1.6
64.98-+2.277
100=00.00
25.774-2.12
$1.824-2110
17.99444.15

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Table 4: Performance comparison of our method with others in IMU datasets for Activity and Step
Methed
UCIHAR
HHAR
Clemson
S-Cons (%)
Acc
F1 1
S-Cons (16)
Acc 1
F1 f
S-Cons (89
MAPE L
MAF.
Baseline
94,07711338
85.39:2.30
83.20:12.04 94
98.27 27:20.33
91 1.3:1.36
91.16:11 38
$4.31 14.40
4:76:0.11
2. 2.74::0.08
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96.58-+0.80
85,4244.50
83.00+6..74
98. 38:00.888
91 971-0.44
91.31-0.49
61.01 1011-4200
4.08==0.14
2 2.29-0.07
LPF
95.05+00.21
83.96+144
81.08 41.08+44.21
98. 10-00-0010
10++0.80
01.43-094
59.77 4.
4.16-0.16
2. 2.38+0.11
APS
96.40.10003
81.7514.11
79.01.5.333
98. 08.30._0.24
91.13.5355
91.01-1.47
45.50.2.09
4.74.00.16
2.6910.07
WaveletNet
94.5611.3]
62.78.1622
80.7311,.59
96.76_10.15
90.7210.38
90.71 0.39
59.14.1..10
5.201.066
2.95.0.41
Canonicalize
97.72.00.37
84. 94.1012.110 10
81.89912.89
98.27.00.07
91.56.11.18
90.73-1.110
35.4711.87
4.54110.46
2.5910.22
Ours
100_0.00
85,67.1.47 11.91
100.00000
91.981114
91.12-1003
100.10.00
4.28.10.34
2.431021
Ours-LPF
100-0.00
84.7812466
82.58.1.62
100100000
92.5110.55
91.80-00 0.62
100100.00
3.75-0.33
2.12.00188
Ours-APS
100:00.0
12.9621
11:10:
100:00000
91.38:10.22
90.64-0032
100100.0
3.87:00119
2.19::0:11
function works as an adaptive linear constraint in the data space, reduces the potential points where samples can exist, thereby enhancing overall performance.
One distinct result from our experiments is that when previous shift- nhit-invvriancy techniques are applied to the heart rate prediction task, the average enror rate of the models increases by 7-10%. This performance decrease can be easily observed in the DaLiA (Table [ for the adaptive sampling technique. The performance discrepancy between tasks be attributed to the dataset and signal characteristics. Since DaLiA contains impulse random noise with multiple periodicities, the nomm- based subsampling can inadvertently emphasize the noisy waveforms instead of the desired pattern during the subsampling of feature maps, leading to decrease in prediction performance.
We conduct detailed ablation experiments to further investigate the impact of various components, with particular focus on the effect of the proposed mapping function under different modifications, i... modified loss for optimization, on the overall model's performance across time series tasks.
4.1 ABLATION STUDY
We present comprehensive investigation of our method and the effect of its components the performance. Mainly, we investigate the effect of guiding the proposed transformation with different loss functions and without any guidance. First, i.e. T(x, 6) is applied with a constant Φ = 0 instead of learning the angle for each sample. We experimented with different values of performance when the mapped manifold is constant for samples. Second, we modify the loss for training the guidance network to increase the variance of angles-increasinn where data can be wotho-wi changing the cross-entropy loss from the classification network as in Equation 9 (Éo). Finally, we train both networks only with the cross-entropy loss (C's = Lc). We compared these three variants of the learning techniques with the original proposed implementation as each represents distinct approaches for manipulating the data space. For example, when all samples are mapped to a single manifold. the variations in samples decrease significantly since there is only one possible phase angle for the chosen harmonic with period To. Additionally, the relationships among all sinusoidal components remain invariant, given that the proposed transformation is a linear function of the frequency. Conversely, optimizing the guidance network to increase the variance of angles, thereby favoring a greater sample diversity, expands the possible variations for samples.
map all samples to single manifold M
(- x]. but observed significant change in the
the possible manifolds
L== = Lc / Varx-BB (fea ([F(x)))
Tables 5 and summarize the results where we exclude the consistency metric from the tables as the models that include the proposed transformation are always completely shift-invariant. The first row (T(x, Φ)) in the tables shows the performance when all the samples are mapped a single manifold
Table 5: Ablation experiments for HR (left) and IMU (right) tasks
Method
IFFE SPC22
DaláAppS
Method
UCIHAR
HHAR
Clemson
MAE
RMSE
et
MAE J
RMSE |
Acc f
F1
Ace
FL
MAPE
MAE
T(x.
11.15
19.18
62.07
4.77
10.13
85.35
T(x
84.67
82.65
92.33
91.56
4.64
2.67
C
9.80
17.16
66.80
1.60
10.10
85.52
Co
84.30
82.49
91.98
91.18
4.42
2.52
17.00
69.10
4.41
9.63
Ea
84.82
91.51
90.83
4.31
Ours
9.45
16.25
70.12
4.39
9.75
86.06
Ouns
85.81
83.81
91.83
91.12
4.28
2.43
Change
+1.70
+2.97
+8.05
+0.38
+0.38
+0.71
Change
+1.14
+1.16
-0.50
-0.44
+0.36
+0.24
8

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Published as a conference paper at ICLR 2025
Table 6: Ablation experiments for EEG (left) and ECG (right) tasks
Method
Sleep-EDP
Method
Chupman
PhysioNet
Asc
FL
W-F1 T
Acc
Fl
AUC
Acc
Fl
AUC
7(x. 6)
Co
77.21+1.511
75.54
67.67-1677
66.96=1178
75.5342.29
76.89+1.71
67.08=0.03
69.39-0.02
91.27
90.10
90.76
98.38
82.81
73.67
73.75
93.45
93.24
La Ours
77.75+1.23
68.04=1.16
77.01+1007
09.94-0.01
90.84
93.51
77.80+1.95
67.01+265
76.7742.58
70.01-1.10
Ours
92.10
91.93
98.40
93.30
Change
+2.26
+0.05
+1.24
+2.93
Change (8)
+1.17
+0.04
+0.03
+0.06
i.e. without a guidance network for learning the mapping. The second row (L') represents the performance when the guidance network is only optimized using the categorical cross-entropy loss. The third row (&g) presents the performance when the variance of angles is optimized increase during training. And, the last row (Ours) is the original implementation of the proposed method. We also report the change when the mapping function is guided using the network fc, and optimized using the loss defined in Equation 7 as opposed to being a fixed, non-learnable function.
As can be seen from the tables, when the models are trained by guiding the transformation function (with fa,), the performance of the models increases significantly up to 8%, except for the HHAR dataset with a marginal performance decrease of 0.5%. Importantly, adding the guidance network does not bring any additional parameters that help the learning, meaning that the model achieves improved generalization with the same capacity. Furthermore, the additional model parameters introduced to the overall framework approximately amount to one percent of those in the classifier. While the performance increase can associated with the decreased possible variations in the signals, our ablation experiments show that decreasing the variations blindly using the transformation with the same angle, decreases performance. Therefore, it is important guide the transformation function for reducing the dimensionalityy i.e.. the space and time variations of a signal, of the whole data space. Over- all, the results obtained from the ablation study and main experiments support the previous propositions and our motivation for introducing a novel diffco-
Table 7: Ablation experiments for Audio
Respiratory
Method
Acc
F1 f
W-FI 1
T(x, o)
21.28:7.43
18.14:1..99 55.03=2.89
Ce
27.17
55.58
21.46:4.07
La Ours
28.5718.31
54.28
23.73
33.10-
4.613
6.53
Change
+11.82
+5.10
+10.19
morphism for preventing the inconsistency of deep
learning models to the time shifts while increasing the generalization capability.
Additional results (i.e.. the extended experiments and ablations) regarding the performance of the pro- posed method can be found in Appendix D Investigations regarding the performance improvements of the proposed diffeomorphism with different model networks are given in Appendix F:] Detailed analysis of the guidance network with its effect is given in Appendix [ We provide an extended discussion of related work in Appendix Gaddd outline limitations and future directions in Appendix H
5 RELATED WORK
Shift-invariant networks Modern deep learning architectures use strided convolution or pooling to decrease the variance to a certain extent (Fukushima] 1980). However, Azulay and Weiss have demon- strated that shift of one pixel in an image can lead to a significant alteration in the output probability of trained classifier (Azulay & Weiss 2018). Previous works showed that the downsampling caused aliasing and used low-pass filtering before the downsampling to prevent information loss (Zhang) 2019 Mairal et al. 2014). However, the used filters have suboptimal frequency responses, and realizing the ideal filter practice is unfeasible. This leads to persistent aliasing, becoming a more significant concern for time series where high-frequency components are crucial for classification.
Adaptive subsampling methods have been recently explored for shift-invariancy (Chaman & Dok- manic; 2021 Xu et al. 2222]) Mainly, these methods perform subsampling on a constant (Chaman & Dokmanic 2021) or input dependent (Rojas-Gone Gomez et al., 2022) grid. This approach has a notable limitation in time series, particularly when nonlinear activation functions involved. The methods tend to overlook variations in boundaries arising from the translation of samples, thereby imposing additional constraints invariance Sonjas-Gomzz.cooo al. 2022). Consequently, the evaluation of

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Published as a conference paper at ICLR 2025
these methods is restricted to narrow range of shifts, covering only a limited subset of the shift space. Moreover, their reliance on a grid scheme for sampling introduces a sensitivity to sampling rates, leading to performance gaps across the entire shift space (Michaeli et al. 2023). Therefore, shift the paradigm and present a bijective transformation to modify the data space. Moreover, unlike existing methods that change network topology by modifying the pooling or adding extra filters without achieving complete shift-invariancy. our method guarantees invariance in neural network models without imposing any restrictions on the model topology or shift range.
Time-delay neural networks Efforts design shift-invariant models for time series predate modern deep learning methods (Hasegawa et al. 1996. Waibel et al. 1989). For example, time- delay neural network (TDNN) network is designed to have the ability to represent relationships between events in time frames where the learned features by the network are aimed to be invariant under translation in time (Waibel et al.] 1989). TDNN is trained with all time-shifted copies of samples and weights are updated by the average of all corresponding time-delayed error values. This is similar to the supervised training of a network with randomly shifted versions of samples. Although this strategy achieved shift-invariance for the first type of networks, they do not include a pooling layer, it was shown that this approach is ineffective for modern architectures where pooling and derivatives are used (Scherer et al. 2010), and the network's 's invariance is limited to patterns seen during training and fails generalization (Azulay & Weiss 2018). In this work, as we learn the mapping for each sample, the proposed transformation ensures that all shifted variants of sample mapped to a single point. Hence, a single data point effectively represents all the augmented variants.
6 CONCLUSION
The inadequacy of shift-invariance in deep learning models, particularly in the context of temporal data, remains a significant challenge. Existing solutions designed for images not only prove inef- fective for time series but also result in performance deterioration for some tasks. To address this, we have introduced a novel differentiable bijective function. Our approach builds on the insight from Proposition 2.1 which states that the shift operation forms an Abelian group for each harmonic of a sample. Leveraging this property, we uniquely represent each point in the shift space using the phase angle of a harmonic whose period is at least as long as the sample length. Our approach ensures that samples, under various shifts, are mapped to unique point in the data manifold without reducing dimensions, preserving task-related information without any loss. We validated our method theoretically and empirically, showing that it establishes ihift-invariance in deep learning models without constraints on the shift range. In extensive experiments across six tasks, our approach consis- tently outperforms oate--oftthe-an methods, demonstrating its effectiveness in achieving complete ihift-invariance without limitations on the model topology.
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APPENDIX
A THEORETICAL ANALYSIS
Here, we present complete proofs of our theoretical study, starting with notations. We assume all the samples (time series) are absolutely summable, and finite.
A.1 REPRESENTATIONS AND NOTATIONS
A.1.1 FREQUENCY DOMAIN
Fourier transform of real-valued sample with a finite duration obtained in Equation 10
= - x(t)e jut
F(x) = =I((ct)))
(10)
where w 30 and w and T are the frequency in radian and period for all sinusoidals in the range of Nyquist rate. [X(e")) and ((w)) denote the amplitude and phase for all frequencies, respectively. Thus, the amplitude and phase angle of a particular sinusoidal are represented as (e }| and
((wo)). Similarly, the period for this sinusoidal is To 2n/wwp The phase difference between
sinusoidal and an angle e is shown in Equation II
A
2T
and (ó(wn) o) % 2nr,
(11)
4wTh
else
2m
where To/2m 1/wn- The calculated phase difference between the sample and given angle normalizes the phase change for the sinusoidal to exactly match the angle di when shifted Adx(o in the complex domain as in equations below.
F(x) = XX (X(e")))e (e (0(()) مم after shifting (X(e"))ee (o(o)))
- (wol(//w))
(12)
(e (j000))))) (=o()(())-d)) /www))
(13)
((eewo))e for the sinusoidal with frequency wi
(14)
A.1.2 TRANSFORMATION AND DIFFEOMORPHISMS
The observed samples x(t) from the sets with the (shift) variants considered elements of manifolds M according to the phase angle ((w)) of the harmonic with a period equal to or longer than the length of the segment i.e., the specific harmonic with period To and frequency wi- In other words, the phase angle of the harmonic wi is oa for a sample x(!), the sample lies in manifold M The mapping function f M M between manifolds is defined as T(x(t), Φb) = (X(t). Ads) where x(t) lies in MM and x(t) lies in M Similarly, the inverse mapping f M + M represented as ((x(t), Q.) (x(t), AQ.).
x() and x() only differ by random time shift t' where this random time shift can be calculated using the phase angles of harmonics with frequency wo. which makes the presented transformation bijective function between time series manifolds. The differentiation of the mapping function at sample x(1) is shown as D T, -> >T.Ma.
'Harmonics and sinusoids are used interchangeably throughout the paper.
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A.2
NOTATION LIST
Notation
Description
x
Time series represented bold lowercase symbol
F(x)
Discrete Fourier transformation of time series
The complex argument for obtaining phase values
Variable that represents the frequencies radian
(e(e)=)
Magnitude components the Fourier transformation time series
((w)
Phase components of the Fourier transformation of a time series
wu
Frequency of the specific harmonic whose period is equal or longer than the sample x
To
2m
Period the specific harmonic with frequency wio
wo
Random variable for phase angles, i..., ø ~ $
((wb)
Phase angle a specific harmonic with the frequency we
(x(()
Variable that represents phase angles all harmonics for sample x(()
(x((--r)
Variable that represents phase angles for sample x(t - -')
(x(e) (wo)
Phase angle of the specific harmonic with the frequency wo for sample x(t)
T(x,0) ②)
The proposed transformation function
Phase difference between two angles
Adx(t)
Phase difference between the given angle in the transformation and the harmonic with frequency wo when sample x(t) decomposed using Fourier transformation
Adx(t-e)
Phase difference between the given angle the transformation and the harmonic with frequency wo when sample x(t - +') is decomposed using Fourier transformation
M
Manifold notation with an angle ø
M
A specific manifold with the angle é.
T(x,
The output of the transformation, i.e., time series that lies on the manifold M
F(T(xx0))
Fourier transformation of the output from the proposed transformation: Since the trans- formation function produces a tuple, we specifically apply the Fourier transformation to the first output, which corresponds to the time series.
fo()
Parametric mapping with parameter 0
Sa, :Rd Φ
The guidance network that outputs an angle to map the sample to specific manifold
fo, X Y
The classifier neural network
Var(.)
Variance function
%
Modulo operation
Table Detailed list of notations used in this work
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A.3 PROOFS
Lemma A.I (Circular Shift), Given a sample x in the interval [0, lina] and its shifted version x (t--1 ('), where the shift is a random value from the finite real numbers, i.e., l' E (-00, oo). The shift is periodic with the signal length tin leading to the same vector representation when the shift is an integer multiple of the signal length.
|0 -0c, 0= tt (mod lins) = x(1) x(t-1) 1)
Proof. From circular shift, we know
x x(1) x(1 (mod lint))
(15)
x(t t¹ x((t t)) (mod tint))
(16)
Therefore, 0 = t' (mod tixs) = x(t) x(t - t))
Throughout the proofs, the length of the samples and their intervals are denoted by the same variable t. In other words, the samples are assumed to start at = and finish at lint
A.4 PROOF FOR PROPOSITION
Proposition A.2 (Time shift as a Group Operation). Shift operation in time domain defines an Abelian Group of phase angles in the frequency domain for each harmonic with frequency Wk-
(#r. + mod 2x), where Φ; & = (δ(w)) + wxl') mod 2n, 1' R
(17)
Proof. Let x(1 + (') be randomly shifted variant of sample x(I) where l' E R. We can decompose these two sequences as in below using Fourier transformation.
FL.)
(XeeW))e and x(t ++) FQ) |X
x(t)
(18)
The phase values of these two time series for harmonic at the given frequency can be expressed as follows.
ex(() (wx) d(wk) and (x(t-e)(wx) ((w) + wat'
(19)
Then, we can define a set of phase angles 2x with shift values t'.
x = ( | Φ (d((w)) + wil') mod 2n, E R}
(20)
This set of phase angles with time shift operation defines the circle group T. The circle group is Abelian (Fuchs) 1960), with time shifts corresponding to multiplication in the complex plane. For completeness, we have shown all the group axioms.
GROUP AXIOMS
The set of phase angles with time shift, Φy. (Ø (Ø(wn) wul') mod 2n, l' RJ. satisfies the five axioms of an Abelian group under modular addition.
Axiom 1: Closure
For any two phase angles Φi, 22 E th. their sum is also in dx.
Let Φi = (Ø(wx) + wal') mod 2n and oi (Ø(wx) + wa(h) mod 2T. Then their sum is:
Φ + 02 (d(w)) + with + d(w)) ) + ww(h) mod 2n
Φ + 62 = (Ó(w)) + WR (t( + t2)) mod 2n
Since t + E R. the sum is also in Φ so closure holds.
Axiom 2: Associativity
For any three phase angles 91, d2, dh tr. their sum is associative under addition modulo 2m.
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((01 + 62) mod 2T) + 0s) mod 2n = (Ø1 + (62 + Øs) mod 2T) mod 2
Let Φ1 (Ø(wx) wxtt) mod 2x, d2 (Ø(wx) wxth) mod 2x, and Φ3 (Ø(wk) + wktz)
mod 2m. Then:
((01 + Φ2) mod 2x) + 03 = (d(wx) + wxtt + wwtt)) mod 2 + Ø3
= (b(wx) + wx(1's + 12) mod 2 + wwll)) mod 2n
Similarly, for the right-hand side:
Φ + ((02 Φ3) mod 2x) Φi + (Ø(wx) + wu(t' tt)) mod 2n)
(Ø(w)) + wxt' wa (t/ + t())) mod 2m
Using (a++b) mod 2n = ((a mod +6 mod 2x)) mod 2n, both sides simplify to (o(w))+
we (f), + 12 + Eg)) mod 2n. Thus, associativity holds under addition modulo 2T.
Axiom 3: Identity Element
The identity element in this group is the phase angle when no time shift has occurred, i.e., t' O.
Φu (Ø(wx) + wk mod 2n x(wk)
For any Φ1 E ti, we have:
Φ + 00 = 01
Thus, @o the identity element.
Axiom 4: Inverse Element
For any phase angle Φ1 (Ø(wx) + wkt') mod 2T,, its inverse is:
Φ = (-wil") mod 2n
Then:
Φ + -- (b(wx) + wht' - wytt) mod 2T = o(wk)
Thus, each element has an inverse.
Axiom Commutativity
For any two phase angles Φ1, &2 E tx, the sum is commutative:
Φ + Φ de + Φ)
This follows from the commutativity of modular addition, the group is Abelian.
Proposition 2.1 states that shift operation in time domain defines Abelian group of phase angles for each harmonic. Proposition [.] holds key role in our algorithm, as establishes an abstract connection between the time-domain shift operation and its effects samples in the frequency domain.
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A.4.1 PROOF FOR THEOREM 2.2
Theorem A.3 (Covering the Entire Time Space Injectively). Given a sample x, the defined function T(x, 6) & x R + le x A is bijective such that all shift variants of a sample can be covered with the unique phase angle of harmonic whose period longer or equal to the length of x.
Voaa Φb EΦ, T(x, Φa) =Txx, Φ6) Oa 6b
Vt' E R. 30 E 0, T(x, 6) = (x(t - Ao),
Proof.
x(t) F(.)
IX(elM)
(21)
x(t - t) F() |X (e
(22)
xx(() = <(w), (x(c--) = (b(w) wll
(23)
Qx(1) - xx((-11) = wt',
(24)
Using Euler's formula,
Qx(1) 0x(t-11 (wt') (mod 2m)
(25)
2T 1' (mod 2T)
xx(1) xx((--1)-
(26)
2n
VTb E T, 00 > To 2 t = Ox(e) - Oxic- (o(t-r)(wo)) t'
(27)
To
Thus,
(x())w0) - (x(1-19)(w0)) (wo) = (x(e)(wo) (x(1-<r)(u0) = =t=t
(28)
Also,
2T
Vo ET, <0 To = (x())(Wo) -xx(t-e)((us) =
mod 2n,
(29)
To
2T
3E' E R,
> 2
(30)
To
:. E'' ER, I, -(x(1) xx(t (wo) = 2x(1) (wo) &xc(-- 1' 1==1),
(31)
which proves injection. Similarly, for surjection using Lemma A.I
2TT
R. 0o <>> T>>>> |t'l = (x())(WO)) (x(e-119 (wo)
t
(32)
To
The final equation completes the proof by showing that the phase difference between the sample and its shifted version can get unique values for any shift, i.e., covering the whole time space.
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A.4.2 PROOF FOR THEOREM 2.3
Theorem A.4 (Guarantees for Shift-Invariancy). Given a sample and randomly shifted variant of it x(t - f), if the transformation, function f Txx, 6) is applied to both samples with the same angle Oo. the resulting time series will be the same while carrying the same information.
T(x(t), (x(t), Adx(o): T(x(t - -1)) (x((), (
Proof.
x(1) F0. {(el))))001) x(1 - 1!) J.) Ix (12"))
(33)
(x(() = o(w), 0x((-() (lw)) wt'
(34)
(x((-1) - Qx(1) = -wt!
(35)
Using Equation 11 the phase difference between samples (x(t), -t') and @u can be obtained as,
[db(w)) - da)
To. A (Ø(w))) - wot' øal To
(xx())
(36)
2x
2m
A0x(1-1) - A0x(1) = -wnt" To
(37)
2T
Using Fourier transform as in Equation
F(T(((t),o.) = X (e'²) (eSs(u) e- jiwdo((t)
(38)
- o.)) [e(e""))e)d(w) - jut -j
(39)
Given that the amplitudes identical, demonstrating equality in phase is sufficient, shown below, (T(x()))))) = <(w) wA)x(t), (T) = d(w) - wt' w wAxuu-
(40)
(T(x(t-1))6o) (T(x(()) -wt w w + w A
(41)
(T(x(=-1))()) (T(x(())0o) -wt w A Adx(t)
(42)
(T((((-1).LL) (T((())) =wt'' witt To 2
(43)
2nt To
(T(((-1))))) (T((())))) = -wt + w
1'
(44)
To 2T
(T(X(--1)8) - (T(()) = wtt + wt'
(45)
(T(((--)))))) (T(X().)))
(46)
F F(T(((t)) 0.)) (X(eew) [
(47)
F [T) (b)) = ( (eeewwlllrrrtteo th
(48)
Therefore, the output time series samples will be the same after applying the transformation.
We complete the proof by showing the phase and magnitude of Fourier transformation of both samples are the same after transformation even though random unknown shift is applied to the sample. Thus, the proposed transformation guarantees shift-invariancy without limiting the range of shifts.
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B ALGORITHM
In this section, we present the pseudocode and the PyTorrc (Paszke et 2019) implementation for the proposed transformation functiion Algorithm [ details each step of the transformation, which takes sample, X, and an angle, ø as inputs and outputs the transformed sample.
Algorithm Algorithm for the proposed diffeomorphism.
Input: x, Pa
2: Output: T(x, ②.)
Calculate the Fourier transformation obtain harmonies 3: [X(e)=)] (((d)) = - x(t)e-
4: ((w)) L (1x(e
Obtain the angle for the harmonic with period To
5: -d(w)) 6.] % 2T
(0-2x).To
i
6: AΦ=
b Calculate the phase difference between the harmonic and the angle
AwTh
else
2x
7: [X (e(e))) - Ix (e (e -ima
Apply linear phase shift each harmonic
8: Return: F
Below, provide the PyTorch implementation of our proposed transformation function, which includes two functions. The first function, di stannecao cale,, computes the phase differ- ence between the harmonic with frequency and the desired input angles. The second function, di oeffremorphii sm, performs the main transformation: takes as inputs the batch of samples x and the angles o, and outputs the transformed samples in the time domain.
def distanceCalculltte (angleDiff): :
theta = angleDiff $ (2 torch.pi)
Calculate the angular distance the unit circle theta > torch.pi]
torch.pi
return theta
def diffeomorphism (ssmomrrphismsmmmpe,,, desiredAngles)
B, 1,, sample. samplesFFT torch, frt. (sample, dim-1) freq = terch. fft. rfft ffrth.fft (n=l.) phAngle torch. (samplesFFT)
Get the phase angle of the harmonic with frequency
angles phwngle/teeeeeeeeeepppeepppppppa atange siaeanggeepa 1, 0 squeeeee Calculate the angle difference
theta - distanceCalculate (andeisssee-ssseeeddnnleelleee
Normalize it to the sepecific harmonic with frequency w_0
dtheta theta (2.torch.pix<rre([]]
Create complex exponentials with specific phase values
linShift = torch. (-1)2 piwfrrg pi*freq[None, [None, xdtheta [: None])
linShift = linShift unsqueeze (dim=2) expand (-1, -1, D)
Apply linear phase shift to all harmonics shiftedFFT Linshift+ (sampleeEET))))))
Return to the time domain cransformedSamples torch. fft. irfft (shi ftedeeT, n-l., dim=1) return transformedsamples
Implementation 1: PyTorch implementation of the proposed transformation
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C EXPERIMENTS
Here, we give a detailed description of datasets, architectures, metrics, and training details for our experiments. We performed our experiments on NVIDIA GeForce RTX 4090 GPUs, involving training with three random seeds for all datasets, totaling upproximately 480 GPU hours including ablation. We reported the mean of three runs with the standard deviation.
C.I DATASETS
In this section, we give details about the datasets that are used during our experiments. Overall, we have used eight datasets with six different time series tasks including heart rate prediction, cardiovascular disease classification, activity recognition, step counting, sleep stage classification, and lung sound classification from six sensor modalities photoplethysmnograpyyy electrocardiogram. inertial measurement units, elecrrenncephalogrppyy and audio. When selecting datasets for our experiments, we prioritized signals that provide meaningful insights into individuals' mental and physical health, where robust inference is particularly critical. Therefore, we specifically choose signals generated by humans. Additional to main experiments, in appendix, we also included lung audio classification from audio signals.
C.I.1 HEART RATE PREDICTION
IEEE SPC The IEEE SPC dataset overall has 22 recordings of 22 subjects, ages ranging from 18 to 58 performing three different activities (Rocha et al. 2020). Each recording has sampled data from three accelerometer signals and two PPG signals along with the ECG data with sampling frequency of 125 Hz. All these recordings were recorded from the wearable device placed on the wrist of each individual. All recordings were captured with two 2-channel PPGs with green LEDs, tri-axial accelerometer, and a chest ECG for the ground-truti HR estimation. We averaged the two channels of PPG for prediction. We choose the last five subjects of SPC22 be used for source domains. Throughout our experiments, we used PPG channels without integrating any inertial measurements.
Dalia The PPG dataset for motion compensation and heart rate estimation in Daily Life Activities (DaLiA) was recorded from 15 subjects (8 females, 7 males, mean age of 30.6), where each recording was approximately two hours long. PPG signals were recorded while subjects went through different daily life activities, for instance sitting, walking, driving, cycling, working, and so on. PPG signals were recorded at a sampling rate of 64 Hz. The first five subjects are used as source domains, similar to Demirel & Holz 2023).
We standardize all PPG datasets as follows, same as the previous works (Biswas et al. 2019). Initially, a fourth-order Butterworth bandpass filter with frequency range of 0.5-4 is applied to PPG signals. Subsequently, a sliding window of 8 seconds with 2-second shifts is employed for segmentation, followed by z-score normalization of each segment. Lastly, the signal is resampled to a frequency of 25 Hz for each segment. We used an 8-block ResNet model with a stride of 2, a learning rate of 44-4, and a batch size of 32. Additional results with further experiments can be found in Appendix D
C.1.2 HUMAN ACTIVITY RECOGNITION
UCIHAR Human activity recognition using a smartphone's dataset (UCIHAR) (Anguita et al. 2012) collected by 30 subjects within the age range of 16 to 48 performing six daily living activities with a waist-mounted smartphone. Six activities include walking, sitting, lying. standing, walking upstairs, and walking downstairs. Data is captured by 3-axial linear acceleration and 3-axial angular velocity at a constant rate of 50 Hz. We used the pre-processing technique the same as in (Qian ct al.) 2021) such that the input contains nine channels with 128 features (it is sampled in a sliding window of 2.56 seconds and 50% overlap, resulting in 128 features for each window). Windows are normalized to a mean of zero and unit standard deviation before feeding it to the models. The experiments are conducted with one of the domains is chosen to be the unseen target (Qian et al., (022). We used an 8-block ResNet model with stride of 2, learning rate of 3e-3. and batch size of 32.
strategy with the first five subjects, where
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HHAR Heterogeneity Dataset for Human Activity Recognition (HHAR) is collected by nine subjects within an age range of 25 to 30 performing six daily living activities with eight differ- ent Smartphones-Allhowee
well, we use data from smartphones- were kept in a tight pouch and carried by the users around their waists (Stisen et al. 2015). Subjects then perform six activities including cycling, sitting, descending stairs, ascending stairs, standing, and walking. Considering the variable sampling frequencies of smart devices in the HHAR dataset, we downsampled the readings 50 Hz. We employed sliding windows with lengths of 100 (two seconds) and 50, using specified step size. These windows were then normalized to a mean of zero with unit standard deviation. In our experiments, we utilized the data from the first four subjects (i.e., a. b. C, d) as source domains, following a similar approach to previous papers (Qian et al. 2022 Demirel & H 2023). We used an 8-block ResNet model with a stride of 2, learning rate of Te- 3, and a batch size of 64. The learning rate 1 for the guidance network, was the same for the activity recognition task.
HHAR includes data from smartwatches
C.1.3 CARDIOVASCULAR DISEASE (CVD) CLASSIFICATION
Chapman Chapman University, Shaoxing People's s Hospital (Chapman) ECG dataset which pro- vides 12-lead ECG with a 10-second sampling rate of 500 Hz. The recordings are downsampled to 100 resulting in each ECG frame consisting of 1000 samples. The labeling setup follows the same approach as in Zheng et al. (020) with four classes: atrial fibrillation, GSVT, sudden bradycardia, and sinus rhythm. The ECG frames are normalized to have
of 0 and scaled to have standard deviation of 1. We split the dataset to 80-20% for training and testing as suggested in /Zheng et al. (2020). We chose leads I, II, III, and V2 during our experiments for both ECG datasets.
PhyioNet 2017 The 2017 PhysioNet/CinC Challenge aims to classify, from 8,528 single-lead ECG recordings (between 30 and 60 in length), whether the recording shows normal sinus rhythm, atrial fibrillation (AF), an alternative rhythm, or is too noisy to be classified, i.e., four classes. We normalize the signals to have zero mean and unit standard deviation. Additionally, zero-pad pad the shorter recordings to ensure they have the same length. We split the dataset into training, validation, and test sets according the patients using 60, 20, 20 configuration.
For both datasets in CVD task, we used an 8-block ResNet model with a stride of 2, a learning rate of 5e-4, and batch size of 32. The learning rate for the guidance network set same as the main classifier architecture.
C.1.4 STEP COUNTING
The Clemson dataset has 30 participants (15 males, 15 females), Each participant wore three Shim- mer3 sensors. We used the IMU sensor readings from non-dominant wrists to predict step count where each sensor recorded accelerometer and gyroscope data 15 Hz. We calculated the total magnitude of the accelerometer and fed it to the model as pre-processing without any filtering. We used window lengths of 32 seconds without an overlap in the regular walking setting. We conducted 10-fold cross-validation, with each fold consisting of 3 subjects for testing and validation. And, six randomly selected subjects were used for training in each fold. We used a layer of FCN architecture, a learning rate of 5e-4, and batch size of 64. The learning rate for the guidance network was set same as the main classifier architecture.
1.1.5 SLEEP STAGE CLASSIFICATION
We used the Sleep-EDF dataset which has five classes: wake (W), three different non-rapid eye movements (N1, N2, N3), and rapid eye movement (REM). The dataset includes whole-night PSG sleep recordings, where we used single BEG channel (i.e., Fpz-Cz with a sampling rate of 100 Hz. We employed the identical data split as presented in the paper (Eldele et al. 2021). accessible online, without applying any additional pro-psscs steps. of 2, a learning rate of 1e-3, 3, and a batch size of 64. We ran three distinct seeds using the same split and reported the mean and standard deviation the test set.
used 16- ResNet model with stride
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C.1.6 LUNG SOUND CLASSIFICATION
We used Respiratory @ TR which contains lung sounds recorded from left and right sides of posterior and anterior chest wall and back using two digital stethoscopes from 42 subjects collected in Antakya State Hospital (Altan et al.) 2017). The 12 channels of lung sounds are focused on upper lung, middle lung, lower lung and costophrenic angle areas of posterior and anterior sides of the chest. The recordings are validated and labeled by two pulmonologists evaluating the collected chest X-ray, PFT and auscultation sounds of the subjects. Labels fall into 5 COPD severities (COPDO, COPDI, COPD2, COPD3, COPD4). The patients aged 38 to 68 are selected from different occupational groups, cocio-cconomic status and genders. We performed 10-fold cross-validation data from 42 subjects, with one fold reserved for validation in each iteration. All 12 channels, combining left and right chest wall recordings from six channels each, were used.
The audio was segmented into 8-second windows with a 2-second overlap to capture temporal patterns. We used an 8-block ResNet model with stride of 3, learning rate of 1e-5, and batch size of 15. Although audio models typically use Mel-frequency spectrograms with different networks, we did not observe a significant performance difference between these models and our initial model. Therefore, we used the original model with temporal data, consistent with our main experiments.
C.2 METRICS
We used the common evaluation metrics in the literature for each task. Specifically, we used mean absolute error (MAE), root mean square error (RMSE), and Pearson correlation coefficient (p) for heart rate prediction. We used accuracy (Acc), macro-F1 score (F1) for activity recognition, and an additional area under the receiver operating characteristic curve (AUC) for cardiovascular disease classification (Kiyasseh al. 2020). We used the mean absolute percentage error (MAPE) for step counting (Yang et al.) 2020; Femiano et al. 2022). For lung audio classification, we evaluated performance using accuracy, macro F1, and weighted F1 scores (F1, W-F1). For sleep stage classification, we used the same W-F1)-along with Cohen's 's Kappa coefficient (x) (Cohen (960).
macro FI, and weighted Fl scores (F1,
C.3 BASELINES
C.3.1 LPF (BLURRING)
In convolutional neural networks, pooling layers or convolutions with strides greater than 2 would conduct downsampling on the feature maps reduce their size. However, since features are averaged or discarded in the downsampling, information may be lost, i.c., aliasing. Traditional pooling aggregates all values within the window to single value. In contrast, Zhang (2019) aims minimize information loss caused by pooling via replacing the traditional kernel with a Gaussian kernel as an low-pass filtering (LPF). Specifically, LPF applies a Gaussian- weighhted function to the neighborhood values around each feature for convolution, and obtains a weighted average result.
Compared to the pooling operations of simply selecting the maximun/average value within the window, LPF uses Gaussian-weighted averages to preserve the relative positions and spatial rela- tionships between features. In our implementations, we used ID version of LPF with a length of 5. We evaluated filter lengths of 3, 5, to determine the optimum size for each task. During our experiments, we observed the best performance with a filter length of 5, except for HR prediction, where length of was optimal.
C.3.2 APS
To make the sampling layer invariant shifts, Chaman & Dokmanic (202))
proposed subsample by
partitioning feature maps into polyphase components and select the component with the highest norm. This approach has a significant limitation when applied to time series, especially with the nonlinear activation functions. It tends to overlook variations in boundaries arising from the translation of samples, thereby imposing additional shift constraints (Rojas- Gomez et al. 2022). Consequently, the evaluation of these methods is restricted to a narrow range of shifts, covering only limited subset of the shift space. Moreover, as this approach selects the component with the highest norm, it requires feature maps to have unique values.
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C.3.3 WAVELET NETWORKS
Wavelet networks (Romero et al, 2024) consist of several stacked layers that respect scale and translation. At the beginning, the network consists of a lifting group convolution layer that lifts input time-series to the scale-translation group, followed by arbitrarily many group convolutional layers. At the end of the network, global pooling layer is used to produce scale-translation invariant representations. Wavelet Networks are proposed for equivariant mappings between input and output. However, to turn equivariant network into invariant network, extra layer that is equivariant in this degencrate sense (in practice, this often means either averaging or creating a histogram of the activations of the last layer) should be applied (Kondor & Trivedi (018). For example, the well-known wavelet scattering network achieves invariance by stacking equivariant layers followed by a final invariant one in that of scattering networks (Mallat 2012). However, our proposed method does not require an additional layer as it operates on data manifolds directly.
We used the original GitHub (https: / t.ub. com dwrome ro wraveee__re works) im- plementation, leaving the dropout and base parameters unchanged. We searched over the learning rate (e.g.. {10-3 10 1000000 ()) using the validation for each time series task.
C.3.4 CANONICALIZATION
We also compared our method with a canonicalization approach which based on learning mappings to canonical samples. Specifically, we used the equiadapt library (https: / //gitttul hotmarraa39////ooo (atnnab3///qiadaattt.. equuadapt) (Mondal et al., 2023] Kaba et al., (023) while adding translation equivarant architectures for shift operations. For translation equivariant canonicalization architecture, we implemented a convolutional neural network with a kernel size of five and three layers. The network avoids pooling layers prevent aliasing and instead employs global average pooling at the final stage to aggregate information across the signal length. The output is reshaped separate the channels corresponding each discrete translation, followed by aggregation over the fiber channels to produce wranslation-cquivarinn activations.
We used a discrete Group representation with number of translations set 16. The canonicalization network and classifier were trained jointly, incorporating a prior regularization loss to guide the learning process. For optimization, we employed the Adam optimizer with learning rate of 1 x 10-3 and no weight decay. We also performed grid search to identify the optimal learning rate for the canonicalization network; however, no significant performance improvements were observed across different learning rates.
C.4 IMPLEMENTATION DETAILS
Here, we have provided the details of the architectures, and hyperparameters. Primarily, we used the ID ResNet (Hong et al. 2020) implementation in the supervised settings. While some alternative deep learning models can perform better in time series such the combination of convolutional and LSTM layers, similar to previous works in shift consistency (Zhang 2019), we focused on deep learning models which mainly composed of convolutional layers.
C.4.1 ARCHITECTURES
Here, we present the details of architectures that are investigated for the performance of shift-invariant techniques. Some details that are not given in the tables are as follows. Batch normalization (loffe & Szegedy (013) is applied after each convolutional block. ReLU activation is employed following batch normalization, in line with (He et al. 2015). We also applied Dropout (Srivastava et al.) 2014) with 0.5 after each activation and before the convolutions. Finally, a global average pooling is implemented before the linear layers.
Tables 9 10a and 10b give an overall for the architectures with the number of parameters for each dataset. From these tables, it can be observed that the number of parameters for the guidance network is much less than the main classifier, where the ratio is close to 4%.
The parameter count of the guidance network could be further reduced by selectively inputting only the important frequencies or the frequency band for each time series task, which can determined using prior knowledge, rather than the entire spectrum. Nevertheless, for the sake of consistency,
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Table 9: ResNet architecture
Repetition Layer
Kernel Size Output Size Stride
1
Input (C,T)
(C. T)
Conv
(5, 1)
(64, T/2) i
Residual Block
R
Conv
(5. I) (5, 1)
(128, T/4)
S
Conv
(128, T/4) 1
Linear
(n_ cl_classes))
# Parameters for dataset (C,T, R. S)
IEEE SPC T=200, R-8, S=2)
211k
DaLiA (C=1, T=200, R=8, S=2)
2110
Chapman (C=4, T=1000, R=8, S=2)
197k
PhysioNel (C=1, T=6000, R=8, S=2)
Rs 197k
UCIHAR (C=9, T=65, R=8, S=2)
2000
HHAR (C=6, T=51, R=8, S=2)
200k
Respiratory (C=12, T=6000, R-8, S=3) Sleep (C=1, T=3000, R=16, S=2)
2000
#3.2M
Table 10: The model topologies of the classifier fc and guidance network fG
(a) FCN architecture
(b) Guidance architecture
Layer
Kernel
Output
Layer
Kernel
Output Size
Size
Size
Size
Input (C,T)
(C, T)
Input (C,T)
(C, T)
Conv (32 kernels)
(8. 1)
(32, T-4)
Conv (4 kernels)
(8. 1)
(4, T-5)
Max Pooling
(2,1)
(32, (T-4)2)
Max Pooling
(2,1)
(4, (T-5)++1)
Cony (64 kernels)
(8, 1)
(64, (T-4)/2-4)
Conv (16 kernels)
(5, 1)
(16, (T-5V2-1)
Max Pooling
(2,1)
(64, (T-4)/422)
Max Pooling
(2,1)
(16, (T-5)4+1)
Conv (128 kernels)
(8, 1)
(128, (T-4)/-66)
Conv (32 kernels)
(3. 1)
(32, (T-5)4+1)
Max Pooling
(2,1)
(128, (T-4)/8-3)
Max Pooling
(2.1)
(32, (T-5/8+1)
Linear
(n_classes.)
Linear
(n (n_classes.)
# Parameters for dataset (C,T)
# Parameters for dataser (C,T)
Clemson (C=1, T=240)
s3432k
UCIHAR (C=9, T=65)
22.5k
HHAR (C=6, T=51)
22k
Clemson (C=1, T=240)
23k
IEEE SPC T=200)
22.4k
DaLiA (C=1, T=200)
22.4k
Chapman T=500)
=6k
PhysioNel T=3000)
Rv 13k
Respiratory (C=1, T=1500)
Sleep (C=12, T=6000)
888
perform the Fourier transform with the number of harmonics same the length of time series and provide the entire spectrum to the guidance network as the input.
One advantage of this input modeling is that the ratio between the number of parameters for the guidance network and the main classifier decreases more when the main classifier has more blocks as they independent. For example, the guidance network has 1000 fewer parameters than the main classifier for the sleep stage classification task where we have used 16 ResNet blocks for the classifier model for all techniques. At the same time, this addition increases the performance metrics up to 3%. Furthermore, increasing the number of parameters can decrease the performance of the models as it can cause overfitting of the training data (Cao al., 2022 Wen al.) (023), which is observed in our case as well (see Table [1 in Appendix D for additional results).
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D ADDITONAL RESULTS
D.1 SLEEP STAGE CLASSIFICATION
Here, we present extended results for the sleep stage classification in Table 11 include the Fl score as an additional metric and employ a larger network to observe impact. Although our method ranked second in Fl metric, it is important to highlight that Fl scores are
Specifically,
Table 11: Performance comparison of ours with other methods in EEG for sleep stage classification
Sleep-EDF
Method
S S-Cons
Acc f
Fl f
W-FI f
R1
Baseline
95.06:0.61
75.41+2.01
65 1.333
1..87=1.92
67.12+2..6
Baseline (2x)
91.09+1.26
73.88+2.10
65.84+3.29
74.32+22.86
65.1442994
Aug.
99.00110.17
74.8911.11
64.71111.55
74.03.11.46
65.8911.81
LPF
9..43
73.56:22.9
68.08:1197
76.01
65.68=3.46
APS
Ours
100.10.00
77.9011.92
67.01
76.7712.58
7..0
Ours+LPF
100=10.00
13.12
17.42:11.99
75.34+1.61
64.98
biased measure of classification quality (Christen et al., 2023 Powers 2015), which is problem when comparing recordings with different prevalence of the classes as sleep staging (Malafeev reported additional metrics for measuring the performance. In particular, we included the kappa score, a metric widely used for evaluating algorithms this task (Malafeey al. 2018 Biswal et al.) 2018). Additionally, sleep stage classification and certain other tasks have empty APS values as the authors' original implementation encountered overflow issues with matrix repetition, particularly for longer arrays.
al.
2018). Therefore,
Overall, our proposed method demonstrates a significant performance improvement three of the metrics, with high kappa and accuracy-both of which commonly used in the medical domain [Biswal et al. 2018) while ranking second in the Fl metric, following the LPF approach. We also performed the same ablation experiments for investigating the behavior of the loss function on the performance of the model for the sleep stage classification and reported the results in Table [2 while excluding the consistency metric as the model that include the proposed transformation are always completely shift-invariant.
Table 12: Ablation experiments for sleep stage classification
Sleep-EDF
Method
Acc
F1
W-FI f
R 1
T(x,
75.54:2..9
1..96
75.53=2.29
67.08+0.03
1..21+1.51
67.67+1.67
76.89=1.71
69.39+0.02
Le
17.75+1.23
68.04+1.16
77.01=1.07
69.94 19.94400.01
Ours
1..8
67.0112.65
76.77=2.58
70.0112.50
Change
+2.26
40.05
+1.24
+2.93
Table 12 also supports the previous experiments and claims regarding the advantages of guiding the proposed diffcomorphism with neural network. When the models are trained by guiding the mapping function, the performance of the models increases up to 3% in Kappa (x) score.
D.2 HEART RATE PREDICTION
Here, we conducted additional experiments to evaluate the models' performance using varied amounts of training and testing data. Initially, reduced the training data while increasing the testing data by dividing the datasets half based on subjects to investigate the performance with less training data. Table [3 presents the results, where proposed method also increases performance by 3-4% compared to the baseline architecture while reducing the variance between different runs and improving shift consistency by 40-60%, even in the low data regime.
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Table 13: Performance comparison of ours and other methods PPG datasets for HR estimation
IEEE SPC22
DaLiA
S-Cons ($) T
RMSE 1
MAE
((6)
S-Cons (9)
RMSE
MAE 4
p ((S) 1
Baseline
55.70=2.61
25.93=0.07
11.3550.08
48 85
31.48:0.41
12.51=0055
5.59+0.05
85.37+0.20
Aug.
70.35=0047
26.21-0.48
53.38=00229
12.31=0009
85.90+0011
LPF
67.43=20.5
25.87=1002
14.03=0.69
47. 22.7622-49
39.82 19.82211.33
12.65=0.09
5.81:0.01
84.874:0.25
APS
60.43=1108
24.83=1.26
11.14=0.83
52.10=2.90
38.99=0.85
12.49=0.11
5.61#0.04
85,68+00117
Ours
100=0.00
24.67=00.6
52.26-00.77
100=00.00
12.30=0.11
5.57:40.33
85,95+0.22
Ours+LrF
100=0.00
26.01
14.04=0.24
47.20 0..26
100:00.00
12.78:00.09
6.18:00.01
84. 84.48:1:0.26
OuTS+APS
100=00000
24.67=0.32
11.38=0.15
51.6420.73
100=0.00
12.40=0.08
5,67#0.02
85,63=20.0
We also performed the same ablation studies for heart rate prediction task in the low data regime and presented the results in Table 14
Table 14: Ablation experiments for HR task with less training data
IEEE SPC22
DaLiAppO
Method
MAE
RMSE
PT
MAE
RMSE
p T
T(x,)) 6)
12.04
25.49
50.72
6.10
12.94
85.03
Co
11.29
25.10
51.10
5.63
12.80
85.10
Lo
11.32
25.08
51.18
5.60
12.71
85.12
Ours
11.10
24.67
52.26
5.57
12.30
85.95
Change
+0.94
+0.82
+1.54
+0.53
+0.64
+0.92
The ablation study results with a smaller training set align with our main findings, where reducing variations uniformly using the same angle negatively impacts performance. Likewise, expanding the potential solution space with LG leads to performance decline compared method.
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D.3 DIFFEOMORPHISMS IN DEEP LEARNING
In this section, we review previous transformation functions and diffeomorphisms used in deep learning models. Invariant classification of input samples with neural networks has a long-standing history, with the Spatial Transformer Network (STN) being introduced to learn transformation functions for invariant image classification (Jaderberg et al. 2015). Similarly, Temporal Transformer Networks (TTN), an adaptation of STNs for time series applications, were introduced to predict the parameter of warp functions and align time series (Lohit et al. 2019, Shapira Weher Freifeld 2023). Recent methods have focused on optimizing known family of diffeomorphism, known as diffeomorphic warping functions (Martinez et al.) 2022) for time series alignment, through deep learning (Detlefsen et al. 2018 Shapira Weber et al. 2019). Thus, significant distinction between our approach and previous techniques lies in the fact that we introduce a novel tailored diffeomorphism that is capable of mapping samples subjected to shifts to the same point in the nigh-dimensional data manifold, to ensure shift-imm innariiinaiii.c...
Although previous techniques were designed for different purposes, such time-warping (Lohit et al. 2019), we also evaluated and compared the performance and shift consistency of their transformation functions. Specifically, we implemented sequence temporal transformations (STN) for clinical time series from Oh et al.] (2018) and TTN (Lohit et al., (2019) where the transformation and the classifier trained together to maximize classification performance by minimizing the cross-entropy loss for both methods. In our implementation of STN, we followed the original design, using neural network with two convolutional layers and two pooling layers, where pooling applied between and after the convolutions. After the pooling operations, two fully connected layers are applied to the resulting feature maps to obtain the transformation parameter. The overall results time series tasks are presented in the tables below.
Table 15: Performance comparison of our method with different transformations in HR estimation
Method
IEEE SPC22
DaLiA
S-Cons (%) 1
RMSE
MAE L
p (%) f
S-Cons (%)
RMSE |
MAE L
(S8
Baseline
61.99:11.19
18.39+2.96
10.284141
62.64415.7
32.08:0022
9. 9.86:2023
4,40:0..3
86.01=0.51
Aug.
76.48:t1.77
18.73+1.115
10.42:0:40
64.0643.70
$2,77+0.39
9.85-+021
4.47:0006
85.9 85.99920:49
Buseline+IIN
67. 67.13:1155
18.4512.73
10.35:1.73
63.4911..10
44.8111:25
9.90
4.4310.04
85.96.00.47
Buseline+TTN
60.12+1..10
20.574-1.25
11,55+187
60.174:3.6
19.1340330
10.23-+0.00
4.45+00.55
84.31=0067
Ours
100::0.00
16.25:00.7
9.45-00.3
70.12+2.10
100::0.00
9.75-=0.15
4.39::0.03
86.06-20.19
Table 16: Performance comparison of our method with different transformations in ECG datasets
Method
Chapman
PhysioNet 2017
S-Cous (%) 1
Acc f
F11 1
AUC (56)
S-Cous (%) 1
Acc
F1
AUC
Baseline
17
91.32=0023
91. 22+00.22
98. 14
15
83,22+0.72
73.50=1199
93.21+0.30
Aug.
99.0010.16
91.9620.19
01.8910.22
98.45:0.188
98.9620.17
82.28:1.18
12.32=2.20
93.2010.12
Baseline+STN
98. 13
91.45-0.20
91.33 19
98.31
98.5540.14
83. 83.12-0.50
73.27=1544
93.2344028
Baseline+TTN
07.60:20.1 15
91.27=0011
90. 90.54-2038
98. 08.23-10.21
97. 02:0222
82.51=0063
41441=143
03.07-10335
Ours
100:-0.00
92.10::0.25
91.93:005
98.47=0.15
100:4000
83.
74.12=1.00
93.28+
Table 17: Performance comparison of method with different transformations in IMU datasets
Method
UCIHAR
HHAR
Clemson
S-Cons (%)
Acc f
Fl
S-Cons (%)
Acc
Fl
S-Coms
MAPE
MAE
Baseline
94.0711.38
85,
83/2011294
08.37_1033
91.871136
91.1611.38
54.31.14.00
4.76100.1
2.74-0088
Aug.
96.5540.30
85.42=4.50
83.69
98.38
91.972044
91.31
61.01+488
4.08=0.14
2.29=0077
Bascline+STN
1.226.11.22
83.5712.14 13.22-1.23
98.301024
889211.10
19.1011.20
58.5611478
4.9410.13
2.50.0.00
Buseline-TTN
03.32
83.27=1.57
82.78::3.12
97,10=077
90.03++1.4
90.18=1.10
43.89223322
5.43:10.20
2.89-0118
Ours
100+00.00
87.71
17.71-1.00 1.08
85.67+2..47
100--0.00
91.93-1.14
91.12-1.03
100-0000
4.28--0.34
2.43-0.21
As shown in the tables, other transformation functions fail to achieve true thift-invrrance While the STN method shows some improvements on certain datasets, it still lacks full bhift-invariance. This limitation arises because STN relies on a neural network to estimate transformation parameters from time series data. However, this temporal transformation has no information about the position of the time series. Thus, when the input is shifted, the output changes.
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Published as a conference paper at ICLR 2025
D.4 EXPANDED COMPARISON
Here, we compared the performance of techniques when random data augmentation (Aug.) is applied during training. In other words, the samples are randomly shifted and fed to the models during training. During inference, the performance of methods with original samples is evaluated without applying any shifts. We present the results in the tables below.
Table 18: Performance comparison of our method and other techniques with data augmentation for
HR estimation
Method
JEEE SPC22
DaLiA
S Cons (%:
RMSE
MAE
(S6)
S-Cons (56)
RMSE, +
MAE.
((%)
Baseline
11.99-1.19
18. 2.99
10.28+141
62.64-5.74
32.08++0.22
9.86++0.23
4,40+0.0
86.01+0055 51
76.48=1.77
18.7311.15
10.4210.40
64.0613.70
52.7710.39
9.8510.21
4.4710006
85.99:1041
LPF
08.83
20.20+1154
13.44=082
65.40=1.92
38.67400300
10.01
4.6740.12
85.68=40.51
73.99-1.06
19.4240.00
12.98=0799
65.27-1155
44.33-+0.1 16
10.45=0440
5.0
84
WaveletNet
51.71=1.95
2..5611.01
14.6110.34
60.74:14.77
36.7113..11
15:46.0644
7.6710.23
1..13.11.86
LPF + Aug.
76.17=1.115
10.39
13.22.00.36
61.7242.87
55.62.10.21
12.57.0.19
6.0210.11
84.9410.40
APS + Aug.
74.88-0.61
18.01+0.115
10.57=00.15 15
66.40+2221
53.70+0.00
12.84+0.10 10
6.08+-0:22
85 85.62-40.65
WaveletNet Aug.
50. 14:1144
20. 1.11
13.41
61. 61.90:33550
36,7113.04
15.46200.4
7,67:0225
76. 13:1 86
Ours
100::0.00
16.25::0.72
9,48:10.03
70.12:22110
100::0.00
9,75
4.39:0.03
86.06:10.19
1001:0.00
20.34:11.62
13.77=00.4
65.60=22.31
100+00000
10.7240.11
5.3010.03
84.12-10.23
OUSS+APS
10010.00
18.81.1.59
12.3240.84
67.01
100100000
10:47:00009
5.1010.03
84.62.10.31
Table 19: Performance comparison of ours and other techniques with data augmentation in ECG datasets for CVD classification
Chapman
PhysioNet 2017
Method
S-Cons (%)
Acc
f
AUC (#61
S-Cons (%)
Fl
AUC
Baseline
98.53+0.11
91.322+0.25
91.22-0.24
98.374+0.15
83.22-0.72
13.50+1.99
Aug. LPF
19
0.289-0.22
98 18:66:111
82 22.28-1.18
72.32+2.20
93,20-10.42
98.6910.11
92.0110.23
91.94-0558
08.50110.24
98.9410.399
84.40_0.16
75.6810.76
93.80.10.32
APS
98.60++0.17 17
90.69+0.89
89 89.44-1.00
98.31+0024
WaveletNer
91.02::.14 14
90.87:11002
90.02 1.00
97,94410.21
65 65.03:10.71
76.06=0064
63.3513.40
87.02:10.29
LPF + Aug.
98. 200
92.05=0013
91.94-0.40
98.53==0.20
99.00+0.50
83.27=0661
74.03+1.56
93,20+0031
APS Aug.
98 00:10011
90.69:10.89
89 1.:04-1.00
98, 98,31:00.24
WaveletNer Aug.
91,02&1.14
90.87:11002
90.02-1.00
97.9410.21
65.03-0.711
78.90=0.57
65,881
88.67.10.22/
100000.0
92.10:10.25
0.893-0.85
98.47.00155
100:10.00
63.15-0065
93.28.10.31 1..12
Ourse
91.96-0.54
+0011
100+0000
85.20-0.40
77.50+1.21
94.20-H0.19
Ous+APS
100.00.00
91.6111111
01.50_00.6
98,36110.00
Table 20: Performance comparison of our method and others with data augmentation in IMU datasets for Activity and Step
Method
UCIHAR
HILAR
Clemson
S-Cons (%))
SoCons (86)
(%)
MAPE
Baseline
94.
14.31
4.7610.11
2.7410088
96. 96.55e0.30 40
85.4 45.42:4.50
23.6926-74
98. 28..8841:2
91.9759.44
91.31=0040
61.01:4885
4.0820114
2.29=1107
LPF
95 05:05100:11
83.661344
81 01.0814221
9% 10100.00
92. 10:000000
91.43:1044
59.77:4400
4.16:10
2.3510.11
APS
96.40 96.40:20.08
81 81.75=4.11
79.01
98. 28.30211.24
91,832135
91.01=1.47
45.50=1.69
4.7420
2.69::1077
WavoletNer
94, 14.56-1:::1
82 62.78:4466
80.73455.5
96.76+0.115
50.7240038 38
90.71:0.30
59.14:3.10
5.20-1066
2.95-0.41
LPF + Aug.
97.65+1 30
84.67=3.445
83.3243.50
98.65e00112
92.45:07M
91.70=0:2
59.77=4.40
3.81
2.23-00100
APS Aug.
96 40:::003
40:53.5
988710.34
92. 42.4011:40
91.49:0.73
45.50-2.69
3,410000
2.500:0007
WaveletNer Aug.
22.8224.62
80.73
96.7620.15
90.726038
91.71:0.29
60.23-22.5
5.16:1102
3.02:0448
100:10.00
10010.00
11.12:1:00
100.0000
2.431021
OuP- Ours-APS
100:2000
84.78=2.44
82.58:2262
100:0000
92.51
91.80=0062
100:1000
3.75:0033
2.12:0118
100:00.0
82 96:11.77
81. 11.10-21:77
100:0000
91. 31.38:00322
90..64:0.32
100:0.00
3.87:50.19
2.19::0.11
From the results, we can see that applying shift data augmentation during training did not consistently improve performance, and in some cases, it led to a decrease. For instance, in the HR prediction task, training the low low-pass filtering method with randomly shifted samples resulted in lower performance. We believe that random shifts may reduce the inter-class separation between samples, causing them to overlap in the feature space (Wang et al. (022).. As a result, even though the number of training samples increases with augmentation, this reduced separation can lead to decline in model performance.
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Published as a conference paper at ICLR 2025
D.5 PERFORMANCE IN DIFFERENT HYPERPARAMETERS
In this section, we vary the training batch size to evaluate the performance of the proposed method. Given that the guidance network includes additional loss function reduce angle variance within a batch, we examined its performance under different batch sizes. The results are presented in
Tables 21 22 and 23
Table 21: Performance comparison of our method and other techniques with different batch size for HR estimation
Method
IREE SPC22
DaLiA
S-Cons (%)
RMSB. +
MAE
(())
S-Cons (%)
RMSE
MAE.J.
((6)
Baseline
6..85
19.28
11.51-0.21
65.1511.43
36.34.0.39
12.47.1..17
5.53.10.06
85.65.10.20
Aug.
78.90-1.12
19.51
11.82-0.49
63.62:11.5
52.77-10.99
12.36.10.09
5.61
85.84.10.16
LPF
1..16=1.13
1.26
13.86-1.06
65.22443.06
42.08
12.82:00.26
5.88+0.07
84.50++0.47
APS
71.25:11199
19.79
12.59-00.8
67.05.0.85
42.01.10.26
12.37.10.18
5.50.0.04
85.75
Ours
100110.00
18.29
11.30-0.57
69.10.1.20
100.10.00
12.49.0.116
5.55.10.07
85.60.10.23
OursaLiv 100+00.00
20.49-00.7
14.18-0442
67 67.28-33.20
100-:0:00
13.17=0023
6.43+0.11
83.63440.57
OurS+ARS
100=-0.00
20.03::0.95
66.40:44.03
12.59.0.17
5.76:0.05
85.22440.7
Table 22: Performance comparison of ours and other techniques with different batch size in ECG datasets for CVD classification
Method
Chapman
PhysioNet 2017
S Conos (%)
Ace 1
f
AUC (%i)
S S-Cons (%)
Acc
F1 F
AUC
Baseline
99.00:20.13
92.56:1..25
91.58:0.26
98.59=20.12
98.41:0026
82.71=1.55
13.16933,39
93.00=0.330
Aug.
99.08-10.16
92.4610.188
91.45.0.16
98.57-10:20
98.73-10.07
82.0311.60
12.61
22.63:10.51
98.88:10.17
92.32:10.15
91.3120.17
98.58:10.14
98.92:1.400
83.0741130
73.9813.85
93.60=0.70
Ours
100::0.00
92.58++0.26
91.50+0.30
98.59-0.10
100-00.00
83.14
74.40+22.28
93.60-0:20
Ours+LPF
100:0.00
92.28:t0.25
91.3420.27
98.5420:20
100+0.00
83.01:1101
72.94=2274
93.0 93.05=0.500
Table 23: Performance comparison of our method and others with different batch size in IMU datasets for Activity and Step
Method
UCIHAR
HHAR
Clemsoa
S-Cons (%)
Acc
F1
S-Cons (9)
Acc 1
F1
S-Cons (R)
MAPE 1
MAE 1
Baseline
95:13:11.2
19.89:1.76
89.37+1882
98, 08.25
91.9911.86
91.20:=0:1
43.61=1.53
4,2740.26
2.41+0.14
Aug.
96 06.55-00.8
1.
90.60+1.50
98. 13:0227
91.70+1.55
90.94=161
63. 63.74-4483
3.81+0.21
2.13+0.11
LPPE
96.88-00.76
92.13-+9.56
92.23+00.6
98. 14+0.09
90.29+0.86
89.45+094
52.60-371
4.24+0.24
2.18++0.15
APS
97.12-00.75
92.43+1.40
92.07+1.21
98. 00..04+0.24
91,83+1.35
91.01+147
16-008
5.00+0.03
2.84+00022
Ours
100.10.00
02.78.10.51
92.9411.333
10000.00
91.77.0..6
91. 1010..00
100.00.00
4.06.10.16
2.2410.10
Ours-LPE
10010.00
89.78.11.25
90.0411.11
10010000
91.4011.20
90.6011.32
100.0.000
4.14.00.9
2.3210.06
Ours+APS
10010.00
90.64.1146
90.4011.22
100.00.00
91.1910.60
90.36.10.55
10010.000
4,41.00.16
2.48.10.10
When we doubled the batch size during training, we observed a performance decrease in some datasets for method, such as Clemson and DaLiA. We believe this could be due to the need to adjust the guidance network's ' learning rate when changing the batch size. Additionally, since the introduced loss function aims reduce overall angle variance for batch of samples, larger batches may pose optimization challenges for the guidance network. It is important to note that despite performance decreases in some cases, method consistently outperformed baseline models with the original batch size used the main experiments.
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Published as a conference paper at ICLR 2025
E IMPROVEMENTS IN PERFORMANCE ACROSS DIFFERENT NETWORKS
In this section, we conduct experiments to observe the performance of our proposed method when is integrated into different network architectures. First, we employed the same ID ResNet architecture for IMU related tasks as we used the fully convolutional network without residual connections in the main results because of better performance. Second, we applied a transformer with positional encoding, which is designed for time series tasks (Qian et al. (022), to all tasks. Specifically, we used linear layers with stack of four identical blocks. The linear layer converts the input data to embedding vectors of 128. Each block is made up of a multi- self-attention layer and fully connected feed-forward layer. We use residual connections around each layer.
We have not included blurring (LPF) and adaptive sampling in the transformer network analysis, as these methods are primarily tailored for convolutional architectures. Additionally, due to lack of convergence in the heart rate prediction task, we have omitted reporting results from the transformers. We reported the results in the tables below.
Table 24: Performance comparison our method with others IMU with ResNet
Method
Clemson
S-Cons (%)
Acc
F1
S- Cons (%)
Acc
F1
S. S-Cons (S)
MAPE І
MAE
Baseline
97 97.40.10.86
83.35133.0
91,7811107 07
84.80--115
6,8311660
3,97-10.96
98 0.3377
86 66.6611.22
85. 12:11.55.53
99 00.31:0:04
02.82:10.35
92.84:0:35
6.621:.10
3.83:10.66
TPP
97.9 -:0.00
84.03-42.67
82,49+2.93
03.01=1.92
91,33++143
01.38:11.4
94.59 94-59-071
4.46+00.4
2.50+0.26
APS
98.02 98.02:+0.46
81.98+3.56
79.23++.20
93.01+1.92
92.13+0.22
92. 14+0..2
93.01-1922
6.61+1.44
3.84+0.84
91,9010.10
Ours-LPF
83.05
100.0000
92,45110.45
92.
10010.00 92.3010.16
4.45100.22
2.4510.13
Ours+APS
84.33.1293
83.01.13.13
92.2510.17
6.0710:47
3,50.10.28
Table 25: Performance comparison of our method with others in IMU with Transformer
Method
UCIHAR
HHAR
Clemson
S-Cons (%)
Acc
Fl1
S-Cons (%)
Acc
FL
S-Cons (%)
MAPE
MAE J
Baseline
90.44:10.48
69.15:4.79
65.64::4.59
96 98.00.20
103
91.04+1992
93.8742.12
6.552083
3.77=:0.88
93 68-60.66
73.23=2275
09. 69.799:3.61
98 +5
89:21-007
89. 16:0.11
95.46+2655
6,54:034
3.76=0.17
100+-0.00
74.02-3001
70.42:3.47
100-00000
91.55-13
91,19-1.19
100-0.00
6.50-0.55
3.77-10.31
Table 26: Performance comparison of ours and others in ECG with Transformer
Method
Chapman
PhysioNet 2017
S-Cons (%6
Acc
F1
AUC (56)1
S-Cons (8)
Acc
F1
AUC 1
Baseline
98.53-0.17
91. 03
22.224
98. 34-10.16
98 35:
83.22-40.2
73.50==1.99
93.21
Aug.
91.89-10.22
98.45-10.18
98.96-20.17
82.28+11118
72.32-42.00
93.20--0.42
Ours
100:00.00
92.10:20.25
91.93-10.85
98. 08.40=0.115
100::0.00
83.35=0065
74.12::180
93.28::0.31
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Published as a conference paper at ICLR 2025
We also integrated proposed transformation into recent neural networks and investigated the performance. Mainly, we employed ModemTCN (donghao wang xue 2024) and T- T-WaveNet (LIU et al. 2022 architectures. When we implemented ModemTTN, tion from We set the patch size and stride to 5 and 2, respectively, while keeping the backbone and dropout rate the same in the original implementation. The stem, downsampling. and FFN ratios were set to 1.
follow the original implementa-
Table 27: Performance comparison of our method for HR estimation using ModernTCN
Method
IFEE SPC22
DaLiA
S-Cons (56) f
RMSE
MAE І
(()
S-Cons (%)
RMSE
MAE v
(%) 1
Baseline
38.62=0:23 50.15=0.07
34.67=0.45
29.26#0.41
3,45#0,54
10 40.24:0.46
20.52=0:46
11,90=0,24
60 10,1671
Aug.
34, 14
28.294+:.04
01 2.22232
39.10+112
15. 1..36=1.16
7.25-0.64
79.97=2.58
Ours
100:0.00
33.45
29.33-1109
08.9611.73
100:00000
15.20=1.22
7.13-0.10
80.05=1.08
Table 28: Performance comparison of our method for ECG datasets using ModernTCN
Method
Chapman
PhysioNet 2017
S-Cons (%)
Acc
Fi
AUC (5)]
S-Cons
Ace
Fl
ACC
Baseline
98.5320.17
55.48=1.34
46.07:1:.03
73.35:10.34
87.77=1.20
4.884=480
22.41
60.34.1.A 1644
Aug. Ours
95.511-3.38
81,93:3.60
79.15414.33
94.88-11.11
98.96:20.11
60.13:2.57
28.86
70.57-16.10
100:00000
83.80=2.06
80.7342.70
95.41::0.33
100::0.00
60.58=1.50
30.73+1.08
72.58=3.76
Table 29: Performance comparison of our method for IMU datasets using Modern TCN
Method
UCIHAR
HHAR
Clemson
Cons (%)
Acc 1
F1
S Cons (8)
Acc
S. Cons (S)
MAPE J
MAE 1
Baseline
95 8912210
8..6811553
85.4512.35
97.23.10.88
92.21-0.77
92.0911.10
73.96112.18
4.0310.11
2.32:10.09
Aug.
96 55-+0-80888
85.42=-450
83.69=6.74
98.38+0.28
91.97-044
91.31=0.49
61.01+488
4.08=0.14
2.29=4.07
Ours
100:00.00
88.73:1447
87.19:1.94
100::0.00
94.12-087
93.43:1110
10000.00
3.88-0.26
2.15::0.16
Table 30: Performance comparison of our method using ModemnTTN for sleep stage classification
Sleep-EDF
Method
S-Cons f
Acc f
FI 1
W-F1 f
K
Baseline
71.8441.78
19.50.11.81
62.8441.39
71.21
60.3942.20
Aug. Ours
95.47+:5.75
72.96+44.72
64.50+4.14
73.61+5.12
64.90++6113
100:10.00
73.36=3.100
65.37+2.88
74.10:
65.42+3.51
From these results, we can see that ModernTCN architecture performs relatively poor compared to ID ResNet architecture in most tasks. However, for the IMU related tasks, ModernTCN outperforms other architectures. Similarly, when we integrate our method into the ModernTCN. the performance increases by 5-10% while decreasing the variation between runs.
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Published as a conference paper at ICLR 2025
We also performed experiments with T- T-WaveNet, a tree-structured wavelet deep neural network. The model decomposes imput signals into multiple subbands and builds a tree structure with data- driven wavelet transforms the bases of which are learned using invertible neural networks. We use the original implementation from https: : /et:://oppeeeee /openrevies net. / forum? dd=U4uF aLyg Following the original implementation, the wavelet functions are learned together with the neural network instead of using stationary wavelet transforms like Haar.
Table 31: Performance comparison of our method for HR estimation using T-WaveNet
Method
IEEE SPC22
DaLiA
S-Cons (%)
RMSE
MAE
((%) 1
S-Cons (%) 1
RMSE
MAE 1
p (())
Baseline
49.73:11.61
2..78
15.77:11.59
60.30::4.16
39.69.1.117
13.38.10119
6.15
83.10.10.47
Aug. Ours
71.11
19.29#2.31
12.16:1151
66.50=6630
53.95=10.28
12.82=00.00
5.89=0.11
83.63-10.66
100.10.00
19.03
12.10.2.10
67.76.6.55
100.00.00
12.65
5.59
84.05
Table 32: Performance comparison of our method for ECG datasets using T- T-WaveNet
Method
Chapman
PhysioNet 2017
S-Cons (%)
Acc
F1
AUC (%6)
S-Cons (%)
Acc 1
F1
AUC 1
Baseline
97.23-0.19
03.17
92.40:0.78
08.8740.18
94.72++191
78.94=1.00
71.43:2.24
92.44=1.03
Aug.
98. 19.6.8
93.6310.33
02.89.10.32
98. 96.10.100
95 95.43.10.89
79.7710.99
1..7241.42
92.33.10.63
Ours
100:0.00
93.4520.40
92.92:10500
99.00:0.15
100:0.00
19.89=1.10
71.61:2.10
92.50=0.89
Table 33: Performance comparison of our method for IMU datasets using T- T-WaveNet
Method
UCTHAR
HHAR
Clemson
S-Cons (%)
Acc
FL
S-Cons (%)
Acc
FL1
S-Cons (%)
MAPE
MAE
Baseline
97 144
72.77=236
70.48+4.16
98. 37.06
91. 91.53+1.00 92.37=0.89
89. 09.50++0.50
6.69::0:5
3.90=0031
98 30:2243
72.82-3334
68. 68.78:13.5
98 08.68-20.65
92. 02.88-1.15
92.05=1225 22
89,2944.72
6.6220039
3.83:10331
Ours
100:00.00
74,04:210
71,04:3.25
100:00000
92.95 1.14
91.60=0.91
1.000.00
6.03-0.50
3.54:0035
Table 34: Performance comparison of our method using T-WaveNet for sleep stage classification
Sleep-EDF
Method
S-Cons f
Acc f
f
W-F1 f
K
Baseline
71.8441..8
69 19.50111.81
62.84411.39
71.21:1166
60.3942.20
Aug.
95.47-15.55
72.96:1.772
10.1..3
73.61:5.12
64.90-6.13
Ours
100:10.00
73.36:15.10
65.90
74.10
65.42:13.51
As shown in tables, the proposed transformation also increases the performance of the different neural networks. One important result from the comparison of these tables is that there is correlation between the model's ability to remain invariant shifts and its performance, up to a certain threshold where the model performs adequately. However, beyond that point, as the model's performance declines, the consistency in shift increases, resulting the model consistently outputting the wrong class. For instance, in the case of step counting, the transformer architecture performs quite worse and fails to distinguish between samples. As result, the consistency of the transformer is higher compared to ResNet and fully convolutional networks while the performance is lower. We believe that investigating the invariance of different neural architectures alongside their performance on time series tasks shed light model networks and the fundamental reasons behind abrupt output changes with small changes in the input signal, i.c., so 10-15 ms shift.
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Published as a conference paper at ICLR 2025
F VISUAL EXAMPLES FOR THE GUIDANCE NETWORK
In this section, we provide some visual examples to show how the proposed transformation function works. First, we show the t-SNE (van der Maaten & Hinton 2008 representations of the embeddings obtained from a trained model with and without applying our transformation in Figure 4
(a)
(b) Baseline model
(c) Embeddings with our transformation
W/o transformation
With transformation
Count
10P
201
Different samples from healthy rhythm Shifted variants of single sample
0
0.1
0.2
0.3 0.
04
Distances
Figure 4: (a) Comparison of pairwise Euclidian distances of randomly shifted embeddings with and without applying our method. (b) t-SNE visualizations embeddings without our method show some shifted samples clustering with opposite class embeddings. (c) With our transformation, all shifted variants of the same signal cluster correctly within their true class label.
For visualization, we selected 50 different ECG (healthy) signals from the test set. We then took a single arrhythmia ECG sample from the test set, applied 49 shifts to it (50 samples with the original), and created variants shown in blue. Finally, we compared the embeddings with and without applying our proposed transformation function. As seen in Figure 4 applying our transformation function maps the shifted samples to a single point in the embedding space, with the maximum Fuclidean distance between embeddings being close 10-6 6
Second, we conducted a simple experiment to investigate how the guidance network works with the proposed transformation. Specifically, we created two-label classification task where the model classifies sinusoids by frequency. The dataset includes two waveforms: (t) cos (wit + Φ1) cos (w2t + Φ2) and (t) cos (wit + Φ1) + cos (wat + Φs). with the model identifying whether the input contains frequency w2 w3. Frequencies were set at 5, 24, and 25 Hz for wi, 42, and w3, respectively, with w1 included both waveforms to increase task difficulty. We set the sampling rate of the signals to 300 Hz.
(a)
coslwryt + (2) + cos(wat d2) + costuyt 6i) coslust 6s)
(b)
ocafare + di) conlwsy + (r) + conforzt 62) conluut + da) +
Amplitude
Amplitude
Timm tst 1
Tima (G)
(c)
cnstast: 61) caslugt (2)
(ct)
Euty 3.) oon fuagt de)
Amplitude
cosfont + 6i) conforst + (x)
Amplitudee
oos foog dia) cos (unst di)
Figure 5: Input waveforms to the classifier in the third epoch (a) without applying our transformation function. (b) with our guidance network (fea). Another experiment with different seed. Input
waveforms (c) without applying transformation function. (d) with our guidance network (foo).
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Published as a conference paper at ICLR 2025
To add diversity, randomly shifted (t) and (2(() by angles sampled from [0, r) and added Gaussian noise (e) with variance of 0.1. We used the FCN similar to that specified in Appendix C.4
the architecture. This experimental setup is inspired by similar experiments exploring neural network behaviors [Rahaman et al. 2019).
Figure iillustrates interesting result: after few weight updates, the guidance network assigns angles @ that maximize the Euclidean distance between inter-class samples. For instance, before applying the guidance network, the distance between pair of samples is 9.12 (Figure (a)). After applying the guidance network, this distance increases by four to 43.6 (Figure 5 (b)). Interestingly, running the same experiment with a different seed (i.e., a new random initialization of the guidance network) shows that the assigned angles differ, but the Euclidean distances between the samples remain almost unchanged, shifting only slightly from 43.6 42.8 (See Figure (d)). (d)).
This experiment can also explain the occasional performance increase when the angle variance increases with a loss term as there is no single solution for minimizing the distance, but there can be infinitely many depending the frequency distribution of the dataset and classes.
(b)
cos(d1) vs cos(és)
cos( va cos(ds)
Distance distribution
-070
0.70
Seed
Seed
Seed
-0.80
Seed
-0.80
Density
0.08
Seed
0.06
W/o T(x,0)
-900
-0.g0
0.04
0.02
10
20
25
-090
-0.0
-0.00
-0.60
-070
Fuclidian Distance
Seed
Seed
Figure 6: (a) Angle assignments across different seeds. (b) Euclidean distances between intra-class samples, compared with and without the proposed transformation (W/o T(x, Φ)). The results show that the Euclidean distances are highly consistent across different seeds, with the curves nearly overlapping. Additionally, when the transformation not applied, the distances between intra-class samples noticeably higher.
We conducted an additional experiment to analyze the angle assignments across runs using the IEEE SPC22 dataset. The experiment was performed with three random seeds, and the results are presented in Figure 6 In Figure (a), we reported the assigned angles in cos (ω) the angle TT and are the same due to the circular property of the angles. While the assigned angles vary slightly between runs, the Euclidean distances between samples consistently converge to similar values. Specifically, the intra-class samples are closer in the transformed space compared to the case when the proposed transformation is not applied.
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Published as a conference paper at ICLR 2025
G EXPANDED REVIEW OF RELATED WORK
Transformation Applying phase shifts to time series is commonly used in signal process- ing (Haykin & Veen 2002] Oppenheim et al. [996). Recently, shifting phase values of harmonics have also been applied in the machine learning community for data augmentation of time series (Demirel & Holz, 2023 Qian et al. 2022). However, our proposed transformation differs from these in two key aspects. First, our work is the first to represent every point in the shift space uniquely with the phase angle of a harmonic whose period is equal to or longer than the length of the sample, i.e. To S I. This observation enables us to design bijective transformation that ensures shift invariance. Additionally, we integrated this observation into deep learning frameworks using a novel loss function, demonstrating that our proposed method enhances model performance while ensuring shift invariance.
Second, we apply linear phase shifts to keep waveform features intact, Since if input signal is subjected to a phase shift that is a nonlinear function of w similar to Qian et al. (2022); Demirel & Holz (023), , then the complex exponential components of the input at different frequencies will be shifted in a manner that results in a change in their relative phases. Superimposing these exponentials can result in signal that significantly differs from the input if special precautions not taken. Thus, this alteration in the waveform (Oppenheim et al. 1996) can potentially affect downstream labels or generate unrealistic signals.
However, our transformation applies the tailored shift linearly to all harmonics while keeping the information content unchanged (Mallat 2012) as the transformation operates as group action.
Shift-invariant Kernels
Learning shift invariant representations from data has long history in machine learning (Grosse et al. 2007 Rahimi & Recht 2007). The initial effort focused on designing shift-invariant kernels for feature extraction (Rahimi & Recht 2007), which were applied to support vector machines. A different approach introduced shift-invariant sparse coding technique, which reconstructs an input using all basis functions across all possible shifts (Grosse et al.) (2007). However, the classification performance of these techniques were significantly outperformed by the modern networks. Thus, recent approaches have focused on integrating shift-invariant kernels into modern convolutional neural networks in a stacked manner while using Gaussian low-pass filters to prevent aliasing MMairal et al. 2014).
However, applying low-pass filters to prevent anti-aliasing completely is not possible (Oppenheim al. 1996). Thus, high-frequency components will always (partially) alias. This is more problematic for time series as the interaction of high and low frequencies are more common (Demirel & Holz (0233/Caooly & Knight 2010). Therefore, applying a low-pass filter can reduce the performance of neural networks in certain time series tasks, as shown by our experiments. The filtering may inadvertently attenuate important high-frequency components, which are essential for distinguishing patterns, leading to suboptimal model outcomes.
Learning based Transformations Methods to standardize inputs have been around for a long time (Yücceer & Oflazer 1993). An important recent work along this direction is the Spatial Trans- former Network (STN) being introduced to learn transformation functions for invariant image classi- fication (Jaderberg et al. 2015). Similarly, Temporal Transformer Networks (TTN), adaptation of STNs for time series applications, were introduced to predict the parameter of warp functions and align time series (Lohit al. 2019 Shapira Weber & Freifeld 2023). Recent studies have utilized canonical equivarant networks to obtain mapping points for inputs (Kaba et al., 2023). However, these methods face significant limitations as the operation order increases. Specifically, higher-order transformations in group equivarant networks require additional filter copies in the lifting layer and an increased number of parameters the subsequent group convolution layers. While this can improve performance, it comes at the cost of significantly higher computational and model complexity. Furthermore, prior works restrict mappings to a finite number of group elements defined by the canonicalization network. In contrast, our proposed transformation eliminates this limitation entirely, enabling each sample to map to any point in the input -anii--ifi many- many-without without relying on neural network. This is achieved by uniquely representing each point in the shift space using a specific harmonic.
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Published as a conference paper at ICLR 2025
H DISCUSSION, LIMITATIONS AND FUTURE WORK
In this work, we propose a new diffeomorphism to achieve shift invariant deep learning models for time series real-world tasks. While existing techniques show promise for images, they fall short in time series, where the interaction of low and high frequencies are an important part of the data generation. The proposed transformation offers novel solution, ensuring that samples will map the same point in the high dimensional data manifold despite random shift. Theoretical and empirical analysis demonstrates its effectiveness across several time series tasks, enhancing model robustness while improving the performance.
While our approach consistently improves the performance of deep learning models for time series data, it is worth noting the potential areas for future investigation and improvement. First, we conducted our experiments on health-related time series tasks from humans since the robustness of models is crucial in those domains. Therefore, extending the proposed transformation to images for shift or rotation invariancy presents an intriguing direction for future investigations. Thus, we believe that future research could benefit on adapting explore shift or rotation invariance further. Second, our approach requires samples to be expanded into the sum of periodic sinusoidals with Fourier expansion followed by using the phase angle of the one whose period equals or exceeds the length of the signal. Input samples should be decomposed the sinusoidals while considering this requirement. Therefore, we believe future work can benefit by detecting the phase of a specific sinusoidal which satisfies the condition and apply linear phase all all-pass filter without performing the operation in the frequency domain. Lastly, we observed notable performance improvements from the additional guidance network when it is used with the proposed diffeomorphism while applying proper loss function. Thus, we believe that further performance improvements can be achieved through refined design incorporating alternative inputs.
approach to diverse domains, including images, to
40


Reviews:
[{"review_id": "Nz5tFdJIu6", "reviewer": "Reviewer_iuou", "summary": "The paper describes an approach to make neural network pipelines for time series shift invariant. The method is based on Fourier theory and follows a canonicalization approach, where the input signal is shifted by picking a reference frequency and its phase, then shifting the signal such that the phase of that reference frequency equals a target phase φ. The target phase φ is predicted using an invariant network that takes the power spectrum as input. The experiments show strong results across 5 different tasks.", "questions": "This is a mix of comments and questions:\n\n* At the beginning of the notations sections why write parenthesis around $\\mathbf{x}$, i.e., why write $(\\mathbf{x})$?\n* Why is proposition 2.1 relevant. This is a well-known result, but I don't see the paper reflect back on the group structure? In essence, each $e^{j \\omega t$ is an irreducible representation of $SO(2)$ and this explains the Fourier shift theorem. The text however, does not really connect prop 2.1. to the rest of the paper.\n* I don't understand why fixing $\\phi$ would not work. In principle that would make the entire pipeline shift invariant too right?\n* The talk about diffeomorphisms is really unclear. What precisely is the diffeomorphism and between which spaces? It seems to be a diffeomorphism of the data-space onto itself, simply by a shift, but I might be misunderstanding something here.\n* The last step of the proof of theorem 2.3 is unclear. The first equation of (6) shows that the difference between the two transformed shifted signals is not equal to 0 (but $[T_0/T \\omage_0 - \\omega]t'$). But then all of a sudden the second equation equates the two signals.\n* It is unclear why a variance loss is necessary? More importantly, what is the random variable in this case. It seems like the output of the guidance function is a single angle, then what is there left to take a variance over? The text says \"reduce variance in the angles\" but the network only returns one angle, right?\n* Fig 3 with the depiction of the \"manifolds\" in (b) is confusing. I still don't understand the manifold viewpoint.\n* The text says \"In other words, if we conceptualize the dataspace of samples as expanding with shift invariants\", what does it mean to expand with shift invariants? And what does it mean \"potential points where samples can be found\".\n* Regarding fully convolutional networks. What precisely is meant with this? I was assuming that this meant that no down-sampling is used, or at least that the output has the same resolution as the input. But then the guidance network does not take a spatial signal as input, but rather the power spectrum, so convolution does not make sense to me, but also even it would, if you return a single phase some sort of pooling or fully connected layer must have been used, right? And thus it cannot be an FCN. Could it be that fully connected is meant here?\n* In the section time delay as adversary, what table is being discussed here? Please clarify this in text.\n* \"However, our results indicate that the model ... the model prediction jumps more than 100%\" what does this mean? How can a model \"jump\" 100%?\n* \"One distinct results ... increases by 7-8%\", relative to what?\n* I don't understand why fixing $\\phi$ does not work. Theoretically seen, doesn't fixing $\\phi$ also lead to a fully invariant pipeline? But in table 5 I see that the first row gives poor performance, why would that be?\n* Equation (9) is, similar to my previous question about variance, puzzling to me. Firstly, what is the random variable the variance is taken over, and secondly why would this even work at all? First you minimize variance, now you maximize variance, and the results are barely different. I would expect one of the two to be extremely unstable.\n* \"Importantly, adding the guidance network does not bring any additional parameters\", this is not true. $f_{G_\\theta}$ has parameters, right?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "* The empirical results are strong and comprehensive across multiple tasks\n* The method has clear practical value for making neural networks more reliable for time series analysis\n* The problem setting is highly relevant to real-world applications\n* The proposed solution addresses an important gap in making neural networks for time series more robust", "weaknesses": "1. Clarity and Presentation Issues:\n* Despite attempts to make concepts intuitive through figures and explanations, the paper raises more questions than it answers\n* The theoretical framework around diffeomorphisms feels disconnected from the practical method\n* Several key concepts and proofs need better explanation or justification\n2. Missing Context:\n* The paper lacks discussion of related canonicalization approaches, particularly work like Kaba et al. (2023) on \"Equivariance with learned canonicalization functions\"\n* The relationship between the theoretical foundations (e.g., Proposition 2.1) and the practical method isn't clearly established\n\n``Kaba, S. O., Mondal, A. K., Zhang, Y., Bengio, Y., & Ravanbakhsh, S. (2023, July). Equivariance with learned canonicalization functions. In International Conference on Machine Learning (pp. 15546-15566). PMLR.```", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HEazeqkvoX", "reviewer": "Reviewer_6oim", "summary": "This paper proposes a method to modify time-series data samples in such a way to make neural network training shift-invariant. Their method involves mapping the signal into the frequency domain via the Fourier transform, applying (learned) phase angle transformations to each of the frequency components in the frequency domain, and then mapping back into the time domain via the inverse Fourier transform. They compare results on mostly health-related time-series datasets and compare to alternative methods of promoting shift-invariance such as anti-aliasing methods which use LPFs to remove high-frequency aliasing effects, data augmentation methods, and more. The authors find on average that their method results in improved accuracy and consistency of models on time-series data where shift invariance is desired.", "questions": "If I understand correctly the model is chosen based on the best validation loss, is the reported number based on that validation loss or a held-out test set evaluated by the model whose parameters come from that point where there is the best validation loss during training?\n\nIs it possible that the guidance network is overfitting to the train set in some way? Specifically, if the numbers reported are when you pick the networks that achieve the best validation performance and the reported numbers are not from a held-out test set then there might be some overfitting reported, although the shift consistency metric does seem to indicate that the models are learning shift-invariant estimators.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The results seem to indicate on average that the models trained in this manner achieve shift invariance (through shift consistency) and better results than existing methods. Theoretical results support that you can achieve a bijective mapping for samples that promotes shift invariance.", "weaknesses": "I am admittedly not the most knowledgeable in this area of research, but the method sounds to me like a form of data augmentation that operates in a model-driven manner given that there is a guidance network that is optimized which selects the phase angle to use in the augmentation? Perhaps it would be worthwhile to compare this method to some model-driven / data-driven data augmentation processes? \n\nSome of the results indicate improvement when averaged over multiple runs (how many runs are these results averaged over? I seem to have missed this information in the manuscript). But in some of those cases based on the standard deviations computed there is a fair overlap between prior methods and your method, it’s possible that the advantage in those cases is not so clear when there is a large variance over the accuracy of different runs.\n\nI could imagine there are cases where the computational overhead of training two networks (classifier + guidance network) to get a small increase in accuracy may not be desirable, but as far as I can tell the shift consistency being 100% using this method would seem to be a benefit to any use-case that requires shift invariant predictors, as compared to whichever existing methods are reported in this paper.\n\nI assume that the guidance network, after training, is fixed meaning that the angle for each sample is fixed? How can I interpret the value of the angle that the guidance network converges to for a given sample? Does it say something about the data itself? How robust is this technique to alternative ways of choosing the desired angle, perhaps via random angle methods (could be an interesting baseline to report) or some other method that doesn’t involve optimization of a guidance network?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cqyyEsNfEf", "reviewer": "Reviewer_Sy9x", "summary": "The paper introduces a diffeomorphism that transforms observations from one high-dimensional data manifold to another of the same dimensionality. The goal is to quotient with respect to random shifts. The method ensures that each sample is mapped to a unique point on the data manifold regardless of its random shift. Keeping the same dimension in the representation space is suppose to retain all task-relevant information without any loss.\nA theoretical analysis is performed as well as intensive experimental demonstrations.", "questions": "Data do not necessaraly lives in Manifold, if it does not (no local representation with metrics), would that be an issue?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This paper is well motivated and with both theoretical analysis and deep experimental studies and comparisons.\nThe modeling is interesting and remarks on the performances are relevant.", "weaknesses": "The model requires that the data lives in a Manifold. This is a strong assumption (although good one to start with) which sounds necessary as it is quotiented by the group of shifts.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZR0VCzeOjf", "reviewer": "Reviewer_Se81", "summary": "The paper introduces a novel method to achieve shift-invariance in deep learning models for time series data. Traditional methods often lack robustness to shifts, which can lead to highly inconsistent performance in tasks such as heart rate prediction, especially when requiring both low- and high-frequency information. The authors propose a differentiable bijective transformation that maps shifted samples to a unique point in a high-dimensional manifold, ensuring shift-invariance without dimensional reduction or architectural changes. This approach is tested across various time series tasks, where it demonstrates improved performance and consistency compared to existing techniques.", "questions": "- Figure 1: while I appreciate the thorough overview, the figure contains a lot of information that is a bit hard to grasp without more detailed descriptions or additional reading, and connections between subpanels (e.g. a&b and c&d) could be grouped more clearly.\n\n- If two time series are merely shifted versions of each other, a purely analytical approach based on the Fourier transform could align them without the need for a neural network to estimate the phase shift (e.g. by saying that one of the two is the “unshifted prototype” of the other time series). It is hard for me to understand why we need a neural network architecture for this, since if the phase shift between two time series can be determined analytically in the Fourier domain, the guidance network’s task of estimating phi seems redundant.\n\nThis connects to the question of how to determine $\\phi_0$, since there is an additional symmetry breaking required in the guidance network by randomly designating certain samples as “unshifted” to train the network. Does this introduce potential inconsistencies in the latent space representation across training runs which might learn different mappings for the \"unshifted\" state, depending on which samples are chosen as the reference? The authors do not explicitly argue that the guidance network is introduced to handle more complex cases with non-uniform shifts or noise, which would make sense to me as a motivating factor. An interesting follow-up question could be what the model actually does if there are non-uniform shifts. How robust is the guidance network’s output for longer/shorter time series?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper is well-written, and the figures all look aesthetically nice and neat.\nThe medical/empirical motivation is clearly laid out, with good intuitive examples of why this is an important and challenging problem.\nThe empirical results appropriately cover a range of real-world empirical datasets with a range of important tasks.", "weaknesses": "- The authors present the proposed transformation as a novel aspect of the method, yet phase-shifting is a well-known approach in signal processing. The architectural novelty here lies primarily in the integration of the guidance network, not the phase-shifting transformation itself. The paper however does not provide an independent evaluation of the guidance network’s specific performance in estimating the phase angle \\phi, which could easily be done when ground truth shifts are known. Most evaluations, including the ablation studies, focus on downstream tasks, which are interesting and practically relevant but do not provide clear insight into what the algorithm is doing at the shift-invariance level. They also add confounding factors. For instance, in the appendix, you argue that \"it can be observed that the number of parameters for the guidance network is much less than the main classifier\", around 2% of the number of parameters, but the classifier is not an inherent part of your shift-invariant approach.\n\n- A natural follow-up question is how well the guidance network performs in OOD settings where the ground truth shifts are known but the time series were not part of the training data. Since the learned latent representations are still just time series, a simple illustration of the approach could be to take in a number of shifted time series and visualize the resulting latent representation of the time series.\n\n- While the shift consistency captures something in this direction, it still requires training the classifier to assign class probabilities. While the S-Cons results look impressive across the experimental results, could one not directly measure the variance or consistency of the output angles (or phase-shifted latent representations) across the shifted samples to get clearer insights into what is happening here?\n\n- In that spirit, there could also be comparisons to other approaches in time series alignment. This could also provide a more direct evaluation of the shift-invariant transformation and isolate the main contribution of the method.\n\nIn summary, while the paper introduces an interesting approach, these concerns prevent me from fully supporting acceptance in the current form. But I am open to adjusting my assessment if some potential confusion is clarified/some clarifying results are provided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)