DeepFRC: An End-to-End Deep Learning Model for Functional Registration and Classification
Authors: Siyuan Jiang
Yihan Hu
Wenjie Li
Pengcheng Zeng
Venue: ICLR 2026 Poster (2026)
Keywords: Functional Data Analysis; Deep Learning; Registration; Classification
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: This paper introduces DeepFRC, an end-to-end deep learning framework that jointly performs alignment and classification of functional data to improve model performance.

Abstract: Functional data, representing curves or trajectories, are ubiquitous in fields like biomedicine and motion analysis. A fundamental challenge is phase variability—temporal misalignments that obscure underlying patterns and degrade model performance. Current methods often address registration (alignment) and classification as separate, sequential tasks. This paper introduces DeepFRC, an end-to-end deep learning framework that jointly learns diffeomorphic warping functions and a classifier within a unified architecture. DeepFRC combines a neural deformation operator for elastic alignment, a spectral representation using Fourier basis for smooth functional embedding, and a class-aware contrastive loss that promotes both intra-class coherence and inter-class separation. We provide the first theoretical guarantees for such a joint model, proving its ability to approximate optimal warpings and establishing a data-dependent generalization bound that formally links registration fidelity to classification performance. Extensive experiments on synthetic and real-world datasets demonstrate that DeepFRC consistently outperforms state-of-the-art methods in both alignment quality and classification accuracy, while ablation studies validate the synergy of its components. DeepFRC also shows notable robustness to noise, missing data, and varying dataset scales. Code is available at https://github.com/Drivergo-93589/DeepFRC.

Published as a conference paper at ICLR 2026
DEEPFRC: AN END-TO-END DEEP LEARNING MODEL FOR FUNCTIONAL REGISTRATION AND CLASSIFICA-
TION
Siyuan Jiang', Yihan Hul, Wenjie Li, Pengeheng Zeng"
Institute of Mathematical Sciences
ShanghaiTech University
Shanghai, China
(j [jiangsy2023, huyh1, 1iw)2022, zengpch |ashnghaiitcch.hee www.:ittechcccc.....oooooo tehh
ABSTRACT
Functional data, representing curves or trajectories, are ubiquitous fields like biomedicine and motion analysis. A fundamental challenge is phuse variabil- temporal misalignments that obscure underlying patterns and degrade model performance. Current methods often address registration (alignment) and classifica- tion as separate, sequential tasks. This paper introduces DeepFRC, an end-to-end deep learning framework that jointly learns diffeomorphic warping functions and classifier within a unified architecture. DeepFRC combines neural deformation operator for elastic alignment, spectral representation using Fourier basis for smooth functional embedding, and class-aware contrastive loss that promotes both intra-class coherence and inter-class separation. We provide the first theoreti- cal guarantees for such joint model, proving its ability to approximate optimal warpings and establishing a data-dependent generalization bound that formally links registration fidelity to classification performance. Extensive experiments on synthetic and real-world datasets demonstrate that DeepFRC consistently outper- forms state-of-the-an methods in both alignment quality and classification accuracy, while ablation studies validate the synergy of its components. DeepFRC also shows notable robustness to noise, missing data, and varying dataset scales. Code is available htt
com Dr Drinm/rrvieeee www.oo//wieiooooooooeeoooo wtw.eerggoo339999eee9a0
INTRODUCTION
Functional data analysis (FDA) is key area for analyzing data that varies continuously over domains like time, space, other variables (Ramsay & Silverman, 2005 Ferraty & Vieu 2006 Srivastava & Klassen (016). Functional data is ubiquitous in fields such as biomechanics, neuroscience, healthcare, and environmental science, appearing datasets like growth curves, wearable device signals, BEGs, fMRI scans, and air pollution levels. Despite its wide applicability, FDA faces challenges stemming from the aninite-dimensiionll nature of functional data, as well as issues related smoothness and misalignment, necessitating ad advanced analytical tools (Wang et al. 2016 Matuk et al., (2022).
1.1 MOTIVATION
Two key tasks in FDA are functional registration (curve alignment) and classification. Registration aligns curves to remove phase variability, enabling meaningful comparisons (Srivastava et al. 2011 while classification assigns labels based on underlying curve features. Traditionally, these tasks addressed separately, with pre-registration followed by classification. However, this decoupled approach is inefficient, as the label significantly influences the progression pace the curves, and alignment can provide valuable insights for classification (Liu & Yang 2009 Tang al., 2022). Jointly analyzing curve alignment and classification accounts for both temporal and structural variations, offering more comprehensive understanding of the data.
These authors contributed equally. Corresponding author: zengpch&shangha tech. edu.cn

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Deep learning has revolutionized data analysis across various domains by enabling automatic feature extraction, representation learning, and scalability (LeCun et al. 2015). In FDA, deep learning presents opportunity to integrate registration and classification into unified framework. However, its application remains underexplored, with existing studies primarily focused on improving either registration accuracy (Chen & Srivastava, (02T) or classification performance (Yao et al. [02T] Wang & Cao [ao) but rarely addressing both simultancously. To address this gap, we propose an end-to-end deep learning framework that combines functional data registration and classification, eliminating the need for separate preprocessing while leveraging neural networks to model the complex, non-lineat relationships inherent in functional data.
1.2 RELATED WORK
Functional registration is critical step in FDA, aimed at aligning functional data to correct for phase variability (Ramsay & Silverman 2005). Traditional approaches include landmark-baad methods (Keeip & Gasser 1992 Ramsay & Silverman 2005), metric-based methods (Wang & Gasser 1997]
Ramsay & Lil 1998 Srivastaaaa et al. 2011 Srivastava & Klassen 2016), and model-based methods
(Tang & Müller 2008 Claeskens et 2010 Lu et al. 2017)). While effective some contexts, these methods face limitations such as high computational cost, manual intervention, and sensitivity to
noise. Landmark-basee methods, for example, require subjective and potentially impractical landmark selection (Marron et al. 2015). Recently, neural network- -baed methods (Chen & Srivastava (021) have been proposed, but they are still typically used as preprocessing steps, disconnected from the downstream tasks.
Functional classification has traditionally relied on statistical methods like generalized functional regression (Jaes]] 2002; Müller (2005) and functional principal component analysis (fPCA) (Hall ctt al. 2000; Leng & Müller (0ül)), which involve dimension reduction followed by classification. However, these methods are limited by their reliance handcrafted features and basis function selection, which restricts their adaptability to complex, non-linear data. The integration of deep learning into functional classification, seen in works by Thind et al. (2020), Yao et al. (2021), and Wang & Cao (2024), addresses these issues, although they typically treat registration as a separate preprocessing task.
Few studies have integrated registration and classification within a unified framework for FDA. Lohit al. (2019) proposed Temporal Transformer Network (TTN) for time series classification, but is not designed for functional data, neglecting the curve's smoothness and diinite-dimensional nature, thus struggling with accurate alignment. More recently, Tang al. (2022) introduced two-level model for joint registration and classification, which employs parametric mixed-effects model for warping functions (e.g., using B-spline bases with Gaussian assumptions) and requires alternating optimization. However, this approach is constrained by its parametric and distributional assumptions, computationally expensive, and is not easily extendable to multi-class settings.
1.3
CONTRIBUTIONS
Our main contributions are: (1) Unified Architecture for Joint Learning. We introduce the first end- to-end deep learning model that integrates a neural deformation operator for diffeomorphic warping, spectral representation for smooth functional encoding, and class-aware contrastive loss, enabling mutual reinforcement between alignment and prediction. (2) Theoretical Foundations. We establish the first theoretical guarantees for such model, proving approximation capabilities and providing generalization bound linking registration fidelity to classification performance. (3) Extensive Empirical Validation. Extensive experiments on synthetic and real-world data demonstrate that DeepFRC consistently outperforms state-of-the-art methods in both tasks, with ablation studies confirming the synergy of its components and further analyses highlighting its robustness and computational efficiency.
2
THE MODEL
We study supervised learning with functional data, where each sample consists of an observed trajectory and a categorical label. Formally, the dataset is ((x,(,,), y.)) where 2x(t,) = (Ia(ta)...., (t(().)))).... observations of underlying trajectory s+()) sampled at imregllr and
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potentially misaligned time points (f41...... ... tin), and ys (1...., C} is its class label. We view functional data as trajectories embedded in an Simmite-dimensional space, where misalignment corresponds to latent time reparameterizations, Instead of treating alignment and prediction two separate stages, we propose to jointly learn a diffeomorphic time-warping map - and a classifier f stt)
in single end-to-end architecture (Figure 1). This allows directly learn representations that are invariant to temporal misalignment while remaining discriminative for classification.
Input
Deformation Operator
Spectral Representation
Classifier Network
Output
7(8)
C1 C2 .2....,.9 CK
# (l.... Cf
Objective = Contrastive Geometric Alignment Loss Classification Loss
Figure 1: Overview of DeepFRC. Multiple raw functional trajectories are first aligned via neural deformation operator (ID CNN) that learns diffeomorphic time warping y(t), producing both warped curves and alignment functions. The aligned signals are then expanded in Fourier basis to obtain spectral coefficients c1,..., CK. which serve as inputs to a classifier network (MLP with Softmax output) for predicting class labels y ...... C. Training is performed jointly by minimizing contrastive geometric alignment loss and a classification loss.
2.1 NEURAL DEFORMATION OPERATOR FOR TIME WARPING
introduce a neural deformation operator. parameterized by a ID convolutional network (Kiranyaz al. 2016 Ince al. [0166] Kiranyaz al., 2021), that learns velocity fields defining diffeomorphic warpings. This perspective differs from handcrafted registration methods: instead of explicitly constructing alignment rules, we parameterize y through a deep network optimized jointly with downstream prediction.
Given input sequence (t,) R", multi-layer ID CNN extracts temporal features (t:) R":
h = ReLU(W()) @ (t-1) + B()), 1= 1= 1,
with h (0) Fy (t.). The final layer is fully connected: (tx; = (c + b(t)). Following Chen & Srivastava 2021), use 4 = 4 convolutional blocks with kernel size 3 and channels 16 32 64. This encoder, parameterized by θ, produces latent features that encode temporal deformation fields.
To construct
diffeomorphism, we transform the latent features into a monotone
E -
cumulative sum: (4az) =
j = 1,.... n,, where t. 0, f.n = 1. and To 0.
Σ T
This ensures (0) (1) and d% 0, satisfying diffeomorphic constraints (Cen & Srivastava 2021). To further improve smoothness and geometric consistency, we apply an additional
normalization:
Ex Tip
Pie
This defines a neural operator that outputs valid, smooth, and data-adaptive warping functions. The construction of (t) via the monotone cumulative sum guarantees valid diffeomorphism. Squaring the latent features T; ensures non-negative increments, which, after two steps of normalization. enforces strict monotonicity (%g > 0) and the boundary conditions (0) = 0, (1( 1. In the continuous limit, this yields a function with a positive derivative, fulfilling the requirements for a diffeomorphism [0.
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2.2 SPECTRAL REPRESENTATION OF ALIGNED FUNCTIONS
Once trajectories are aligned by 7/ we interpolate the aligned function (.(1) on points ((x(())) using stable ID linear interpolation:
a(trr)+ + xill(r+1)) 2z(ler)
(t &
(t m(trr)) ((atte) ((((++11) - -7((tr)
Rather than vectorizing (t) into a high-dimensional grid, which ignores smoothness and leads to inefficiency, we embed aligned functions into compact spectral basis:
(.(
(1)
(4)
where { Fourier basis functions and Gij are coefficients estimated by least squares:
f [a:(1)
Σ a di G
c; arg min
(5)
{ayt
with = (t)) (t)) dt and d, (t) dt. This spectral representation, inspired by classical functional data analysis, acts as a form of neural Fourier features, yielding compact, smooth, and regularized embeddings aligned functions. We employ 1D linear interpolation (Fq. (3) for
its numerical stability and computational efficiency, and a Fourier basis expansion (Fq. (4) as it introduces no additional regularization hyperparameters. Crucially, both methods satisfy the Lipschitz continuity condition required by Theorem 3.3
2.3 CLASSIFIER NETWORK
The spectral coefficients é E K are fed into a fully connected classifier with l2 hidden layers and ReLU activations:
t0 ( + bb),, 1= +,, I,....,,4 + +/1+12211 1
(6)
followed by softmax layer: ((()) (a (1+12-11) + b(l++)))). We use three-layer MLP (12 = 3) with hidden units of size and 4, parameterized by O2.
2.4 OBJECTIVE: CONTRASTIVE GEOMETRIC ALIGNMENT AND CLASSIFICATION
We optimize the model with parameters θ {O1, O2] by combining geometric alignment and discriminative classification.
Class-aware Contrastive Alignment. To align functional trajectories while preserving class struc- ture, we adopt the square-root velocity function (SRVF) representation g(() = < Sri vastava & Klassen 2016). For warped trajectory <()(t)), the SRVF is
(t)()) (t).
For sample 1, the observed SRVF vector is (+(%)) ((g: (tai), ..... (g: (t).)) We detine a contrastive alignment loss:
(i(0)) = Σ EiyrE (1Q(()) - @66
Σ (Q(h) - -011-1
NO)
#=1
ISucesC
where Q66) is the class-wise SRVF mean. The first term encourages intra-class alignment; the second is a contrastive separation term that increases inter-class margins. Together, this yields a new contrastive-geometie alignment objective.
Classification Loss.
The classifier is trained using standard cross-cntropy:
(2(
Uij log viyj.
(8)
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Algorithm Training DeepFiCC
Require: Training data (t,) 9))
})
1: Set Hyperparameters: Size basis function K, loss-related {a, 3). and training parameters f (epochs E, batch size, learning rate, etc.)
2: Initialize Parameters θ Omitial
3: for to E do
4: Forward Propagation:
(I) Compute 7: (t.) for each (t,))
(2) Warp (tz) (t)) to obtain (t), calculate its SRVF Qi(%)) and extract coefficients č;
(3) Pass e; through the classifier compute (b())
(4) Compute the loss L())
Backward Propagation: Update θ via AdamW optimizer using #C(e) 0
6: end for
7: Return Trained DeepFRC with optimized parameters O*
Joint Objective. The full objective integrates both alignment and prediction:
( (1))) (L2(O),
(9)
where 3 balances geometric aligmment and classification accuracy.
Summary. Integrating Sections 2.1 2.4 obtain DeepFRC (Deep Functional Registration and
Classification), an end-to-end framework coupling (i) neural deformation operator for alignment, (ii) spectral embedding for smooth functional representation, and (iii) a classifier guided by contrastive contrastiv--gommetr geometric loss (Figure []]. To the best of our knowledge, DeepFRC is the first model to unify alignment and prediction of functional data through diffeomorphic neural operators and contrastive geometry, offering new paradigm for learning invariant representations of misaligned functional trajectories. A natural extension of our framework allows it handle d-dimensional functional inputs (d 2) by simply considering (f,) E Rnd under the assumption of shared warping process across dimensions, without requiring additional structural assumptions.
2.5 OPTIMIZATION, MODEL SELECTION AND COMPUTATIONAL COMPLEXITY
The objective in Eq. (9) is optimized via stochastic gradient descent over the model parameters θ using the AdamW optimizer (Ilya & Frank 2019) for efficient and stable convergence, particularly in deep architectures (Diederik Lei (015). Gradients with respect each 0 θ are computed via the chain rule, as detailed in Eqs. (A. (A22) of Appendix A The complete training procedure is outlined in Algorithm
The number of basis functions K is empirically set to 100 and shown to be robust across tasks (Section 4.2). Hyperparameters a and 8, which balance alignment and classification loss terms, are selected via data-spliting methods following (Wang et al. 2023); further model selection details are provided in Appendix B
DeepFRC achieves linear time complexity O(N)) with respect to sample size N and sequence length n. contrast, traditional alignment techniques such as dynamic time warping (DTW), which rely on dynamic programming, incur quadratic cost O NN k) (with i n2), rendering them computationally infeasible for long sequences (Sakoe Chiba
1978
Chen & Srivastaval 202T.
THEORETICAL ANALYSIS
We now discuss the theoretical properties of the proposed model, demonstrating that it achieves low registration error under elastic functional data analysis (EFDA) (Srivastava & Klassen 2016) framework and low generalization error under mild regularity conditions. Detailed proofs are provided in Appendices Caddd] D
The phase variability is modeled using monotone and smooth transformations within the set I {r: [0, [0, (0) (1) 1, 0}. Under the framework of elastic functional data analysis
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(EFDA)
(Srivastava et al. 201 1. the optimal warpings are defined as: " argminQuet (7) -
(x)
Σ ()
Q
argmin (mr)
Ta E T, where "reg" means "registration". For any other estimated
NG
j=1
warping Y, the registration error can be quantified as AQre((r"; 4)
Theorem 3.1 (Low Registration Error). Assume that: (i) The proposed deep neural network has sufficient capacity, (ūi) Each warping function 7(t) belongs the admissible set I and is continuously differentiable, and (iii) The SRVF of qt)) of each curve xxt) is continuous and bounded.
Then, for any e 0, there exists an estimated 4y produced by the model such that:
10 5) <.
Remark 3.2. Theorem 3.1 provides theoretical foundation for our approach, establishing that neural networks can approximate smooth warping functions in SRVF space, thereby justifying the use of learnable diffeomorphic registration module. While direct empirical verification of the approximation eπror is challenging due to the inability to obtain globally optimal warpings Y in real data, we validate the practical observability of this result through controlled simulation studies where the ground-truth warpings are known.
We further show that the proposed model achieves a small generalization error. Let fe x + represent the mapping of the proposed DeepFRC model with a fixed architecture. Define the population risk and empirical risk as R(()) = Ef((f( (x),, y)] and Rt (+) きे. Σn ((fe( vi), respectively, where denotes the individual loss function. The generalization error is given by
A Rgann (6) |E5, (R(O) R, (+)]l,
where is the parameter set estimated via a random algorithm A based on a random sample S. During training, we assume the AdamW W optimizer uses an inverse time decay or a custom decay function to ensure the learning rate ae is monotonically non- non-increasse ren-iccreasing with a, I co/t, for some constant Co > 0, and that the algorithm runs for To steps. The following result builds upon Theorem 3.8 from Hardt et al. (2016) and Theorem 2 from Yao et al. (202T).
Theorem 3.3 (Low Generalization Error). Assume that: (i) There exists en > O, such that (Q(u) C and thig 2 co, and (ūi The weight θ is restricted to compact region. Then, for the weights estimated by the proposed model, there exists a constant eo > 0, such that
Q for
To T1-1/c0
A Rggn
(
N
Remark 3.4. The assumptions of Theorem 3.3 are empirically satisfied and guide model design.
Class separation IQ Q(") 2 en supported by distinet SRVF means (Figure Table 1. The probability floor thi 2 is enforced via additively smoothed softmax, keeping all probabilities above 10-4, Although non-constructive, the bound informs hyperparameter tuning: a is chosen to maximize separation, while 3 balances classification and alignment.
Discussion of Assumptions. The assumptions in Theorems 3.1 and 3.3 (Lipschitz continuity, boundedness, compactness) are standard for theoretical analysis. our framework, these hold by construction: the ID-CNN uses continuous activations, and training (e.g.. with weight decay) implic- itly constrains parameters, ensuring the deformation operator yields a Lipschitz-continuuus velocity field. The resulting warping function y(t), constructed via the cumulative sum of squared velocities, therefore lies in the admissible set I' with (t) > O. Violations (e.g., from unbounded weights) would break diffeomorphic guarantees and degrade empirical performance, but our experiments confirm that standard training practices suffice to satisfy these conditions, ensuring both theoretical validity and practical robustness.
4 EXPERIMENTS
We present the results of functional data registration and classification using DeepFRC, evaluated on both simulated and real-world datasets. Simulated data provides insights into registration, recon- struction, and classification progression during training. Real-world datasets allow for performance comparisons with state-of-the-ne methods.

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Evaluation Metrics. We evaluate method using metrics for registration, reconstruction, and classification. Registration quality is assessed by the alignment error (AQrey) between estimated and true warping functions on simulated data, and by the Adjusted Total Variance (ATV) (Chen & Srivastava, 2021) on both simulated and real data, where lower values indicate better alignment. 'The fidelity of the recovered smooth process is measured via the correlation between true and estimated basis coefficients (simulated data only), Classification performance is reported using accuracy (ACC) and the macro-averaged Fi score (Sokolova Lapalme 2009), with higher values being better. Detailed metric definitions are provided in Appendix E.1
4.1 SIMULATION
Synthetic Data Generation. We generate a balanced two-class functional dataset (a;(t), L.V. where each function x;(() combines amplitude and phase variation. Amplitude modeled by sums of Gaussian bumps, while phase variation is introduced via nonlinear warping functions % (t). Class-specific parameters and full generation details are provided Appendix E.2
Results. Figure 2 demonstrates the iterative improvement of joint registration and classification on a simulated dataset. Initially, the raw curves of two classes are visually inseparable (Fig. aa)). Through optimization, DeepFRC progressively refines the alignment, leading well-separated, class-specific templates (Fig. (()()).. This visual improvement is quantified by a rapidly increasing Pearson correlation p(c", .)). indicating accurate reconstruction of the true, unwarped functions (Fig. (e)). Concurrently, both registration and classification metrics improve steadily as the combined loss converges (Fig. fh))),), illustrating the synergy between the two tasks. The model's strong generalization is confirmed on a held-out test set (Figure A1 Appendix H.2
10.
M
MM
38
60
60
1.0
1.0
16
1.2
1/2
Epech
Epoch
3.0
Repi inratts loss
0.4
5.
Pi 00111
Loss
epachs
Figure 2: Contribution of íterative optimization in training DeepFRC on simulated two-class (yellow and blue) data. (a)-(d) depict the progression of registration Epochs 0, 5, 15, and 30. (e)-(h) show the improvement in reconstruction, registration, classification, and loss over SGD epochs.
4.2 REAL DATA APPLICATION
Real-World Datasets. We evaluate our method on five publicly available datasets, selected for their
prevalence in functional data analysis (Ramsay Silverman] (002) 2005) and relevance to phase variation: Wave, Yoga, Symbol, and MotionSense (Malekzadeh et al.) 2019). Wave, Yoga, and Symbol are sourced from the UCR Time Series Classification Archive, while MotionSense is from Kaggle For the Symbol dataset, use both binary and three-class subsets. The Wave, Symbol, and MotionSense datasets are class-balanced, whereas Yoga is imbalanced. In terms of dimensionality. Wave, Yoga, and Symbol are oie-dimensional, and MotionSense diceeddmensional. Further details are provided in Appendix E.3
Baseline Models. We benchmark DeepFRC against several deep learning models for functional data analysis, including a joint regissrationnlllssffccciiin model (TTN (Lohit al. 2019)), registration-
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only model (SrviRegNet (Chen & Srivastaval 2021 ), and five classification models: FCNN raw. fourees FuncNN (Thind et al. 2020). ADAFNN (Yao et al. (02T), and TSLANet (Eldele et al.) 2024). To ensure fair comparison,
pipelines by combining SryfRegNet with each classification model (excluding TTN, which is already joint). A de-
evaluate sequential iegistration-classifcctttt
tailed description of all baseline methods and implementation details provided in Appendices E.4
and E.5 respectively.
Performance Comparison. Table reports both alignment and classification metrics across five real-world datasets. Among all models, only DeepFRC and TTN jointly perform registration and classification. DeepFRC outperforms TTN by using elastic FDA for enhanced functional smoothness, delivering superior registration across all datasets and significant classification improvements on Symbol and MotionSense data. Unlike pipelines combining SrviRegNet and classification models which disregard label information during registration DeepFRC achieves superior alignment, leading to more interpretable transformations and improved downstream classification accuracy. We observe that DeepFRC achieves classification accuracy comparable TSLANet, recent state-of-the-art time-series model that outperforms many transformer- alternatives, across all datasets. This highlights important trade-off: while powerful classifiers can learn invariance phase variability from amplitude features alone, DeepFRC's explicit class-aware registration actively enforces this invariance, simplifying the classification task and yielding more interpretable alignments. Figure A2 (Appendix E.5 plots the training loss versus epochs for DeepFRC across all datasets, confirming convergence.
Table 1: Quantitative comparison of registration and classification performance with state-of-the-ar approaches across five real datasets. Bold indicates best results.
Ware
Yogn
Symbol classes)
Symbol (3 classes)
MotionNease
Model
ATV
.BCC
ATV
ACC
ATV
ACC
ATV
F.
JCCC
DerpFRiC
566
96.4%
0.95
16.2
89.8%
0,900
4.8
96.8%
0.989
3,2
96.3%
0.963
25.0
95.0%
0,852
TTN
6.3
94.7%
0:348
57.7
83.4%
0,904
3.5
92.0%
0.918
4.5
93.39
0.33
35.1
85.0%
0157
SevRegNiss RCNN.
7.3
94.6%
0.947
136.0
81.0%
0,830
14.8
94.5%
0.942
6.5
94.79
0.947
37.7
90.09
0.91)
Sex RONNNN
7.3
94.9%
0.950
136.0
$4.0%
0,852
14.5
95.05
0.949
6.5
96.09
0.951
37.7
90.09
0.919
Stefleeg Nei4FFiccN
7.3
95.7%
0:987
196.0
89.0%
0.968
14.8
93.5%
0.93
6.5
94
74
0.947
37.7
90/74
0.889
7.
94.69%
0.949
196.0
73.4%
0,753
14.8
89.05
0.894
6.5
94
0.941
37.7
85.0%
08877
Sev TeIL-SLAN
7.3
96.4%
0.961
196.0
80.3%
0.884
14.8
95.5%
0.955
6.5
96.3%
0.966
37.7
95.0%
0.952
Alignment Visualization and Interpretability. Figure alignment quality on the Sym- bol (3 classes) dataset. DeepFRC produces smooth, class-separated functional alignments, while TTN distorts class-specific trajectories (e.g., purple), and SrviRegNet loses inter-class separation (e.g., blue vs. yellow). Both baselines significantly degrade interpretability (see also Figures A3 and A4 Appendix E.5)). DeepFRC's superior alignment improves inference reliability and preserves generalization performance critical for real-world applications. Beyond quantitative metrics, these alignments offer significant practical interpretability. For MotionSense, DeepFRC's warps syn- chronize biomechanical events like heel-strikes and mid-swings, producing crisp, physiologically meaningful activity profiles. For Symbol, it explicitly corrects for variable writing speeds and stroke orders, normalizing user-specific kinetics to reveal canonical symbol shapes. This ability to distill noisy, misaligned observations into clean, class-specific functional templates demonstrates the method's utility not only for boosting classification accuracy but also for providing domain experts with intelligible and representative data patterns.
Symboota
380
140
200
100
200
100
400
Raw
Aligneeett by DeepFRC
(c) Alignment by TTN
Alligmment by SrvfRegNet
Figure 3: Visual comparison of alignment the Symbol (3 classes).
Ablation Studies. The ablation study in Table 2 evaluates the individual contributions of the core components of our method: the Neural Deformation Operator (N.D.O.) for registration, the
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Spectral Representation (S.R.), and the Classifier Network (C.N.). The results demonstrate clear synergy between these modules. Removing the registration component (DeepFRC w/o N.D.O.) significantly degrades classification performance, particularly the Yoga, Symbol (2-class), and MotionSense datasets. Conversely, removing the classifier (DeepFRC w/o C.N.) impairs registration accuracy. The removal of the spectral representation (DeepFRC w/o S.R.) adversely affects both tasks across all datasets. These findings underscore the importance of the spectral representation and the tight coupling between registration and classification within DeepFRC's unified architecture. The performance advantages of the full DeepFRC model are statistically significant. As reported in T paired f-tests across 10 random seeds confirmed that removing any core component leads to significant performance degradation (p < 0.05 for all relevant metrics across all datasets; p < 0.01 for the vast majority of tests). This provides robust statistical evidence that the Neural Deformation Operator, Spectral Representation, and joint optimization are all essential to the framework's success.
Table 2: Ablation study: contributions of three components in DeepFRC.
Symbol classes)
Symbol
MoticaSSuue
Model
Fi-ccme
DespFRC
0.965
0.989
488
96.0%
0,599
0.963
95.0%
0.952
DespFRC
94.4%
0946
53.3%
94.7%
0.947
DegFRRC
175
5.
DepPFRC
Table 3: P-values from paired t-tests (10 runs) comparing full DeepFRC against ablated variants.
Bold indicates significance (p < 0.05).
Hypptteesi Tests (evalue)
Ware
Symbol clases) ACC
Symbol clasess)
MotionSense ACC
ATV
ACC
Fr-sces
ATV
ACC
Fi-ecrr
ATV
F Fysorre
.TVV
ACC
F F-scur
ATV
Fi-sces
Full vs. wio N.D.O.
8,0045 8,0000
BJ0551
TUNES
0,0033
ILINIS BU1180
0,0086 0,0179
0,0010
IJR00
0,0000 0,8091
0,0009
Full vs. wio S.R.
0,0011
BJ0880
0,8770 0,8000
0J1549
0,848
BJU82 IJ0000
8.0956
0,0001
IJN003
0,0025
0,0009
Fall vs wio C.N.
8.0000
0,0000
Sensitivity Analysis. We assess DeepFRC's robustness the choice of basis family and size K across five real datasets in Figure 4 The left panel shows performance is stable across basis types (Fourier, B-spline, polynomial). We attribute this robustness to two factors: (a) joint training adapts alignment to the inductive biases of each basis, and (b) class-aware registration loss guides the model toward discriminative structures, reducing basis-specific limitations. The right panel shows that increasing K improves performance up to around K 100, after which it plateaus. We believe this invariance arises from (a) upstream registration reducing functional variability, and (b) downstream layers adaptively re- reightgh pruning redundant basis functions.
ATV
18
Wove
Yoga
ALV
Nodiondense
42
6 Yoga
Wave
Symbol dlasses)
Symbol classes)
MotionSense
Figure 4: Left: Sensitivity to basis family. "F":: Fourier; "B": B-spline; "P": Polynomial. Right: Sensitivity to number of basis functions K.
5 DISCUSSION
Computational Efficiency. DeepFRC offers significant computational advantages during training, with average runtime of 60s on our real-world datasets dramaticcly faster than the traditional registration method DTW (1000s) and competitive with deep baselines like TTN (15s) and SrviRegNet hybrids (90-100s). This efficiency is achieved by eschewing pairwise dynamic programming. Although training time is on par with other deep models, DeepFRC achieves superior accuracy, striking excellent trade-off between performance and computational cost. A detailed analysis of

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training complexity shows DeepFRC requires an average of 1000 TFLOPs with 200k parameters, representing a favorable trade-off between the high accuracy of SryfRegNet hybrids (1500 TFLOPs, 200k params) and the lower capability of TTN (200 TFLOPs, 100k params).
Robustness to Noise and Reconstruction Accuracy. We evaluate robustness by injecting Gaussian noise (o O, 0.05, 0.10, 0.20) into synthetic data and measuring reconstruction quality via the Pearson correlation between true and estimated basis coefficients, As shown in Table A3] DeepFRC maintains high correlation under low-to-moderate noise, outperforming TTN and exhibiting greater stability than Nevt Even under high noise (ar = 0.20), its performance degrades gracefully while surpassing competitors, indicating that the smooth basis representation and joint optimization confer strong resilience.
Scalability to Large and Small Data. DeepERC scales efficiently to large datasets with O(Nn) complexity (Section 2.5) On the augmented Symbol dataset (100 x). it achieves near-raw per- formance (ATV=4.8, F 11-score=0094 vs. 4.8, 0.959) and outperforms all baseline models in both alignment and classification (Table A4p. This demonstrates that its performance advantages are maintained at scale. Simulations under sparse data conditions (samples, time points, or both) confirm robust classification, though registration requires sequence length 100 for reliability (Table A5 Appendix E.5
Robustness to Missing Data. We evaluated DeepFRC's tolerance to missing observations by randomly removing 5- 6-10 of data points from half of each real-world dataset (Section 4.2). imputing
gaps via Fourier splines (Wahbal 1990). Under identical hyperparameters, DeepFRC achieved comparable registration and classification performance complete data (Figure A5 Appendix E.5
Robustness to Non-Diffeomorphhc Warpings. To assess the sensitivity of DeepFRC to violations of the diffeomorphic assumption, we followed the same synthetic data generation protocol as in the main experiments and constructed additional test sets in which a proportion of curves were warped by mappings containing flat regions (i.e., intervals where ((t)) = O) covering a specified fraction of the domain. These perturbations mimic temporal "jumps" and constitute clear departures from strict monotonicity, As reported in Table A6 (Appendix E.3 DeepFiRC exhibits graceful degradation under such violations: classification accuracy remains above 90% even when 20% of the curves contain flat regions spanning 60% of their domain. This indicates that the model learns a smooth, diffeomorphismm-lik approximation the mis-specified warpings. Although alignment quality (ATV) naturally more sensitive to these distortions, the discriminative structure captured by the spectral features remains largely preserved.
CONCLUSION
We introduced DeepFRC, unified framework for joint functional registration and classification with theoretical guarantees and comprehensive empirical validation. ReepFRCC consistently outperforms state-of-the-art baselines and remains robust across real-world and simulated settings with noise, missing values, and scale variation. While empirically invariant to the choice of basis functions, its theoretical underpinnings remain open.
Our work also highlights key limitations and considerations for joint modeling, DeepFRC's perfor- mance may degrade on highly volatile trajectories with sharp, discontinuous changes, as relies on smooth, diffeomorphic warping assumptions. Furthermore, the class-aware alignment can be sensitive to significant label noise. A joint framework not universally optimal; separate processing remains preferable when phase variability is minimal, label information is unreliable, powerful classifier already exhibits strong inherent invariance to misalignment.
Future work will focus on adaptive architectures and robust loss designs that explicitly capture signal complexity to enhance generalization, particularly for challenging non- smooth functions and noisy label settings.
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ACKNOWLEDGMENTS
This work was supported by National Natural Science Foundation of China (NSFC) Youth Science Fund 112301353]. We used DeepSeek and ChatGPT to polish the language and improve the clarity of this manuseript.
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A OBJECTIVE FUNCTION GRADIENT
By the chain rule, the gradient of the objective function with respect to any 0 E is derived as follows:
Σ
(0.(%)) (Q.())) the
#E(yi=5)
8
Mii
0 E O1,
aC(O)
NO!
(A.1)
00
N
Ma
BA 39
E 22
Here, the gradient of the warped SRVF with respect to the warping function % is given by:
8Qi(n)) [K] ((v((t))) +((((t)))))) 0%
7(4x))
k L. L.....n
(A.2)
2 2V7(())
When computing the gradient of (1)) with respect the warped SRVF (Q)(7:)) the mean SRVF (Q15) is treated constant, updated using the arithmetic mean the warped SRVF from the previous iteration (Cen & Srivastava 2021p. For 0 E θj, the gradient
dr can be expressed as the product Dess is
89
of gradients from integration, fully connected layers, and the ID-CNN module. Similarly, 20 derived from the gradient of l2 fully connected layers, interpolation, integration, and the ID-CNN module Yann et al. 2015). For 0 E 22, 0yss involves the gradient from fully connected layers in the "00"
prediction module Yann al. 2015).
B MODEL SELECTION OF DEEPFRC
The model's hyperparameters include the size of basis functions K, loss-related parameters fa, B), and training-related parameters (such as epoch size E, batch size, and learning rate, denoted by vector nn). For the basis representation module, set K 100 with Fourier basis, and the sensitivity analysis will be studied in real data analysis. To select the other hyperparameters, we follow the procedure proposed by Wang al. (2023): the dataset {<x(tt), } is firstly split into two subsets with a 4:1 ratio. For each combination of hyperparameters, the model is trained by minimizing Crain (Eq. (9) on the larger subset, and testing error Liea is computed on the smaller subset. The combination minimizing fres is selected.
In the ID-CNN module of our model, we fixed the setting of the network structure (set 1 4, and set kernel sizes as 3-3-3 and the channel dimensions as 16-32-64 for the three hidden convolutional layers). To improve efficiency, robustness, and interpretability, we apply max-pooling (Nagi et al.. 2011) and 1D batch normalization (loffe & Szegedy (2015) after each layer to downsample inputs and stabilize training, and use global averaging (Krizhessyy et al., (017) followed by the final fully connected layer further reduce features and noise. In the classifier module,
set le 3 and nodes number n = (8, 4). The above configuration for ID-CNN module and the fully connected neural network classifier is widely used Chen & Srivastava (021): Lohit al. (2019), and works well across all the datasets in this paper.
C PROOF OF THEOREM 3.1
where [0, 7' is the observed time range, in the following proof. Consider the function f(z) = /I, which is continuous on [o, 1]. For any >> 0, there exists 51 > such that for all |r] - r2l < 51, we have |/=1- /x2/ < E. Similarly, since g is continuous, for any 8 > there exists 82 such that for all [2] x2 82. we have (g(x1) - (2x2) <
For simplicity, we assume 6; -Q. ......,47 F).
Let L be the domain of the neural network's output function, and I' be the domain of the alignment functions 7. Denote 4 9 - I' as the mapping from the neural network output to the aligmment functions. Since both input and output of the network are discrete, we consider values only at discrete
points. We define (*) g; (+) to maintain consistency between discrete vectors and continuous functions. The mapping is defined as:
Ti - 7/ + 7h
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(0) 2
k
k
(+)
T
Σ
(x)
2'
7
T
0=
Let T. be such that +(T;) = 7/-- Since (((T)) generality, that ||g:122 = 1.
= Φ (T:)) for all 1 + 0, we can assume, without loss of
By the approximation property of convolutional neural networks (Yarotsky) 2018) and classical
universal approximation results (Cybenko] 1989; Funahashi 1989 Horni 1991; Stinchcombe
1999), for any da > 0, there exists neural network NNg with parameters O such that
sup Ne.)) - -rill2 < 64.
Let = = NoNNNNN()) (x.). We now aim to show that for sufficiently small |7 - frl. we also have lyv - -xl sufficiently small.
4:
Le=o 7ं (+)
E (+)
T
-
T
=
Σ 7ी (7)
Σ. (+)
Σ x= 72 (7)
Eh (t)
Σ 72 (+)
Es 78 (+)
x
+
Σ v= ?? (7)
T
Σ (+)
Σ 7. (+)
Σ == #8 (7)
T
8 T
(+)
04 + (1 + 02) (2 + 64)60
+ f:
(1-6a)²
84 + 276 = -M2"0 8s.
To T
+ ft
Therefore, we have
T
11% - Rill2 S
M = MLP05.
k=
By Hölder's inequality 1lf S 115111 S VT flls we have
||=- - f S VT + I 5s.
Note that |/rvlla
2 5
f
1. On the other hand, for ys we have
k
Σ/ Σ 2a (+)
Σx R/ (+)
7
7/4
T
T
Σ 0 7s (+)
Σx= 9 (4)
Σ 0 2 (4)
Ex 7x (4)
Σn 9 (i)
9o (+)
ITo i (+)
2/ (4)
_ 7o (+)
n 2 (+)
Σx n io (4)
Σ+1 0 2 (+)
- Rr (+)
+
0 7o (7)
Σn 7n (f)
= 3o (7)
Σラ
Σ E (7) Rr (+)
Σ (+)) 3/ (7)
+ +llilll
ΣT 7s (+)
EET=o to (i)
(Er Ra (7)
k T + 4:
179111 + (176ll1 + VT TMM b.)
54
l1balla (1 - vT + 410
< 11 on for some (
Such that we have
117 - -Bill2 S
Σ
M = -Mº?0a 6s-
k=0
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Recall that Qi(1s) (+) = (a o (+)) v/: (4). Using the conditions that qi and % are bounded, we can bound the difference between Q (7)) and Q((5x) by:
(r(2s)
(.(%))
T
k
h:
k T
h:
k
k
9: %
T
7/2
T
V
+
V
7/
T
4:
T
qi
ft
T
k
k
k
)
sup [q:(t)
|
As
+
sup Vir(t)
qe
qt
T
T
T
T
3.(
442
7/(
We use
to approximate the derivative for 9%. so we have
%
1:
3/4
h:
7/8
h
Y
k+1
T
%/
(
(A7²)
Yi
k+1
T
7/
1=1
T
[++1
T
f/a
T
T
T
T
2
T
T
T
k
k+1 T
-%
( (K=11
k:
k
7/
T
2
T
sup
74
T
-/8
T
T
For 0 < dg < 1, there exists a T > 0 such that:
k T
7/
7i
As
< 83. Vk.
2 T
Here can choose 83
81
2,
82
51 }. and de < 82. Then, we have min(d2, 27
h:
h: T
< <s,
9: %%0% %
q: o 9
<
T
T
T
such that
Qi(7s) - (+(t))
(4)
(
s M0 where M is a constant.
This means at each fixed point $. the difference between (1(7) and Q4 is bounded by M (3) e. and it is straightforward to have
- s VTM²,s.
We finally get
AQrog("", ,)) (r)((7") Qmg(
c 1
1Q(()))) - -2(0)))00012
1
Σ
11((()) - 2 (1) 112
NG
NO
iE (
j=1
(E(#==>)
- - (1)
Σ
[Q(()))
12
NO
iE ():
We now simplify NUT Eie(u= [1Q(6))) ((Q.(t)) -0' 12 as
1
||a:112 + ||e|| + 2(a;, b.) + 2(bbi, c) + 2( e)/
NO
where a; = Qi(7)) Q,(1)),, Q((+)) - "0) _c -== --
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inequality: By the Cauchy-Schwarz inequallytyyy (xr u) S || have ||a:||2 S VTM and ||c|2
NoT Eie(n=s) a.11 - SVMM E. We also have ||bزه|2 S MM(4) due the boundness of q and %
Thus, have
Σ 110(%) -Q
2"1 1 AT SA () E M Σ 11Q(())
NO
NG)
$E(==}}
:E(==)}
Therefore, is straightforward to have
AQrey Theeg("; (7) 5 (M e² +4CCVMM
D PROOF OF THEOREM 3.3
Based on Theorem 2 from Yao al. 202T), we only need to verify that the loss function and its gradient both Lipschitz continuous. Since the mean SRVF (Q())s) are treated constants when computing their gradients, and we have the condition that there exists eg >> 0, such that (Q(") Q 2 eo for 1 I u < v I C, we can omit the term Liguresoallo alli(u) - Q(1)) 11-,, and simplify the loss function DeepFRC for any individual as
C
((fe( 91)
Σ 1| 1
Σ Wiy log Paj.
The loss function is obviously continuous for 1i 2 Ep. as 1 is continuous everywhere except at zero. Similar to Eqs. (A.1) and (A.2), the gradient of / with respect to any 0 E θ can be easily derived. These gradients can be expressed as products of gradients from ReLUU, max pooling, etc., which are Lipschitz continuous. We note that both the ID linear interpolation and Fourier basis expansion in the stage of spectral representation in DeepFRC satisfy the Lipschitz continuity condition. Therefore, the theorem follows directly from Theorem 3.8 in Hardt al. (2016).
E EXPERIMENTAL DETAILS
E. EVALUATION METRICS
Registration Metrics:
Registration Error (AQue This metric, defined within the EFDA framework, quantifies the discrepancy between the estimated warping f and the ground-truth y". It is applicable only to simulated datasets where y is known. Lower values indicate better alignment.
Adjusted Total Variance (ATV): For both simulated and real-world data, we use ATV defined as: ATV
to assess alignment without ground-truth warping functions. where TV.. (Chen & Srivastava] 2021 is the total variation
7cr Σigucesc
TV
between classes u and v, and me) is the mean function of the aligned curves. ATV accounts for inter-class distance, making it suitable for evaluating alignment of similar classes. Lower ATV indicates better alignment.
Reconstruction Metric:
Coefficient Correlation (p): To evaluate the recovery of the underlying smooth process, we compute Pearson's 's correlation p(c", between the true (c*) and estimated (é) Fourier basis coefficients. This metric is used only in simulations where c" is accessible. Higher values indicate better reconstruction.
Classification Metrics:
Accuracy (ACC): The proportion of correctly classified samples.
Macro-averaged Fi -core: Provides a balanced measure of performance across classes, especially important for imbalanced datasets (Sokolova & Lapalme 2009.
Higher values for both ACC and Fioce indicate superior classification performance.
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E.2 DESCRIPTION OF SIMULATED DATASETS
We construct a balanced two-class functional dataset (z:((), yi Ji=1. where each function z;(() for I [0, 1] is generated by composing amplitude and phase components. The amplitude function is defined as the sum of two Gaussian bumps:
1 (t 103 (-2
21 dis 14, 01) = exp
where and are independently drawn from uniform distributions, and μ is fixed constant. To introduce phase variation, we define warping function (t(() as
exp(bit)-1 exp(b;)
if b;
+ with by U(-1.5,1.5) if b; O,
(l)
Such constructions are widely used for modeling functional data with phase variability (Kneip & Ramsay 2008 Tucker et al.) 2013). We then generate each function as:
Σ =1 (1)((t)) a a
(05)
(0))
(05)
if yi = 0. if yi
zt( =
Eソ= (1)((t) a;
(15) p00)) (1)).
1,
where the parameters {a. ,f (30 for c = 0, 1 are summarized in in the following table. We simulate N 6000 functions, each evaluated at 1000 time points. We split the balanced dataset into training, validation, and test sets with sizes 1600, 400, and 4000, respectively.
Table A1: Parameter setting for simulated data
Index
(01)
(02)
(11)
(12)
13 U( 0.5) 0.250
12.5 UC 1,
12 UT 1,
13 Ui 1,5, 1.5) 0.695
0.715
0.225
0.06
(-000003 0.003)
0.075
(0000003,, 0.003)
0.06 + U( (00.003, 0.003)
0.1 ('-0.003, 0.003)
2.0
15:
15-
10
0.3
05-
0.0
-03
-05
700
400
500
800
1003
200
400
800
1200
Test
Raw
Test Aligned
Figure A1: Visualization of alignment by DeepFRC on simulated test data, with evaluation metrics P 0.976, AQreg = 0.056, ATV=0.504, ACC 1.000 and Fi-score 1.000.
E.3 DESCRIPTION OF REAL DATASETS
The first four time series datasets one-dimensional functional data, from the UCR Time Scries Classification Archive www.timnsericcclasffcttoooo
and the last dataset is three-dimensional
functional data, from Kaggle (www.kaggle.com/
The dataset (Liu & Yang 2009) consists of eight simple gestures generated using accelerometers, collected through particular procedure. For participant, gestures gathered when they hold device and repeat certain gestures multiple times during a time period. The dataset includes X, Y, and Z. dimensions with classes. Here, selected the X dimension and classes and 8 for the
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experiment. There are 1120 samples with label partition 591/529, and the train/validdaiiontess split is 320/80/720, with each sample having 315 time points.
The Yoga dataset consists of images of two actors (one male, one female) transitioning between yoga poses in front of a green screen. The task is to classify the images based on the actor's 's gender. Each image transformed into one-dimensional series by measuring the distance from the outline of the actor to the center. The dataset contains 3300 samples with label partition 1770/1530, and the train/validation/test split 800/200/2300. Each sample has 426 time points.
The Symbol dataset involves 13 participants, who were asked to replicate a randomly appearing symbol. There were 3 possible symbols, creating total of 6 classes, and each participant made approximately 30 attempts. The dataset contains X-axis motion data recorded during the process of drawing the shapes. We conduct experiments on both the 2-class and 3-class cases using the Symbol dataset. In the 2-class case, there are 343 samples with label partition 182/167, and the train/validation/tess split is 115/28/200. the 3-class case, there are 510 samples with label partition 182/167/161, and the train/validaatinn/ess split is 168/42/300. Each sample has 398 time points.
The MotionSense Malekzadeh et al., 2019) dataset involves multivariate time-series signals recorded from smartphone sensors during six daily activities performed by 24 participants. From this dataset, we selected three signal channels (G.x, G.y and G.z) from two actions (walk and jog) to construct binary classification task, containing 96 samples with 200 time points cach. The train/validation/testt split is 60/16/20.
E.4 INTRODUCTION OF BASELINES
Here we provide brief introduction of the compared baseline methods:
TTN (Lohit et al.) 2019) ; A method for joint alignment and classification of discrete time series but does not account for curve smoothness.
SrvfRegNet (Chen & Srivastava 2021): An unsupervised method that aligns raw functional data via ID CNN.
FCNN A model that discretizes raw functional data into vector, inputting it into fully connected neural network.
FCNN A model that transforms the functional data into Fourier basis coefficients and then feeds the resulting vector into fully connected neural network.
FuncNN Thind al.
2020 A model that inputs entire functional curves directly into fully
connected neural network.
ADAFNN (Yoo et al.
2021 ): A model that inputs entire functional curves, adapting the bases
during learning.
TSLANet (Eldele et al.
2024): A model that inputs entire functional curves, with ASB (Adaptive
Spectral Blocks) and ICB (Interactive Convolutional Blocks) as core structure.
E.5 MODEL IMPLEMENTATION DETAILS ON REAL DATA
For all real-world datasets, the functional input is Z-score standardized entry-wise based on the mean function and standard deviation. To handle missing values (such as NAs) in the functional input, we use the Bas sSmoother function from the skfda. wetaaappproossessee innicesssinn..e nng... smooth h hing module in Python to impute missing values. All of these baseline models are trained until the best performance is achieved. All the experiments are conducted on NVIDIA GeForee RTX 3090 32G. The details of implementation for all models provided as follows:
The DeepFRC network is implemented as a PyTorch neural network. The hyperparameter settings for DeepFRC are shown in Table A2 In the discussion stability with respect to basis expansion for DeepFRC, we compute Fourier, B-spline, and Polynomial scores as follows: The Fourier scores are obtained using the fast Fourier transform function rfft from torch. nn. fft, extracting the first 100 scores. The B-spline scores derived from the cubic B-spline basis with 98 knots uniformly distributed over the interval [0, 1), resulting in exactly 100 B-spline basis functions. The Polynomial scores obtained using the Chebyshev polynomial orthogonal basis, taking the
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Published as a conference paper at ICLR 2026
Table A2: Hyperparameter setting for DeepFRC on four real datasets
Case
a
Irieg
Trelass
Wave
3000
50
le-3
1e-3
Yoga
600
15
le-3
5e
Symbol (2 classes)
300
15
le-3
le-3
Symbol (3 classes)
10
1e-3
le-3
MotionSense
15
1
1e-3
1e-3
first 100 Chebyshev polynomial functions. The B-spline and Polynomial scores are both calculated using the MasisSmmothee function from the prefdaappepzocesiing smoothing module in Python.
The model Srv Re processes functional data through learnable pre- nee-wapping block with three ID convolutional layers (16, 32, 64 channels, kernel size 3, RcLU/BBtchhNormpooiiinn followed by a fully connected linear layer to generate warping functions. It applies time warping by minimizing the SRVF loss under an MSE criterion. The input data is the full functional data requiring alignment. The output of this network is the time- time-warping function that can generate aligned functional data. We use the default architecture for the models SrvfRegNet.
Both FCNN raw and FCNN rourier have the same architecture: 3-layer MLP, with the input layer consisting of n observations or 100 Fourier scores (by default), followed by hidden layers with dimensions [16, 8], and finally outputting the classification results. Each layer is equipped with Layer Normalization, the ReLU activation function, and a dropout rate of 0.1. The input of FCNN is the data, while the input of FCNNrocriar is derived from the first 100 Fourier coefficients of the functional data.
The models FuncNN and ADAFNN both take the entire curve as input. FuncNN uses a manually chosen basis expansion, while ADAFNN employs adaptively learned basis layer for representing curve data. We use the default architecture for the models FuncNN and ADAFNN.
The model TSLANet takes univariate or multivariate time series as input, first processing them via patch segmentation and positional embedding to retain temporal order. Its core structure consists of Adaptive Spectral Blocks (for dependency capture and denoising) and Interactive Convolutional Blocks. We use the default architecture for the model TSLANet.
A note on handling multidimensional functional data is necessary. Models including DeepFRC, TTN, TSLANet, and FuncNN natively support such inputs. For other baselines, we implemented the following adaptations: SrvfRegNet was extended to process multidimensional data analogously to DeepFRC. For F aaN and ADAFNN, we concatenated signals from channels into a single mee-immeniionll vector. For FCNNgouries from each channel as the model input.
used the concatenated basis expansion coefficients
Training loss DeepFRC
500
Wave
Yoga
Symbol(2 classes)
400
Symbol/ clesses)
MotionSense
200
100
50
100
150
175
200
epochs
Figure A2: Training loss vs. epochs by DeepFRC across the five real-life datasets.
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Published as a conference paper at ICLR 2026
Wave
26
360
100
100
Yogn
4
400
249
140
Raw Data
Alignment by DeepFRC
Alignment by TTN
Alignment by SrvfReghett
Figure A3: Visualization comparison of alignment performance by DeepFRC, TTN and SrviRegNet across three two-class real datasets
Hotionsenss
164
154
180
380
Raw Data
Allgnment by DeepFRC
Allgnment by TTN
Allgnment by SrvfRegNett
Figure A4: Visualization comparison of alignment performance by DeepFRC, TTN and SryfRegNet on MotionSense dataset in 3 channels G.x, G.y and G.z.
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Published as a conference paper at ICLR 2026
40
F1.score
1.0
30
0.8
--score
20
0.6
0.4
10
0%
5%
10%
0% 5% 10%
0% 5% 10% Symbol (2 classes)
0% 5% 10%
0% 5% MotionSense
10%
Wave
Yoga
Symbol (3 classes)
Figure A5; Performance of DeepFRC with different missing rates for raw data across five real- wollw datasets.
Table A3: Representation coefficients correlation on different noise level on synthetic data
Noise level æ ~ N(0, o2)
Model
0
0.05
0.10
0.20
DeepFRC
0.983
0,982
0,979
0.969
TTN
0.541
0.529
0.501
0.490
SrvfRegNet
0.969
0.966
0.961
0.950
Table A4: Quantitative comparison of registration and classification performance on large scale real dataset: 2-class Symbol dataset (100x) ). Bold indicates best results.
Model
ATV
ACC
Fi-scoee
DeepERC
4.8
94.47%
0.944
TTN
6.4
90.46%
0.897
FCVReegNet+FNNN....
12.8
93.29%
0,933
StviRegNet+FCNNonnd
12.8
93.08%
0,930
SrvfRegNat+FuncNN
12.8
92.23%
0,921
StvRRegNet+AAANNN
12.8
90.08%
0,942
StviRegNet+ TSLANet
12.8
94.02%
0,940
Table A5: Robustness under data scarcity. N and n represent sample size and sequence length, respectively.
Sparse samples (n
1000)
Sparse time points (N
6000)
Sparse samples and time points (N 200)
Evaluation
N = 200
N = 100
N = 50
n 200
100
50
(100. 100)
(50, 50)
ATV
0.580
0.784 100.0%
0.867
0.605
2.105
2,571
1.097
2,454
12.492
ACC
100.0%
100.0%
100.0%
99.3%
98.2%
100.0%
95.0%
80.0%
Fi-score
1,000
1.000
0.992
Table A6: Robustness to non-diffeomorphic warpings on synthetic data, prob: proportion of curves that are warped using functions with flat regions (zero derivative). flai_frac: flat fraction of the domain.
Original version
prob 10%
prob 20%
Evaluation
flat frue
Mat_frac- 0.4
flat_fruc frac 0.6
flat flat_frue 0.2
flat flat_frac frac 0.4
flot_faac 0.6
ATV
0.50
6.82
13.91
16.57
14.72
25.12
32.15
p ACC
0.976
0.940
0.920
0.911
0.881
0.847
0.828
95.25%
94.15%
90.43%
Fy-score
1.000
0.958
0.952
0.943
0.935
0.932
0.913
22


Reviews:
[{"review_id": "doxkYtirTr", "reviewer": "Reviewer_PLqL", "summary": "This paper presents DeepFRC, a deep learning framework that jointly performs functional registration and classification for functional or trajectory data. It integrates a neural deformation operator with 1D CNN to learn diffeomorphic time-warping functions for temporal alignment, a spectral representation module based on Fourier bases for smooth function embedding, and a classifier trained with a contrastive–geometric loss to align within classes and separate between classes.\n\nTheoretical analysis shows that DeepFRC can approximate optimal warpings and achieves bounded generalization error. Experiments on both synthetic and real-world datasets (Wave, Yoga, Symbol, MotionSense) demonstrate improved alignment and classification accuracy over baselines such as TTN and SrvfRegNet.", "questions": "1. Theorems 3.1 and 3.3 rely on many assumptions such as smoothness, bound and compactness. How realistic are these for real-world datasets with irregular sampling or noise?\n2. How would the baseline models perform when scaling to the large datasets (time complexity, ATV, ACC, etc.)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper presents a well-motivated problem by targeting the joint challenge of phase variability and classification in functional data analysis (FDA).\n2. Empirical results are consistent: DeepFRC outperforms alternatives across several datasets, enhancing both alignment and classification and confirming the effectiveness of joint optimization.\n3. Theoretical discussions and included proofs, effectively situate the model within the mathematical landscape of FDA.", "weaknesses": "1. The neural components (1D CNN, MLP, Fourier basis) are standard. The main contribution is integrating known elements instead of developing a new architecture or loss function.\n2. The baseline models are too few and outdated; it would be better to include more recent baseline models for comparison.", "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": "YCjYnRlutP", "reviewer": "Reviewer_q1ar", "summary": "The paper proposes DeepFRC, an end-to-end framework for function/sequence alignment and representation-driven registration/classification. The method builds a deformation operator for time warping, a spectral/spectral-coefficient representation of aligned functions, and a classifier that uses contrastive geometric alignment objectives.", "questions": "1. add more comparisons to current methods.\n2. explain why transform the latent features into a monotone cumulative sum can guarantee diffeomorphism?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The authors provide theoretical guarantees, proving that the model can approximate optimal warping functions and establishing a data-dependent generalization bound that links registration fidelity to classification performance.", "weaknesses": "1.\tThe authors claim “but rarely addressing both simultaneously”, however, there are several works addressing registration and classification simultaneously, for example,\n[1] Zhang, Y. and Telesca, D., 2014. Joint clustering and registration of functional data. arXiv preprint arXiv:1403.7134.\n2.\tWhy transform the latent features into a monotone cumulative sum can guarantee diffeomorphism?\n3.\tNovelty is limited, for registration, only introduce neural deformation operator for alignment\n4.\tThe claim that DeepFRC is \"efficient\" is poorly supported. The authors only report inference time, which is already longer than other learning-based methods. Crucially, they omit training time, computational complexity analysis, and a comparison of model parameters (number of weights). \n5.\tThe theoretical results rely on assumptions (e.g., Lipschitz continuity, compactness) that should be discussed more critically.\n6.\tThe comparison methods are from 2021, the authors should compare most recent methods, like the methods published in 2025 and 2024.", "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": "jqIpduq6qN", "reviewer": "Reviewer_UBjh", "summary": "This paper introduces DeepFRC, an end-to-end deep learning framework that jointly addresses functional data registration (alignment) and classification. Its main contribution is a unified architecture that integrates a neural deformation operator for diffeomorphic warping, a spectral representation for smooth functional encoding, and a class-aware contrastive loss. This approach eliminates the need for separate preprocessing steps and allows the alignment and classification tasks to mutually enhance each other. The work is further supported by theoretical guarantees on its approximation capability and generalization error, and is empirically validated to outperform state-of-the-art methods on both synthetic and real-world datasets.", "questions": "1. The paper uses a 1D CNN to parameterize the neural deformation operator (the registration module) but does not explain why Transformer-based models were not chosen. Can you explain the core reasons for this selection?\n2. It is recommended to supplement the experimental results with statistical significance analysis to prove that the performance improvements are statistically significant.\n3. The current computational efficiency comparison in the paper only mentions runtime. Would supplementing it with the total training time, model parameter count, and FLOPs provide a more comprehensive demonstration of DeepFRC's efficiency advantages?\n4. Can you further clarify the sampling methods of the four real-world experimental datasets? Additionally, could you explain how the proposed method handles highly irregularly sampled functional data?\n5. I would suggest adding more application analysis to demonstrate the effects of the proposed method.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper demonstrates good originality by proposing the first end-to-end unified framework for joint functional data learning. \nThe research quality is solid, well-supported by comprehensive theoretical analysis and systematic experimental validation.", "weaknesses": "1. It is not clear about the network architecture choices.\n2. Current computational complexity analysis is not sufficient.\n3. Lack of a detailed description of the datasets, which helps to understand the possible applications of the proposed method.", "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": "ZSjXMYZe9V", "reviewer": "Reviewer_gJ51", "summary": "The paper introduces a novel end-to-end deep learning model DeepFRC that integrates deformable registration with classification, tasks that are so far adressed usually separately. This is achieved utilizing a diffeomorphic neural registration operation, Fourier spectal representation for smooth functional encoding and a class-aware contrastive objective. Experiments on both synthetic and real datasets demonstrate consistently improved alignment together with competitive or superior classification performance.", "questions": "Please see above.\n\nIf the authors address some of the discussion points above I am willing to increase my score in the rebuttal.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Offers and end-to-end method to jointly learn functional registration and classification in a unified architecture rather than sequentially.\n2. Theoretical guarantees are provided that limk registration fidelity to classification generalization.\n3. Improved performance across real world and synthetic dataset and comprehensive ablation for each architectural component.\n4. Robustness is shown against noise and missing data.\n5. The language of the paper is clear. \n6. Git link to code is available.", "weaknesses": "**Weaknesses and Questions**\n\n1. Since TTN is also addressing jointly the registration and classification tasks, can the authors elaborate and highlight a bit more the architectural and conceptual differences between these methods?\n2. How sensitive is the model to mis-specified diffeomorphic constraints if real warpings violate the diffeomorphic guarantees?\n3. In line 082: the paper claims that the paper of Tang et al. is heavily reliant on assumptions,…\nCan the authors clarify which assumptions those are and whether they avoid making these assumptions in this work?\n4. In the experiments the paper claims that simulated data provide insights regarding registration, classification and reconstruction. I would like to understand why is it important to draw conclusions on the reconstruction along with the other 2 tasks discussed thoroughly in the paper.\n5. In table 1 we sometimes see that several methods demonstrate high acc and F1 score while their registration performance is not optimal. Is there any intuition behind this? Does it mean that the SrvfRegNet was not tuned properly? Does this not make those methods robust against misregistration which can be an interesting and useful property? What is the intuition behind SrvfRegNet always not being able to recover the registration as accurately as the proposed method?\n6. Another question regarding the results is whether there is any intuition why in the case of Symbol 3 the SrvfRegNet + TSLANet is delivering higher classification accuracy compared to Symbol2. Is there any correlation to the number of classes?\n7. I would like to request some addition of limitations of the current method and of why one should not consider to addressing classification and registration jointly.", "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)