Rethinking Time Encoding via Learnable Transformation Functions
Authors: Xi Chen
Yateng Tang
Jiarong Xu
Jiawei Zhang
Siwei Zhang
Sijia Peng
Xuehao Zheng
Yun Xiong
Venue: ICML 2025 poster (2025)
Keywords: Time Encoding
Dynamic Graph
Time Series
Deep Function Learning
Primary area: deep_learning->everything_else
TL;DR: Generalizing time encoding in diverse domains through learnable transformation functions

Abstract: Effectively modeling time information and incorporating it into applications or models involving chronologically occurring events is crucial. Real-world scenarios often involve diverse and complex time patterns, which pose significant challenges for time encoding methods. While previous methods focus on capturing time patterns, many rely on specific inductive biases, such as using trigonometric functions to model periodicity. This narrow focus on single-pattern modeling makes them less effective in handling the diversity and complexities of real-world time patterns. In this paper, we investigate to improve the existing commonly used time encoding methods and introduce **Learnable Transformation-based Generalized Time Encoding (LeTE)**. We propose using deep function learning techniques to parameterize nonlinear transformations in time encoding, making them learnable and capable of modeling generalized time patterns, including diverse and complex temporal dynamics. By enabling learnable transformations, LeTE encompasses previous methods as specific cases and allows seamless integration into a wide range of tasks. Through extensive experiments across diverse domains, we demonstrate the versatility and effectiveness of LeTE.

Rethinking Time Encoding via Learnable Transformation Functions
Xi Chen' Yateng Tang² 2 Jiarong Xu 3 Jiawei Zhang*
Siwei Zhang'
Sijia Peng' Xuehao Zheng² Yun Xiong'
Abstract
vations in weather forecasting (Murat et al., 2018; Neumann et al., 2024). Adopting time series models and dynamic graph models to handle time-related data are two common approaches (Wu et al., 2023; Yu et al., 2023). In both cases, effectively incorporating time information is crucial for mak- ing accurate predictions. To achieve this, existing research works typically employ time encoding methods to capture and represent time information, with the resulting time em- bedding being treated independent feature in time series forecasting and dynamic graph representation learn- ing models.
Effectively modeling time information and incor- porating it into applications or models involving chronologically occurring events is crucial. Real- world scenarios often involve diverse and complex time patterns, which pose significant challenges for time encoding methods. While previous meth- ods focus on capturing time patterns, many rely specific inductive biases, such using trigono- metric functions to model periodicity. This nar- row focus on single-pattern modeling makes them less effective in handling the diversity and com- plexities of real-world time patterns. In this pa- per. we investigate improve the existing com- monly used time encoding methods and introduce Learnable Transformation-basee based Generalized Time Encoding (LeTE). We propose using deep function learning techniques to parameterize non- linear transformations in time encoding, making them learnable and capable of modeling general- ized time patterns, including diverse and complex temporal dynamics. By enabling learnable trans- formations, LeTE encompasses previous methods as specific cases and allows seamless integration into a wide range of tasks. Through extensive experiments across diverse domains, we demon- strate the versatility and effectiveness of LcTE.
Early studies represent time by using hand-crafted temporal features designed specifically for downstream tasks (Choi et al., 2016; Baytas al., 2017: Kwon et al., 2018). A prominent example the time encoding method illustrated in Figure 1 (a), which is widely used in existing time se- ries processing work (Wang et al., 2023; Wu et al., 2023), Such methods typically involve manually splitting times- tamps into components (e.g., month, day, etc.), assigning a embedding to each component, and adding these embed- dings to form the final time embedding. However, these methods are resource-intensive and often rely on domain expertise, which may limit their abilities to capture only specific pre-defined time patterns (Kazemi et al., 2019).
With the rapid development of attention mechanisms, which offer advantages such as better handling of long-range de- pendencies and adaptive weighting of time-related infor- mation, subsequent research on time series and dynamic graphs has increasingly leveraged these mechanisms (Xu et al., 2020; Yu et al., 2023; Liu et al., 2023). To better model time and ensure compatibility between time encod- ing methods and self- attentoenn Functional Time Encod- ing (FTH) methods were proposed, with two representative works: Functional Time Representation (Xu et al., 2019) and Time2 (Kazemi et al., 2019), as illustrated in Figure (b). Nearly all subsequent dynamic graph representation learning research employs these methods to encode time (Yu et al., 2023). These techniques transform time input into
1. Introduction
Time-related data are commonly observed in real-world ap- plications, such as user transaction data in financial institu- tions (Kazemi et al., 2020; Lezmi & Xu, 2023), purchase be- havior sequences e-commerce (Kang & McAuley, 2018: Rossi al., 2020; Skarding et al., 2021), and climate obser-
'Shanghai Key Laboratory of Data Science, College of Com- puter Science and Artificial Intelligence, Fudan University. Shang- hai, China "Tencent Weixin Group. Shenzhen, China 'Shoool of Management, Fudan University. Shanghai, China *IFM Lab, Uni- versity California, Davis, CA, USA. Correspondence to: Yun Xiong <yunx @ fudan.edu.cn".
time embeddings by applying mul- tiple linear transformations followed by pre-defined non- linear transformation functions. Due to their reliance on pre- defined non-linear transformations functions to capture periodic patterns- -here methods are inherently limited to capturing fixed, specific time patterns.
as trigonometric
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Rethinking Time Encoding via Learnable Transformation Functions
Time Encoding
Hand-Crafled Time Encoding (HCTE)
Functional Time Encoding (FTB)
LeTE (Ours)
Pateems Capturing
Pre Pre-defined
Pre-defined
Formula
TEinl/] mimsut///naniiii Eayy
TE(A
sintonr + vy or oy th. sinteay + vt).
TEM s. LayerNorm (di 9:))
"weekday/ + houur(hourl00) Fannaall000iiimmmmmmmmm/
iisd
(8)
(b)
(e)
Time Embeedding
Embedding
Embeddiing
Non-lineer Transbraaton
LayerNorm
Architecture
Ieboking and
sintory 91)
Leamable Nam-lincar
Teansforatton
do 91)
Voae Var Sronder
(mi + 9?)
(mi 9t)
Extrac Features
Tranchimation
Figure I. A comparison of previous time encoding methods and proposed LeTE
As result, they often struggle to represent more complex, non-linear temporal dynamics (Kazemi et al., 2019; Wu et al., 2023) and require additional dimensions to account for diverse periodic components (Xu et al., 2020; Rossi et al., 2020; Zeng et al., 2024). Furthermore, these encoding methods frequently lack the capacity to effectively model non-periodic patterns, such trends, irregularities.
To better encode time information and simultaneously capture diverse time patterns, we propose Learnable Transtornation-basel Generalized Time Encoding, ab- breviated as LeTE-a simple yet effective time encoding method. Instead of hand mrandicr time encoding or relying on pre-defined non-lineat transformations, we draw inspira- tion from deep function learning, which is known for its gen- eralizability, interpretability, and reusability (Zhang, 2024; Liu et al., 2025), and propose to use learnable non-linear transformations for time encoding. Specifically, we parame- terize non-linear transformation functions using techniques derived from deep function learning. This parameterization makes the transformations learnable and jointly optimizable with the model's parameters under supervision from down- stream tasks, allowing them to flexibly adapt both linear and arbitrary non-linear forms. With learnable transforma- tions, LeTE adaptively models time information, enabling different dimensions of time encoding to capture complex time patterns such as irregular trends, abrupt changes, and overlapping periodicities that are commonly encountered in real-world scenarios and beyond the capabilities of pre- vious methods. This generalization also allows our method to encompass previous approaches specific cases. An illustration of LeTE is in Figure (c).
We observe that previous time encoding methods-whether hand-crafted or functional primmarlllppimmallly introduce a strong inductive bias rooted in the periodic nature of human behav- ior and natural phenomena (Li et al., 2017; Xu et al., 2019; Kazemi et al., 2019). They mainly focus on capturing pre- defined periodic patterns, often struggling to capture more complex ones, such as non-periodic and mixed patterns.
However, in real-world scenarios, data often exhibits com- plex interplay of mixed patterns, making accurate modeling more challenging. For instance, in financial risk control, periodic patterns- succ as daily transaction peaks, weekly spending habits, and seasonal trends around holidays or salary payments-offe valuable insights into predictable behaviors. On the other hand, non- non-periodic events, such sudden spikes in transactions caused by market fluctuations, regulatory changes, potential fraudulent activities, neces- sitate flexible and adaptive modeling techniques. Moreover, different patterns often coexist. For example, fraudsters may blend regular periodic transactions with abnormal non- periodic activities to evade the detection by regulators. To illustrate the presence of complex mixed time patterns in real-world data, conduct extensive investigations in this paper, with partial results in Appendix G.I.
LeTE also offers following advantages (cf. Section 3.2). Since time can be measured on different scales, its rep- resentation should be invariant time rescaling (Kazemi et al., 2019). We prove that LeTE satisfies this property (cf. Appendix C.3). Furthermore, we prove that LeTE is generalized version of previous methods and can integrate seamlessly with various models (cf. Section 3.2). By em- ploying an interpretable deep function learning approach, LeTE achieves high degree of interpretability (cf. Ap- pendix G.3). Additionally, experimental results demonstrate that LeTE achieves superior results with fewer dimensions than previous time encodings, as the learnable transforma- tions capture part of the complexity that would otherwise require higher-dimensionall embeddings (cf. Section 4.5).
This motivates us to rethink the design of time encodings. We argue that an effective time encoding method should adhere to a key principle to enable comprehensive and accu- rate analysis: Capacity for modeling diverse and complex time patterns, i.e., the method should be capable of cap- turing wide range of time patterns, including periodic, non-pen non-periodic, and mixed patterns.
2

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Rethinking Time Encoding via Learnable Transformation Functions
We highlight our contributions as follows:
2.2. Deep Function Learning
Deep Function Learning refers to the approach of learn- ing target functions by optimizing parameterized functions, such as polynomials, sinusoidal functions, or splines. This method leverages the flexibility of parameterized functions and optimizes their parameters using deep learning frame- works to approximate complex functions (Zhang, 2024).
We reinvestigate the design of the existing time encod- ing methods, highlighting their limitations in handling real-world data and propose LeTF,, a generalized time encoding method that allows the entire encoding pro- cess, including both linear and non-linear transforma- tions, fully parameterized and learnable.
Fourier Series Expansion: The Fourier series expresses target function (() as combination of sine and cosine functions:
LeTE has the capacity to model diverse and complex time patterns, and it offers additional benefits, includ- ing invariance to time rescaling, plug-and-play func- tionality, enhanced interpretability and improved di- mensional efficiency.
N
f(x) = ag + (an, cos(nwr) + b,, sin(nwx))
(4)
Through extensive experiments across diverse do- mains-ineluding event-based image classification, time series forecasting, dynamic graph representation learning and real- wallworld& applications- we demonstrate the effectiveness and versatility of LeTE.
Here, a, and bn are learnable coefficients, and w is the fun- damental frequency. By optimizing ao. an and b,, through a learning process, the function fz) can approximate com- plex patterns (cf. Appendix B. I for details). Unlike fixed sine functions, parameterized functions adjust their ampli- tude, frequency, and phase through downstream supervisory signals, enabling them to effectively model wider range of patterns.
2. Preliminaries
2.1. Functional Time Encodings
Spline Functions: Spline functions approximate a target function f(x) using a sum of piecewise polynomial basis functions:
Functional Time Encoding (FTE) methods can be viewed as feature mappings from 1-dimensional input time to a high- dimensional time embedding:
: E R' + TE E Rd where E [0, fmax] is from the value range bounded by 1max- Two representative works of FTE are Functional Time Rep- resentation (FTR) (Xu al., 2019) and Vec (T2V) (Kazemi et al., 2019). Although these methods construct time encodings from different perspectives, they are math- ematically nearly identical (the only difference is that a separate dimension that undergoes only a linear transfor- mation is used by T2V to capture non-periodic patterns). We state the following proposition, with details of the two methods and the proof provided in Appendix C.I.
f(x) = Σ (x)
(5)
Here, o, are control points, and (f) are the basis functions (e.g., B-splines). By learning and optimizing the control points, knot positions, and weights (ef. Appendix B.2 for details), splines provide smooth and accurate represen- tation of diverse functions. Their piecewise and localized structure makes them highly adaptable for complex function modeling.
Proposition 2.1. Mathematically, with selected values for wi and Pt, the aforementioned FTR and T2V can be unified into the following forms:
3. Methods
3.1. LeTE
Including the first dimension:
To address the limitations of previous time encod- ings-specificallly, their restricted capacity to model fixed or pre-defined time patterns- we propose Learnable Transformation-based Generalized Time Encoding (re- ferred to as LeTE). To capture diverse and complex patterns in time-related data, we propose techniques that make non- linear transformations learnable. This approach allows the model to dynamically adapt its transformations, enabling more precise representations of time patterns. To achieve this, we employ two distinct approaches for constructing learnable transformation functions: Fourier series expan- sion and Spline functions. Both methods share the ability to effectively capture and model complex, non-linear temporal patterns while maintaining flexibility in handling various
TE(M = sin (wat + (p)) or wit + (p ifi 1, sin (wit + sp),
if2 < is d.
(1)
Or excluding the first dimension:
TE(U = sin (wit + sp)),
(2)
or
TE(1) = (win + (i)) ... sin(wal + sP))
For simplicity, we use Functional Time Encoding (FTE) to refer to both the FTR and T2V throughout the paper.
3

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Rethinking Time Encoding via Learnable Transformation Functions
Combined LcTE
41())
1A = Σ BB(o
(eg cos(kx,) + b2 sin(ks,)
[& LayerNorm (00 07)
850 850 8J0 8J0 BGA
Fouricr-based
pascnd-basod
of
Figure 2. An illustration of Combined LcTE: the first dimension is parameterized by Fourier series expansion and the last dimension is parameterized by B-Splines.
time dynamics. Based on these methods, we propose three variations of LcTE. First, we construct the learnable transfor- mation functions using these two approaches, categorizing them Fourier-basse LeTE and Spline-based LeTE cording to their respective construction methods. We then integrate these two variations to develop a more generalized version, referred to as Combined LeTTE, which leverages the strengths of both approaches.
where the number of B-spline basis functions, and (x) is the j-th B-spline basis function. LeTE can then be expressed as:
LeTE
Σ j=1
9i).
(10)
Comparing these two approaches for constructing Φ., the Fourier series expansion in LeTE enforces periodicity in di while the B-spline approach provides flexibility to model more complex d; functions, including both periodic and non-periodic patterns.
For a scalar timestamp input t, LeTE for t, denoted LeT)(()) d-dimensional time embedding vector:
(e) = ów + sp)),
(6)
Recall that neglect the first dimension of the linear trans- formation of time in Equations (2) and (3), which models the non-periodic patterns for Time2 However, in LeTE, by allowing Φ to be learnable, different Φ at different di- mensions can capture more complex non-periodic patterns based on the supervision signals from downstream tasks.
where and (o; are learnable parameters, and Φ; are learn- able functions that can be parameterized by either Fourier series expansion or B-splines.
Fourier-based LeTE: This method assumes that Φ; are parameterized by Fourier series expansion:
To enhance the capability of the time encoding to capture diverse and complex time patterns and to build a more generalized version of LeTF, we further pro- pose straightforward extension: applying Fourier-basse LeTE to a portion of the time encoding dimensions and Spline-based LcTE to the remaining dimensions. The pro- portion of Fourier-based LeTE and Spline-based LeTE used can be controlled by a hyperparameter p. To address po- tential differences in the output scales of Fourier- based and Spline-based LeTF,, we introduce a Layer Normalization layer followed by a learnable scaling weight for the time encoding. For a d-dimensional LeTE, the time embedding is formulated as:
Combined LeTE:
(x) = 0g +
(ax cos(kr) + by sin(kx)),
(7)
where ag dg, and by are the parameters to be learned, and K represents the number of terms in the expansion. LeTE can then be expressed as:
K
LeTE(t)l 0400
0i, cos(k(wut + p.))
+ b sin(k(wss pi))
LETE(t)i = =; LayerNorm (o:(w,t + vi)),
(11)
This method assumes that de pa- rameterized by B-splines:
Spline-hased LeTE:
where [st] is a d-length learnable scaling weight vector, and Equation(7),
M
if i < pp d], Equation(9),
6(xx= = (,B(()),
(9)
(x(x)
(12)
if > Ip p..d.
j=1

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Rethinking Time Encoding via Learnable Transformation Functions
When p- = 1, the method corresponds Fourier-based LeTE, and when p 0, it corresponds Spline-based LeTE. For the remainder of this paper, unless otherwise specified, LeTE will refer to the Combined LeTE, where is set to 0.5. An illustration of LeTE is shown in Figure 2, and the implementation details are provided in Appendix D.
sition with its corresponding proof demonstrate this.
Proposition 3.1. For an arbitrary input t, the network can learn set of parameters such that LeTE can replicate the effects of previous time encodings, making previous time encodings specific cases of LeTE.
Proof. For Equations (8) and (10), we only need to find a set of coefficients for Equations (7) and (9) to approximate the sine function, respectively. By selecting K =1, 01,0 = 0.
3.2. Properties of LeTE
In this subsection, we present the properties of our method from the perspective of theoretical analysis. Specifically, we discuss the strengths of Fourier-based LeTE and Spline- based LeTE individually. Since Combined LeTE integrates these two variations, it naturally inherits their respective properties.
01,1 = 0, and 1 = 1, Equation (8) becomes:
(e] 0 ++ (o- cos (1 (wit + pii)
+ 1 sin (1 (wet + si)))
(13)
Generalizability: Compared with previous methods, which can only capture pre-defined time patterns-usuall periodic ones-our method offers greater generalizability, enabling to capture a wider range of diverse and complex patterns, including periodic, non-periodic and mixed ones. Naturally, Fourier-based Le (as formulated in Equation (8), the di functions are parameterized by Fourier series expansion) can model periodicity, as it resembles a Fourier series expansion with weighted sums of sine and cosine terms at different fre- quencies and phases. By learning appropriate values for Wi and incorporating different harmonies k, the function can ap- proximate complex periodic patterns and capture repeating structures in time. The learnable parameters w; and (p; en- able the model to adapt to various periodic characteristics in the data. Although the Fourier series is inherently periodic, non-periodic patterns can also be modeled: the learnable parameters a, as bi,k, Wi, and yo: provide flexibility to approximate non-periodis behaviors. By using very small or large values for Wi, the model can fit signals with long or slow-varying cycles, effectively creating non-repeating patterns over finite interval. Additionally, the combination of learned frequencies, phases, and amplitudes can produce complex patterns that do not repeat over the observed range, thereby approximating non-periodic signals. For similar reasons, and given the generality of functions formed by splines, Spline-basee based LeTE (Equation (10), the de are pa- rameterized by B-splines) can also model both periodic and non-periodic patterns. Naturally, through multi-dimennional encoding, both Fourier-based and Spline-based LeTE are capable of capturing mixed time patterns. Although Fourier- based Le'TE can capture non-periodic patterns, its inherent periodicity makes it particularly effective at modeling the periodicity of time. Conversely, while Spline-based LeTE is also capable of capturing periodic patterns, it exhibits a stronger ability to model non-poriodic patterns. Therefore, by combining Fourier-based LeTE and Spline-based LeTE, the resulting Combined LcTE achieves enhanced capability to capture diverse patterns. Intuitively, the previous FTEs are special cases of LeTE; we present the following propo-
sin(wut sp:)-
Thus, sin(wat + (p. is indeed a special case of the more general formula in Equation (8). The proof continues in Appendix C.2, which demonstrates that sin + (ps) is also special case of Equation (10).
Moreover, since Xu et al. claim that absolute position encod- ing is special case of functional time representation (Xu et al., 2019; 2020), it is straightforward to see that absolute position encoding is also a special case of our LeTE.
Invariance to Time Rescaling: Since time can be repre- sented on various scales (such as days, hours, or seconds), a key characteristic of time representation is its invariance to rescaling (Kazemi et al., 2019; Tallec & Ollivier, 2018). Similar to FTE, our proposed time encoding is also invariant to time rescaling, as shown in the following proposition with proof provided in Appendix C.3.
Proposition 3.2. LeTE is invariant time rescaling.
Plug-and-Play: LeTE is designed in a plug-and-play man- ner, ensuring seamless compatibility with various models and architectures. By producing d-dimensional time em- bedding vector similar previous time encodings, it can be easily integrated without requiring significant modifications to existing frameworks. Unlike prior methods that rely on fixed non-linear transformation functions. LeTE employs parameterized and learnable transformations, enabling it to capture additional information and complexity. This design allows LeTE to achieve superior performance even with diwen-dimeniionll time encodings compared to traditional methods (see Section 4.5 for experimental results).
Interpretability: Previous time encodings exhibit natural interpretability because they use a fixed non-linear activa- tion function, i.e., the sine function, which has obvious periodicity. Our proposed time encoding uses a learnable non-linear transformation function. However, by examining
5

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Rethinking Time Encoding via Learnable Transformation Functions
the learned parameters, we can reconstruct these transforma- tion functions, enabling our method to also achieve strong interpretability, A visualization of our proposed time encod- ing is provided in Appendix G.3.
We then apply an LSTM with 32-dimensional learnable embedding for the time input and compare it models where the FTE or our LeTE is used for encoding time. The results, shown in Figure 3, indicate that the FTE achieves testing accuracy comparable to that of the LSTM with- out any time encoding method applied. However, our LeTE achieves significantly higher image classification ac- curacy. This simple experiment demonstrates that LeTE can efficiently encode time information for models. Next, we present experiments applying LeTE to time series tasks, dynamic graph tasks, and real-world applications.
3.3. Use of Time Encoding
In time series forecasting research, time embeddings calcu- lated by time encoding modules are usually directly added to feature embeddings and fed into the attention mechanism or Transformer (Vaswani et al., 2017). As a result, they typi- cally share the same dimensions as the feature embeddings:
4.2. Experiments on Time Series Tasks
TokenEncode(x)) + TE(() R..
(14)
For time series forecasting tasks, we select 5 bascline mod- els where we can directly replace the time encoding meth- ods with LcTE: vanilla Transformer (Vaswani et al., 2017), Pyraformer (Liu et al., 2021), Non-stationary Transformer (Liu et al., 2022), MICN (Wang et al., 2023), and TimesNet (Wu al., 2023). We conduct long-term forecasting tasks these baseline models using 4 datasets: ETT, Weather, Exchange (Lai et al., 2018), and Electricity, covering various real-world scenarios. Implementation details and introduc- tions to baselines and datasets are provided in Appendix E.2. We apply LeTE and adjust the hyperparameter p in all ex- periments to capture more comprehensive time information. We report the results in the multivariate setting, as shown in Table I and 5, Because the time embeddings need be added to the feature embeddings, they must have the same dimensions as the feature embeddings. Baseline models commonly apply hand-crafted time encoding (HCTE) with Date-Time Format inputs (e.g.. ISO 8601 format, YYYY- MM-DD HH.msss)). Since our method, like FTE, takes UNIX timestamps as input, we include FTE in our experi- ments for comparison. In our approach, we transform the Date-Time Format timestamps into UNIX timestamps, en- code them with our proposed LeTE, and feed the resulting time embeddings into the models in the same manner as the baselines. In this context, the input consists of abso- lute timestamps, and the time encoding can therefore be regarded as an absolute time encoding.
Here, x represents the input, TokenEncode denotes token encoding function, TE denotes the time encoding, and d represents the dimension of both feature embeddings and time embeddings.
In dynamic graph representation learning research, time embeddings usually concatenated with node features and edge features as part of the input. This allows for more flexibility in the choice of time embedding dimensions:
x = Node Features| Edge Features] (E(+) E +d
(15) Here, || denotes the concatenation operation, while du, des and d represent the dimensions of node features, edge fea- tures, and time embeddings, respectively.
4. Experiments
4.1. Time as the Only Input
To evaluate the performance of the time encoding method in scenarios where the only input is time, and to compare dif- ferent time representations while minimizing the influence of extraneous variables, we follow (Kazemi et al., 2019) and create a sequential (event-based) MNIST dataset (Fatahi et al., 2016; Campos et al., 2018; Bellec et al., 2018) and conduct image classification task (more details are shown in Appendix 1).
From the experimental results, we observe that: (1) When applying different time encoding methods to baseline mod- els for time series forecasting, LeTE outperforms the bench- mark in most cases, achieving an average win rate of 98% on MAE (Mean Absolute Error) and 95% on MSE (Mean Squared Error) across all baseline, dataset, and prediction length combinations, highlighting the effectiveness of the proposed time encoding. This demonstrates that our method can be seamlessly transferred to time series models, reliably achieving strong performance. (2) The improvements on basclines are considerable. For instance, applying LeTE to the Transformer model reduces the average MAE and MSE across all datasets by 25.1% % and 46.5%, respectively. This
LS
12:
LSN
04-
L5T
02
10
130
30
10E
200
Epcch
(a) Testing Accuracy
(b) Testing Loss
Figure 3. Time as the only input: Comparison of time encodings on sequential MNIST
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Rethinking Time Encoding via Learnable Transformation Functions
Table 1. Time series prediction: multivariate long-term forecasting task. The past sequence length is to 96, while the prediction lengths are (96, 192, 336. 7203. The results are reported in terms of MAE, where lower values indicate better performance. HCTE Mhan-Craftted Time Encoding) is a method widely adopted time series research. FTE stands for Functional Time Encoding. The win rate represents the percentage of cases where LeTE outperforms the HCTE. The best results for each baseline, dataset and prediction length combinations are in bold. ETT consists of subsets. Here, we present the average results across these subsets, with the full results provided in Table 7.
MAE
Transformer
Pyraforner
NS Trans.
MINC
TimesNet
Win
TE
HCTE
FTF
LeTE
HCTE
FTF
LeTE
HCTE
LeTE
HCTE
LeTT
HCTE
LeTT:
Rate
96
0.797
0.803
0,550
0.642
0.720
0.583
0.405
0.435
0.377
0.405
0.367
0.350
0.355
0.362
0.352
ETV
192
1.139
0.916
0.712
0.899
0.924
0.738
0.445
0.478
0.414
0.445
0.423
0.395
0.385
0.400
0.388
95%
336
1.119
0.938
0.821
1.043
1.038
0.863
0.478
0.539
0.449
0.478
0.486
0.448
0.421
0.413
720
1,070
1.146
0,878
1.196
1.189
0.959
0.526
0.557
0.490
0.526
0.561
0.505
0.455
0.455
0.429
Weather Exchangs Electricity
96
0.357
0.375
0.347
0.376
0.375
0.365
0.273
0.275
0.265
0.269
0.263
0.254
0.272
0.272
0.267
336
192
0.367
0,370
0.402
0.425
0.353
0.357
0.391
0.399
0.385
0.401
0.372
0.369
0.286
0.304
0,300
0.292
0.278
0.293
0.285
0.304
0.278
0.298
0.271
0.294
0,300
0.289
0.308
0.291
0.277
100%
720
0.374
0,453
0.363
0.390
0.394
0.380
0.321
0.330
0.317
0.321
0.330
0.317
0.320
0.363
0.316
96
0,575
0.705
0.547
0.570
0.641
0.624
0.237
0.261
0.237
0.235
0.233
0.203
0.234
0.237
0.230
336
192
0.945
0.747
0.791
1.123
0.879
0.744
0.903
0.803
0.991
0.815
0.859
0.786
0.476
0.335
0.501
0,369
0.439
0.319
0.407
0,316
0.472
0.332
0.402
0,289
0.448
0,344
0.472
0.339
0.446
0,332
95%
720
1.329
1.147
1,066
1.075
1.046
0.938
0.769
0,901
0.612
0.658
0.710
0.622
0,746
0.756
0,751
96
0.422
0.257
0.245
0.303
0.296
0.267
0.223
0.222
0.221
0.229
0.258
0.225
0.220
0.221
0.215
192
0.523
0.308
0.295
0.336
0.317
0.311
0.285
0.271
0.260
0
0.306
0.261
0,261
0.263
0.253
100%
336
0.607
0.355
0.365
0,403
0.377
0.349
0.338
0.321
0.308
0.331
0.335
0.295
0.306
0.302
0.299
720
0.690
0.459
0.429
0.434
0.417
0.415
0.410
0.357
0.349
0.356
0.387
0.339
0.359
0.350
0.348
Win Rate
100%
944%
100%
100%
949%
98%
Table 2. Dynamic graph link prediction task: The results are reported in AP, where higher values indicate better performance. The better results in bold. Here, we present the performifgg results across variations of LeTE, with the full results provided in Table 9. FTE represents Functional Time Encoding which is commonly used in dynamic graph research.
AP
Wikipedia
Reddit
MOOC
LastFM
TE Transductive
Inductive
Transductive
Inductive
Transductive
Inductive
Transductive
Inductive
TGAT
FIE LeTE
96.95 0.24
96.33 a 0,26
98.53 0.04
97.01 0.05
85,34 a 0.19
84.94 0,04
72.73 0.11 76.22 =: 0.25
77.78 a 0.13 81.32 + 0.14
97.82 + 0.09
97.34 = 0.08
98.56 # 0.01
97.05 = 0.06
88.31 + 0.10
88.37 # 0.12
FTE LeTE
98.45
0.06 98,78
97.83 =: 0.04
98.63 0.006
97.50 2 0.07
89.15 a 1.60
89.04 a 1.17
77.07 #: 3.97
81.45 4.29
TGN
98.19 =: 0.09
98.74 0.00
91.41 * 0.55
90.87 0.83
83.64 +
87.55 + 1.88
TCL
FIE LeTE
96.47 1: 0.16
96.22 =: 0.17
97.53 a: 0.02
94.09 a 0.07
82.38 a: 0.24
80.60 a 0,22
67.27 a: 2.16
73.53 a 1.66
98.19 0,04
97.89 + 0.03
97.78 0.03
94,99 0.07
84.24 0.10
82.72 0.12
76.08 0.79
80.68 + 0.70
DyG-
FIE
99.03 0.02
98.59 a 0.03
99.22 0.01
98. 84
87.52 a: 0.49
86.96 0.44
93.00
94.23 a 0.009
Former
LeTE
99.13 = 0.02
98.73 a 0.00
99,24 +: 0.01
98.86 = 0.01
88.70 = 0.21
88.39 = 0.15
93.64 = 0.10
94.69 = 0.12
illustrates that our method can be applied to various time se- ries forecasting models, consistently achieving strong perfor- mance. (3) FTE can occasionally outperform benchmarks; however, it also fails in many cases, whereas LcTE steadily outperforms benchmarks in such situations. This demon- strates our method's capability to model diverse time pat- terns, including periodic, non-periodic, and mixed, high- lighting its generalizability across different models and data.
baseline methods and datasets are in Appendix E.3. The results are reported in both transductive and inductive set- tings, as shown in Tables 2 and 6. In this context, the time encoding module takes the relative time difference between the current edge and the most recent previous edge, and can therefore be regarded
relative time encoding.
As shown in the experimental results, our proposed LeTE surpasses the benchmark results on all combinations of baselines and datasets, regardless of transductive or induc- tive settings, achieving tfte-of-thh-ar (SOTA) performance. This strongly demonstrates the effectiveness of our proposed time encoding and highlights its potential for improving the representation learning of dynamic graphs. The dimensions for the main experiments are set to 100, following the origi- nal settings in previous work (Rossi et al., 2020; Yu et al., 2023). However, since time embeddings are concatenated with node and edge features in dynamic graph models, this provides significant flexibility in setting their dimensions.
4.3. Experiments on Dynamic Graph Tasks
FTEs are widely used in dynamic graph representation learn- ing models. Representative works include TGAT (Xu et al., 2020), TGN (Rossi et al., 2020). TCL (Wang et al., 2021), and DyGFormer (Yu et al., 2023). Thus, we apply these models as basclines and replace their time encodings with our LeTE. We conduct link prediction experiments on 4 real- world datasets: Wikipedia, Reddit, MOOC, and LastFM (Kumar et al., 2019). The details of the implementation,
7

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Rethinking Time Encoding via Learnable Transformation Functions
670
20
5100
Ru Time Doending
M0
55.30
600
182
152
180
23
650
58.33
20.16
LiT
Latk beln
64.5
55.59
36
6000
2
109
6..
140
21
(a) Wikipedia/TGN
(b) MOOC/TGN
65.5
AUC-8000))
T192
Figure 5, Average Precision results comparing different dimen- sions of the FTE and Spline-based LeTTE on Wikipedia/TGN and MOOC/TGN.
Figure 4. Results evaluated by AUC-ROC, TP1O and Recall @ 10 on real business datasets.
4.5. Dimensions of Time Encoding
We compare the effects of time embeddings with different dimensions in Section 4.5. A comparison and analysis of the variations of LeTE are also provided in Appendix G.2.
Compared to the previous FTE methods, our LeTE takes step forward by making the non-linear transformation learnable, thereby generalizing the time encoding. Since part of the information from the data captured by the learnable non-linear transformation, we hypothesize that us- ing lower-dimensional LeTEE may still outperform the FTE (which relies on fixed non-linear transformation). There- fore, we conduct experiments with lower-dimensional time encodings. Since the dimensionality of time encoding in dynamic graph tasks is more flexible, we conduct exper- iments on dynamic graph link prediction tasks, with the results presented in Figures 5, 6, and 7. As illustrated in the results, models using the FTE suffer from severe per- formance degradation as the dimension decreases, whereas models with LeTE demonstrate more stable performance and consistently outperform those using the FTE, even at lower dimensions. Notably, models using Le'TE with sig- nificantly lower dimensions (e.g., 2. or 16) outperform models with the 100-dimensional FTE. This demonstrates the effectiveness and generalizability of our method.
4.4. Experiments on Real- World Application
Time information plays a crucial role in many real-world fields. We apply our proposed LeTE in a real-world finan- cial risk control scenario to demonstrate its effectiveness in practical applications. In financial risk control, a user's historical transaction data is typically used to predict their credit risk, which can be framed classification problem based on historical transaction information. However, in this scenario, users' transaction behaviors often exhibit combination of complex periodic and non -erriodo patterns. For instance, users may regularly receive salary deposits and purchase daily necessities, whereas peer-to-peer transfers may lack strong periodicity. Using financial risk control data from Tencent Mobile Payment 1, we conduct compar- ative experiments without time information, with the FTE, and with LeTE to encode time information. The backbone model treats the time embedding as feature, concatenates with the user's raw features, and takes the concatenated features as input. The objective is to use users' historical transaction data to predict whether they have default risk. Details of the dataset are provided in Appendix E.4. The results are presented in Figure 4. The results indicate that the model without time encoding performs the worst, as it completely ignores time information. With FTE, the peri- odicity of user behavior at different frequencies is captured, resulting in improved performance compared to the case without time information. Using LeTE yields the best per- formance, as our time encoding effectively models periodic, non-periodic and mixed patterns in a more general manner.
4.6. Additional Experiments
We provide the complete experimental results for the ex- periments mentioned in the main text in Appendix Ad- ditionally, we conduct further experiments to analyze the complex time patterns in real-world data (cf. Appendix G.1); compare different variants of LeTE (cf. Appendix G.2); illustrate the interpretability of Le'EE (cf. Appendix G.3); demonstrate LeTE's ability to simultaneously cap- ture diverse time patterns, including periodic, non-periodic, and mixed ones (cf. Appendix G.4); and assess LcTE's capability fit various functions (ef. Appendix G.5).
'The data used these experiments are properly sampled only for testing purposes and does not imply any commercial information. All users' private information is removed from the dataset. Moreover, the experiments were conducted locally Tencent's server by formal employees who strictly followed data protection regulations.
5. Conclusion
In this paper, we propose a effective time encod- ing Time Encoding Transformation based Generalized to accept both absolute
8

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Rethinking Time Encoding via Learnable Transformation Functions
timestamps and relative time differences inputs, depend- ing the specific requirements of different models, abling it to function as either an absolute relative time encoding method. Through comprehensive analysis, we demonstrate that our proposed LeT'E is capable of model- ing diverse and complex time patterns, including periodic, non-per non-periodic, and mixed patterns. is invariant to time rescaling, sufficiently simple for integration with various backbone models, and exhibits good interpretability and di- mensional efficiency. Extensive experiments on event-based image classification, time-series forecasting tasks, dynamic graph link prediction tasks, and real-world financial risk con- trol applications demonstrate the superior performance and generalizability of our method across various application scenarios.
S.-F. Skip rnn: Learning to skip state updates in recurrent neural networks. In The Sixth International Conference on Learning Representations, 2018.
Chen, X.,, Liao, Y., Xiong, Y., Zhang, Y., Zhang, S., Zhang, J., and Sun, Y. Speed: Streaming partition and parallel acceleration for temporal interaction graph embedding. arXiv preprint arXiv:2308. 14/29, 2023.
Chen, X., Xiong, Y., Zhang, S., Zhang, J., Zhang, Y., Zhou, S., Wu, X., Zhang, M., Liu, T., and Wang, W. Diformer: A transformer- based method for discrete-time dynamic graph representation learning. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pp. 301-311, 2024a.
Chen, X., Zhang. S., Xiong, Y., Wu, X., Zhang, J., Sun, X., Zhang, Y., Zhao, F., and Kang, Y. Prompt learn- ing on temporal interaction graphs. arXiv preprint arXiv:2402.06322 2024b.
Acknowledgements
This work is partially supported by the Noncommunicable Chronic Diseases-Nationas Science and Technology Major Project (NO. 20242D0532400 and NO. 2024ZD0532403), the National Key Research and Development Plan Project 2022YFC3600901. This work is sponsored by the Tencent Rhino-Bird Focused Research Program. This work is par- tially supported by NSF through grant IIS-2106972.
Cho, K. Learning phrase representations using mn encoder- decoder for statistical machine translation. In Proceed- ings of the 2014 Conference on Empirical Methods in Natural Language Processing, pp. 1724-1734, 2014.
Choi, E., Bahadori, M. T., Schuctz, A., Stewart, W. F., and Sun, Doctor ai: Predicting clinical events via recurrent neural networks. In Machine learning for Healthcare Conference, pp. 301-318. PMLR, 2016
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Fatahi, M., Ahmadi, M., Shahsavari, M., Ahmadi, A., and Devienne, P. evt mnist: A spike based version of tra- ditional mnist. In 1st International Conference on New Research Achievements in Electrical and Computer Engi- neering, 2016.
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Rethinking Time Encoding via Learnable Transformation Functions
A. Related Work
Currently, commonly used time encoding methods or strategies for modeling temporal information can be broadly categorized Time Encodings (HCTE) and Functional Time Encodings (FTE).
into two types:
HCTE involves manually designed temporal encodings tailored to specific downstream tasks. These methods rely specific design choices and incorporate various inductive biases to capture fixed periodic patterss, constructing hand-crafted temporal features. These features are typically fed into models such as RNNs (Cho, 2014) or sequential architectures (Liu et al., 2021; 2022; Wang et al., 2023; Wu et al., 2023) to meet specific modeling requirements (Choi et al., 2016; Baytas et al., 2017; Kwon et al., 2018), as illustrated in Figure (()). Such approaches are often employed to address particular challenges in time series tasks.
Additionally, some methods in this category integrate time encoding directly with attention mechanisms (Vaswani et al., 2017), simplifying temporal modeling by adopting position encoding strategies (Gehring et al., 2017). Others embed discrete events into continuous vector space to better capture event contexts in attention-base models (Bengio et al., 2013; Li et al., 2017). These methods, while effective, are often limited to representing fixed or narrowly defined temporal patterns.
FTE represents an advanced and generalized version of time encoding, designed to overcome part of the limitations of Hand-Crafter Time Encodings. Two representative works in this category are functional time representation, proposed by (Xu et al., 2019), and Time2 Vec, proposed by (Kazemi et al., 2019), as shown in Figure ((b). Importantly, the previously mentioned position encoding methods integrated with attention mechanisms can be considered a simplified version of FTE.
Both functional time representation and Time2Vec adopt similar implementation methods, which resemble mie-dimensional- to-d-dimensional MLP with specially designed trigonometric non-linear activation function. In Time2 experiments comparing different non-linear activation functions demonstrate that the sine function performs best across various down- stream tasks. Despite limitations in modeling restricted aspects of time, FTE is widely adopted in dynamic graph representa- tion learning due to its ease of application and effectiveness (Rossi et al., 2020; Zhang et al., 2023b;a; Chen et al., 2023; Yu et al., 2023; Chen et al., 2024b; Zhang et al.. 2024; Chen al., 2024a; Zhang et al., 2025).
Time encodings can be directly applied to sequential models such as RNNs and LSTMs (Graves & Graves, 2012), or easily integrated into attention-based architectures. In time series forecasting, for instance, many models now employ transformer-basee structures (Vaswani et al., 2017). where time encoding is often treated similarly to position encoding, This is usually achieved by adding it to the input of the attention mechanism (Liu et al., 2021; 2022; Wu et al., 2023).
Dynamic graph representation learning models also require precise temporal modeling. For example, TGAT (Xu et al., 2020) directly replaces position encoding with functional time representation within its attention mechanism. Subsequent methods, such as TGN and TIGER (Rossi et al., 2020; Zhang et al., 2023b), have adopted similar approaches. DyGFormer (Yu et al., 2023), which applies Transformer to dynamic graph representation learning, uses the same encoding method by concatenating it with node and edge features before processing them with a bransformer-based model.
B. Methods of Parameterize Continuous Functions
B. Fourier Series Expansion
function (<x that is periodic with period T and satisfies certain conditions (Dirichlet conditions) can be represented as a Fourier Series. This series represents fx) as an infinite sum of sines and cosines (or, equivalently, complex exponentials) with specific coefficients. The series takes the form:
2TTE
2TTE
((x) 0g
On COS
+ bn sin
(16)
T
T
Here, ag is the average value of the function over one period, and an and bn are Fourier coefficients that can be calculated by integrating f fx) over the interval [0, T].
These coefficients are given by:
2 T
/
2nnx
an
f(x)
cos
dx,
(17)
T
T
T ///
2 T
2xnæ dx.
bn_
f(x)
sin
(18)
T
12

---

Rethinking Time Encoding via Learnable Transformation Functions
Under these conditions, the Fourier Series converges f(x) all points where / is continuous and converges the average of the left-hand and right-hand limits at points of discontinuity.
B.2. KAN and Spline Functions
The Kolmogorov-Arnold Theorem (Kolmogorov, 1961; 1957; Braun & Griebel, 2009) states that for any continuous multivariate function ((11,22,.... on the unit cube [0. 1)", there exist continuous functions Φ; and day such that:
2n+1 Σ
Σ (
f(21, 22 ................................ 2nd
(19)
where Φ; are continuous functions of a single variable, enabling dimensionality reduction, and Way are continuous functions mapping each input variable z; to a single output, contributing to the superposition structure. This theorem implies that every continuous function of multiple variables can be represented as a sum of compositions of univariate functions.
Building on the Kolmogorov-Amus splines construct learnable non-linear activation functions for neural networks (Liu al., 2025). We briefly introduce
Theorem and the advantages of splines for function fitting, Liu al. propose using
B-splines here. Given knot vector T to, fm with non-decreasing values, the basis functions (1, for B-spline of degree p are defined recursively as follows: For degree 0:
- {o
1f S < 4,+1
Nao(2)
(20)
otherwise.
For higher degrees p >
I - t4 Nip- (x) + h+p+1 I litp - 4i
(x)
NI+1--1(2))
(21)
1 li-pt1 li+1
The B-spline curve C(x) of degree p with control points (Po.PI..... .I...... Pn is given by:
12 C(x) Σ [Np(o)).
(22)
Here, (2)) are the B-spline basis functions of degree p, and P. are the control points that influence the shape of the curve.
C. Proofs
C.1. Proof of Proposition 2.1
C.1.1. DETAILS OF FUNCTIONAL TIME REPRESENTATION AND TIME2VEC
FTR is designed to use the time difference i = I 1,. where 0 S 1 < 4 < maxx as input. For the input time difference, a learnable frequency parameter is first applied. Next, a non-linear transformation is applied, using the cosine function on the odd dimensions and the sine function on the even dimensions. This method is mathematically represented as follows:
cos (wit), if i is odd,
TE()
(23)
sin (wit), if i is even,
where d is the dimension of the time encoding, i S d, and w; are learnable parameters representing the frequency of the trigonometric functions. Since this time encoding uses time differences as input, it can be considered as a relative time encoding.
T2V is designed to use timestamps / as input. A linear transformation applied to the first dimension to capture non-periodic time patterns. For the remaining dimensions, linear transformation is followed by sine-based non-linear transformation to model periodic time patterns: Mathematically, this method is represented as follows:
wit + Gi,
if Ti=l 1,
TE() =
(24)
sin (wit + sc), if 2 d,
13

---

Rethinking Time Encoding via Learnable Transformation Functions
where [ is the hd element of the time encoding, and wi and yp, learnable parameters representing frequency and phase-shift of the sine function, respectively. Since this time encoding takes timestamps input, it can be considered absolute time encoding.
C.1.2. PROOF OF PROPOSITION 2.1
Proof. Since Equation (23) can be written as (t))) = [cos(w) t), sin(w) t)...... cos(wat), (int))sssi((aa)) sit(())))), we show that the vector [cos(wit), sin(wit),..... .... cos(wat), sin(wat]] can be expressed in the form (w,t + spx) with suitable phase shifts Pp: Recall the trigonometric identity:
cos (0) sin (0 2
(25)
Applying this identity, each cosine term in the vector can be rewritten as:
cos(wit) = sin (iit + TT) 2
for i= 1,2,...,4
(26)
The sine terms are already in the desired form with zero phase shift:
(intt sin (wit 0), for i= 1, ...,,
(27)
With these transformations, the original vector becomes:
(E( sin (1i " ) sin (wit + +0), O),,... ,..., sin wat + 2 sin (wat + 0)]
(28)
or equivalently,
TE(() = [sin (wit + (p))
(29)
where the phase shifts so are defined as follows:
if odd
2
0, if i is even.
C.2. Continued Proof of Proposition 3.1
Proof. The function (0) is continuous and infinitely differentiable (i.e., C*) on R. Thus, it is continuous any closed interval (a, b].
B-spline basis functions of degree l form a basis for the space of piecewise polynomial functions of degree l with continuity Ck-1 at the knots. By the Weierstrass Approximation Theorem, any continuous function on closed interval can be uniformly approximated by polynomials to any desired degree of accuracy.
Since B-splines are piecewise polynomials, they can uniformly approximate any continuous function on [a, 6j. Specifically, for any 0, there exists linear combination of B-spline basis functions that approximates sin(O) within over (a, b).
To build the approximation of sin(w;t + ve) using B-spline basis functions, we first select knot vector T ffo. ft....,tatb+) } that partitions the interval [a, b] appropriately. The choice of the knot vector determines the placement and spacing of the knots, which in turn affect the flexibility and local support of the B-spline basis functions.
Next, we choose the degree & of the B-spline basis functions based on the desired smoothness and approximation quality. higher degree allows for smoother basis functions, potentially improving the approximation at the cost of increased computational complexity.
With the knot vector and degree specified, we generate the B-spline basis functions (B) of degree k using standard recursive definitions. These basis functions possess local support and satisfy the partition of unity property, making them suitable for approximating functions over [a, bj.
To determine the coefficients Cty that yield the best approximation of sin(w,t + (P)). we formulate an optimization problem. Specifically, we set up minimization problem that seeks to minimize the squared difference between the sine function and
14

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Rethinking Time Encoding via Learnable Transformation Functions
the weighted of B-spline basis functions over the interval [a,0]] b]:
]2
2
f" sin(urt + pi) Σ
M
min
Cia By + pi)
dt.
(30)
G,1-% GL.M
j=1
This minimization problem is a standard least squares problem, where the objective is to find the coefficients Ctj that minimize the integral of the squared error. Solving this problem can be accomplished using numerical methods such as the normal equations or singular value decomposition, leading to the optimal coefficients for the approximation.
By leveraging the properties of B B-spline basis functions and the Weierstrass Approximation Theorem, we can assert that, for any >> there exists a sufficiently large M and uppropriate coefficients ot such that:
M
sup (
sin(wrl + pi) - Cij By(wrl + sp)) <e.
Σ
(31)
This inequality indicates that the maximum deviation between the sine function and its B-spline approximation over [a, bJ less than e, satisfying the condition of uniform approximation.
Since 0 is arbitrary, coefficients C.g. Therefore, the sine function sin + sp;) can be represented as a sum of B-spline basis functions, making it special case of Equation (10) in the limit as M - co.
make the approximation accurate desired by increasing M and choosing appropriate
C.3. Proof of Proposition 3.2
A class C of models is considered invariant to time rescaling if, for any model Mi E C and any scalar >> O, there exists model M2 E C that responds to at (where is scaled by a) in the same way that Mi responds to the original t values. We provide the following proof to show that LcTE is invariant to time rescaling.
Proof. Consider time encoding M₁, mapped by LeTE:
(tTE()) (wst + 9p)).
(32)
If we replace t with - (where > 0), the time encoding updates as follows:
LeTE(( - t) = o, (a t) + pz))
(33)
To preserve the behavior of the original model M₁ under time rescaling, consider a new time encoding M2 with adjusted frequencies w/ di With this frequency adjustment, M2 behaves identically to Mi on a I, demonstrating that LeTE invariant to time rescaling.
D. Implementation Details of LeTE
Previous implementations of FTES can be summarized as inputting timestamp or time difference between events into a single-layer MLP with a fixed trigonometric function as the non-linear activation function. Inspired by this, our method can be viewed as making the fixed activation function learnable by parameterizing it with a Fourier series expansion or B-spline functions. Following KAN (Liu et al., 2025), which makes activation functions deep learning models trainable, we use similar implementation method.
D.1. Fourier-based LeTE
The implementation of Fourier-based LeTE straightforward and given by:
x,(x)
Σ
(wrlos) cos(ma;;) + W-(ein) j.a.m sin(ma;)) + by,
(34)
mIml
where i = 1, D indexes the input dimension, 1,2..... M indexes the output dimension (with = M in our method), and m = =1,2 2,,,.... K indexes the Fourier frequencies. Here, K is a hyper-paramen that determines the grid size. The parameters (cos) T MXX Wisin) E MxDx and b E M are learnable weights and biases.
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Rethinking Time Encoding via Learnable Transformation Functions
D.2. Spline-based LcTE
By using B-spline functions, we make the Φ; functions in Equation (6) learnable as follows:
(x (xx + spline, (x))
(35)
sinh(x)
e -
b(x) Tanh(x) cosh(2)
(36)
e0
spline, Σ cit (x)
(37)
where Cij are learnable. Unlike the original KAN, we use Tanh as the basis function here, as we found it performs better in practice.
E. More Details for Experimental Setting
E.1. Details for "Time as the Only Input" Experiments
Following (Kazemi et al., 2019), We generate sequential event-based version of MNIST by flattening the images and recording the positions of pixels with intensities greater than threshold (0.9 in our experiment). After this transformation, each image is represented as an array of increasing numbers, such as [t, 12, f3, f....., fm]- These values are treated as event times and can be used for image classification task. The backbone model we used is 128-dimensional LSTM, with a batch size of 512, aligned with the settings in (Kazemi et al., 2019).
E.2. Time Series Baselines and Datasets
E. EXPERIMENT IMPLEMENTATION DETAILS
Our experiments setting on time series tasks aligns with the definition of long-term forecasting. The baseline results for Transformer and Pyraformer are based on our implementation, while the results for the other baselines are taken from their original papers. Baseline models typically use hand-crafted time encoding method that applies date and timestamps represent various time inclluinn minutes, hours, weekdays, days, and months. The mapped vectors are then added together and added to the feature embeddings and fed into the models. The results are evaluated using MAE (Mean Absolute Error) (Table 1) and MSE (Mean Squared Error) (Table 5), both of which are widely used metrics in time series forecasting research.
E.2.2. BASELINES
We select commonly used time series prediction Daselines-Trrasiiie 2021), Non-stationary Transformer (Liu et al., 2022), MICN (Wang al., 2023), and TimesNet (Wu et al., 2023)-and replace their original hand-crafted time encodings with our proposed LeTE to demonstrate that LeTE can be effectively applied to time series prediction models and improve the performance of downstream tasks. A brief introduction to these baseline models is provided below:
(Vaswani et al., 2017), Pyraformer (Liu et al.,
Transformer (Vaswani et al., 2017) leverages the self-attention mechanism to model long-range dependencies in sequences, making a powerful tool for time series forecasting, especially in cases with complex time patterns. Its global context modeling capability enables it to capture intricate relationships between time steps effectively.
Pyraformer (Liu et al., 2021) introduces pyramid attention mechanism that hierarchically reduces the computational burden while preserving the ability to model both local and global dependencies. This design makes it particularly well-suited for handling long time series with improved efficiency and scalability.
Non-stationary Transformer (Liu et al., 2022): addresses challenges in forecasting non-stationary time series by incorporating dynamic feature adjustments and context-awar attention mechanisms. This allows the model to adapt to evolving data distributions, ensuring robust and accurate predictions in dynamic environments.
MICN (Wang et al., 2023) integrates multi-scale architectures to capture both short-term patterns and long-term dependencies in time series data. By combining local convolutional operations and global attention, it provides a balanced approach to handling diverse temporal characteristics.
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Rethinking Time Encoding via Learnable Transformation Functions
TimesNet (Wu al., 2023) innovatively models time series data in the frequency domain, leveraging discrete Fourier transformations to capture periodicity and trends. This approach enhances its ability predict time series with prominent seasonal and cyclical behaviors efficiently.
E.2.3. DATASETS
We utilize 4 real-worid datasets to evaluate the effectiveness of our method on time series prediction tasks, encompassing various real-world scenarios. The dataset statistics are presented in Table 3, with detailed descriptions provided below.
Table Time Series Dataset Statistics: dataset size organized (Train, Validation, Test). Please refer to (Wu al., 2023) the
original table.
Dataset
Dim
Series Length
Dataset Size
Information (Frequency)
ETTml, ETTm2
(96, 192, 336, 720)
(34465, 11521, 11521)
Electricity (15 mins)
ETTh1, ETTh2
7
(96, 192, 336, 720)
(8545, 2881, 2881)
Electricity (15 mins)
Electricity
321
(96, 192, 336, 720)
(18317, 2633, 5261)
Electricity (Hourly)
Exchange
(96, 192, 336, 720)
(5120, 665, 1422)
Exchange rate (Daily)
Weather
21 (96, 192, 336, 720)
(36792, 5271, 10540)
Weather (10 mins)
ETT dataset includes time series data for oil temperature and power load measurements from electricity transformers, collected between July 2016 and July 2018. Specifically, the subsets ETTml and ETTm2 are sampled at 15-minute intervals, while ETThl and ETTh2 are recorded hourly.
Electricity dataset provides hourly electricity consumption data for 321 clients, spanning the period from 2012 to 2014.
Exchange (Lai et al., 2018) dataset offers daily panel data on exchange rates from eight countries, covering the years 1990 to 2016.
Weather' dataset contains meteorological time series data, comprising 21 weather indicators recorded at 10-minute intervals in 2020 by the Weather Station of the Max Planck Biogeochemistry Institute.
E.3. Dynamic Graph Baselines and Datasets
1. EXPERIMENT IMPLEMENTATION DETAILS
The syper-parameters are based on the best configurations reported in the papers, and we keep them unchanged across different experiments for each baseline model to ensure a fair comparison. We rerun the baseline models TGAT with batch size 100 and reuse the baseline results reported in the DyGFormer paper for other baselines. The results are evaluated using Average Precision, i.c., AP (Table 2) and Area Under the Receiver Operating Characteristic Curve, i.e., AUC-ROC (TTble 6), both of which are widely used metrics in dynamic graph representation learning research.
E.3.2. BASELINES
We select 4 commonly used continuous dynamic graph representation learning baselines TGAT (Xu et al., 2020), TGN (Rossi et al., 2020), TCL (Wang et al., 2021), and DyGFormer (Yu et al., 2023)-ann and replace the Functional Time Encoding methods (Kazemi et al., 2019; Xu et al., 2019) with LeTE to demonstrate its optimal performance on the dynamic graph link prediction task. brief introduction to these baseline models is provided below:
TGAT (Xu et al., 2020) introduces a temporal attention mechanism aggregate information from temporal-topological neighbors, thereby generating temporal node representations in temporal graphs. Additionally, it proposes trainable
"nttps://githuk com/ http://www....o//oooaa.././///// wttam//hoooooyyy/aaaaaaaa
i caa. uci http.///dm/mm/eeeee////e/ /dataseta/aaaaeae httt///tteeerrrtrrriiiiiiiiiitii 2 agrams201 12014 1211
*https: http://www...oooo .tt..app...e.eerr.//////m./m/...
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Rethinking Time Encoding via Learnable Transformation Functions
time encoding function capture distinguishable temporal information, which has been widely adopted in subsequent dynamic graph network architectures.
TGN (Rossi et al., 2020) integrates key ideas from previous models and introduces a memory module that maintains a state vector for each node. The memory is updated dynamically whenever nodes participate in interactions. Additionally, TGN incorporates a message paassagg module, memory update module, and a temporal embedding module to generate effective temporal representations for nodes within temporal graphs.
TCL (Wang et al., 2021) utilizes a breadth-first search algorithm to construct a temporal dependency interaction sub- graph, extracting interaction sequences. It employs a Transformer encoder that integrates both topological and temporal information to learn representations of central nodes. Additionally, TCL introduces cross-attention mechanism within the Transformer model the interddependencies between interacting node pairs.
DyGFormer (Yu et al., 2023) leverages 1- l-hop neighbor information for learning temporal graph representations. It employs a Transformer encoder enhanced with a patching technique to effectively capture long-term dependencies among nodes in temporal graphs. Furthermore, DyGFormer incorporates a Neighbor Co-occurrence Feature to preserve the correlation information between source and target nodes.
E.3.3. DATASETS
We utilize 4 real-works datasets (Kumar et al., 2019) to evaluate the effectiveness of method dynamic graph representation learning tasks, encompassing various real-world scenarios. The dataset statistics are presented in Table 4, with detailed descriptions provided below.
Table 4. Dynamic Graph Dataset Statistics: Dimn and Dime represent the dimensions of node features and edge fcatures, respectively, For mon-attrihuted graphs follow previous studies (Xu al., 2020; Rossi al., 2020) and use 172-dimensional zero vectors padding.
Dataset
Dim,
Dim
# Nodes
# Edges
Information
Duration
Time Granularity
Wikipedia
172
9,227
157,474
Social
month
Unix timestamps
Reddit 172
10,984
672,447
Social
month
Unix timestamps
MOOC
7,144
411,749
Interaction
17 months
Unix timestamps
LastFM
1,980
1,293,103
Interaction
month
Unix timestamps
Wikipedia records editing activities on Wikipedia pages over a one-month timeframe. Nodes in this graph represent users or pages, and temporal links with timestamps capture the edits. Each link is associated with a 172-dimensional feature vector based on LIWC (Linguistic Inquiry and Word Count) (Pennebaker. 2001).
Reddit captures user activity across subreddits over one-month period. In this dataset, nodes represent users or subreddits, while timestamped links denote posting actions. Fach link is further characterized by a 172-dimensional feature vector derived from LIWC.
MOOC captures the interactions of users widely used MOOC platform, structured as directed, temporal network. In this representation, nodes correspond to users and course activities (referred to targets), while edges denote the actions performed by users on these targets
LastFM records interaction data where users listen to songs over a month. In LastFM, nodes correspond users and songs, and the links represent listening activities performed by users.
E.4. Real-World Application Dataset
The dataset used for real-world application experiments is a financial risk control dataset, containing records of 483,379 users' transaction behavior at various merchants over 60-day period. It includes a total of 26,850,000 transactions. Each user is represented by a disensinnli feature, each merchant by a 128-dimensional feature, and each transaction by a 202-dimensional feature, with all transactions labeled with UNIX timestamps. The ratio of positive users (with default risk) to negative users (without default risk) 1:9.9 the training dataset and 1:20.25 in the test dataset. The backbone model
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Rethinking Time Encoding via Learnable Transformation Functions
employs a specially designed Transformer- architecture to aggregate users' historical transaction features, merchant features, user features, and optional time embedding into user embeddings, which are then used predict whether the user has default risk.
F. Additional Experimental Results
Here, we provide the complete results of the experiments discussed in the main text.
F.1 Results of multivariate time series long-term forecasting task evaluated using MSE
The results of the multivariate time series long-term forecasting task, evaluated using MSE, are presented in Table 5. These results are organized in the same manner as those in Table 1.
Table Time series prediction: multivariate long-term forecasting task. The past sequence length is set to 96, while the prediction lengths are (96, 192, 336, 720}. The results are reported in terms of MSE, where lower values indicate better performance. HCTE CradfCrd Time Encoding) is a method widely adopted in time series research, FTE stands for Functional Time Encoding. The win rate represents the percentage of cases where LeTE outperforms the HCTE. The best results for each baseline, dataset and prediction length combinations are in bold. ETT consists of 4 subsets. Here, we present the average results across these subsets, with the full results provided in Table 8.
MSE
Transformer
Pyraforner
NS Trans.
MICN
TimesNot
Win
TB
HCTE
FTE
LeTE
HCTB
FTE
LeTE
HCTE
FIE
LeTE
HCTE
FIE
LeTE
HCTE
FTE
Le'IE
Rate
96
1,219
1.204
0.568
0.794
0.960
0.650
0.392
0.486
0.347
0.392
0.310
0.29%
0.312
0.322
0.305
192
2,601
1.643
0.959
1.667
1.812
1.058
0.446
0.548
0.410
0.446
0.396
0.357
0.365
0.390
0.368
95%
336
2,438
1,557
1.192
1,981
1.962
1.297
0.492
0.651
0.467
0,492
0.487
0,430
0.419
0.422
0,404
720
1.998
2.217
1.313
2,589
2.442
1.612
0.552
0.660
0,531
0.552
0.630
0,517
0.467
0.468
0.441
Electriity
96
0.258
0.280
0.252
0.285
0.281
0.267
0.169
0,171
0.163
0.164
0,156
0.150
0,168
0.168
0.164
192
0,266
0.310
0.259
0,298
0.288
0.274
0.182
0.192
0.178
0.177
0.170
0.166
0,184
0.179
0,179
95%
336
0.275
0.339
0.265
0.307
0,306
0.274
0.200
0,201
0.190
0.193
0.189
0.184
0,198
0.210
0,193
720
0,288
0.395
0.276
0.304
0.304
0,293
0.222
0,237
0.226
0.212
0.228
0.208
0,220
0.287
0,220
Weather Exchinge
96
0.545
0.788
0.464
0,505
0.616
0.584
0.111
0.137
0.102
0.102
0.098
0,080
0.107
0.108
0.103
192
0.950
1.028
0.903
1.015
0.981
0.904
0.219
0.193
0.172
0.186
0.152
0.226
0.217
0.211
95%
336
1 1,62
1.866
1.114
1.263
1,420
1.056
0.421
0.463
0.342
0.272
0.356
0.265
0,367
0.413
0.367
720
2.569
2.002
1.736
1.762
1.714
1.316
1.092
1.546
0.682
0.714
0.833
0.636
0,964
0.989
0,958
96
0.393
0.184
0.172
0.225
0.207
0.181
0.173
0.172
0.168
0.161
0.198
0.166
0.172
0.172
0.166
192
0.547
0.249
0.220
0,252
0.238
0.230
0.245
01.224
0.215
0.220
0.243
0.209
0..119
0.221
0.212
95C
336
0.678
0.328
0.309
0.362
0.324
0.284
0.321
0,295
0.277
0.278
0.285
0.251
0,280
0.278
0.274
720
0.844
0.480
0.425
0.411
0.394
0.383
0.414
0.350
0.338
0.311
0.350
0.303
0.365
0.354
0.352
Win Rate
100%
94%
94%
94%
94%
95%
F.2. Results of dynamic graph link prediction task evaluated using AUC-ROC
The results of the dynamic graph link prediction task, evaluated using AUC-ROC, are presented in Table 6.
Table 6. Dynamic graph link prediction task: better results in bold. Here, we present the top-performing results across variations of LeTE, with the full results provided in Table 10. FTE represents Functional Time Encoding which is commonly used in dynamic graph research.
The results are reported in AUC-ROC, where higher values indicate better performance. The
AUC
Wikipodia
Reddit
MOOC
LastFM
TE Transductive
Inductive
Transductive
Inductive
Transductive
Inductive
Transductive
Inductive
TGAT
FTE LeTE
0.26 96.69
95.95 = 0,33
98.48 0.04
96.90 0.07
86.44 1 0.24
86.04 0.19
70.89 0.10
76.11 1 0.11 79.63 a 0.14
97.63 + 0.11
97.07 + 0.10
98.51 + 0.01
96.96 = 0.05
89.46 + 0.08
89.50 0.05
74.24 :: 0.28
FTE LcTE
98.37 + 0.07
97.72 = 0.03
98.60 1: 0.06
97.39 2 0.07 97.55
91.21 + 1.15
91,24 2 0.99
78.47 + 2.94
82.61 + 3.15
TGN
98.73 0.07
98.11 = 0.10
98.72 0.00
92.68 + 0.37
92.41 0.73
83.94 + 1.85
87.74 + 1.75
TCL.
FTF LcTE
95.84 + 18
95.57 =: 0.20
97.42 + 0.02
93.80 + 0.07
83. + 0.18
81.43 + 0.19
64.06 + 1.16
70.84 + 0.85
97.84 0.06
97.56 0.02
97.68 0.03
94.63 0.05
84.73 + 0.13
83.25 0.23
70.17 0.47
75.86 0.44
DyG- Former
98,91 0.02
98.48 0.03
99.15 0.01
98.71
87.91 + 0.58
87.62 0.51
93.05
94.08 0.08
LcTE
99.04 0.01
98.67 + 0.02
99.17 0.00
98.74 0.01
89.18 + 0.21
89.16 0.30
93.65 0.07
94,52 + 0.08
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Rethinking Time Encoding via Learnable Transformation Functions
E33 Full results of the multivariate long-term forecasting task on 4 ETT subsets
We present the full results of the multivariate long-term forecasting task on the 4 ETT subsets in Tables and 8, as Tables I and 5 report the average results.
Table Time series prediction: multivariate long-term forecasting task on subsets of ETT. The past sequence length is set to 96, while the prediction lengths are (96. 192, 336, 720}. The results reported in terms of MAE The best results for each bascline, dataset and prediction length combinations are in bold.
MAE
Transformer
Pyraformer
NS Trans.
MINC
TimesNet
TE
ICTTE
FTE
LcTE
ICCTE
FTE
LeTE
IICTE
FTE
LeTE
IICTE
FTE
L+TE
IICTE
FTE
LeTE
96
0.621
0.601
0.507
0.581
0,522
0.521
0.398
0.409
0.389
0.398
0.376
0.372
0.375
0.389
0.378
E1Tm2 ETTml
192
0.703
0.550
0.526
0.577
0.547
0.531
0.444
0.427
0.417
0.444
0.393
0.399
0.387
0.418
0.404
336
0.795
0.755
0.596
0.675
0.637
0.610
0.464
0.496
0.439
0.464
0.425
0.426
0.411
0.418
0.418
720
0.798
0,782
0.663
0,760
0,700
0.645
0,516
0.498
0.470
0,516
0.467
0.465
0.450
0,456
0.454
96
0.506
0.451
0.420
0.458
0.756
0,411
0.274
0.304
0.278
0.274
0.287
0.271
0.267
0.268
0.262
192
0.908
0.700
0.558
0.649
0.611
0.625
0.339
0,379
0.325
0.339
0.349
0.321
0.309
0.306
0.302
336
0.796
0.806
0.753
0.811
0.902
0.775
0.361
0,400
0.353
0.361
0.439
0.364
0.351
0.351
0.342
720
1.192
1,301
0.851
1,416
1.491
1.137
0.413
0.446
0.420
0.413
0.506
0.432
0.403
0.409
0.399
96
0.739
0.814
0.524
0.637
0.613
0.593
0.491
0,616
0.452
0.491
0.413
0.409
0.402
0.425
0.408
E1Th2 El
192
0.762
0.815
0.632
0.738
0.778
0.647
0.504
0.631
0.483
0.504
0.465
0.452
0.429
0.457
0.439
336
0.772
0.786
0.679
0.794
0.758
0.736
0.535
0.730
0.541
0.535
0,511
0.502
0.469
0.469
0,453
720
0.800
0.878
0.724
0.776
0.804
0.782
0.616
0,792
0.610
0.616
0.598
0.565
0.500
0.490
0.421
96
1.323
1,349
0.752
0.892
0.989
0.806
0.458
0.413
0.390
0.458
0,392
0,349
0.374
0.367
0.359
192
2.184
1,599
1.133
1.632
1.760
1.150
0.493
0.475
0.493
0.485
0.408
0.414
0.406
336
2.113
1,405
1.256
1.893
1.856
1,330
0.551
0.528
0.463
0.551
0.569
0.500
0.452
0.458
0.438
720
1.488
1.623
1.276
1.832
1.761
1.274
0.560
0,492
0.460
0.560
0.673
0.558
0.468
U.466
0.443
Table 8. Time series prediction: multivariate long-term forecasting task on 4 subsets of ETT. The past sequence length is set to 96, while the prediction lengths are (96, 192, 336, 7203. The results are reported in terms of MSE. The best results for each haseline, dataset and prediction length combinations are in bold.
MSE
Transfommer
Pyrafonner
NS Trans.
MINC
TimesNet
TE
HCTE
FIE
LeTE
HCTE
FIE
LeTE
HCTE
FIE
LeTE
HCTB
FIE
LeTE
HCTB
FIE
LeTE
96
0.713
0.667
0.511
0.708
0.581
0.584
0.386
0,430
0.370
0,386
0.329
0.325
0.338
0.365
0,339
192
0.866
0.592
0.573
0.693
0.621
0.560
0.459
0.464
0.418
0.459
0.364
0.373
0.374
0.430
0.392
336
1,063
1.035
0.692
0.848
0.754
0.689
0.495
0.651
0.451
0.495
0.403
0.396
0.410
0.412
0.412
720
1.075
1.062
0.788
1.009
0,855
0.777
0.585
0.584
0.510
0.585
0.469
0.463
0.478
0.477
96
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0.354
0.327
0.384
1.047
0.310
0.192
0,246
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0.192
0.190
0.187
0.187
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0.182
ElTm2
192
1.499
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0.280
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0.266
0.280
0.271
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0,253
0.248
336
1.107
1.113
0.949
1.144
1.377
1.023
0.334
0,387
0.315
0,334
0.398
0.307
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0.327
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720
2.609
2,931
1.104
3.624
3.708
2.195
0.417
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0.417
0.525
0.414
0.408
0.427
0.408
96
0.876
1.032
0.535
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192
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0
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720
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96
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Er
192
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336
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720
3.279
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2.516
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0.513
0.448
0.562
0,880
0.615
0.462
0.456
0.426
F44 Dimensions of Time Embedding
We present the AUC-ROC results for Wikipedia/TGN and MOOC/TGN with different time embedding dimensions (for both FTE and LeTE) in Figure 6. To cover broader range of scenarios, we also include the results for DyGFormer on the Wikipedia dataset in Figure 7.
G. More Experiments
G.1. Statistic Analysis of the Complex Time Patterns in Data
Time-related data often contains mixed and complex patterns, which can primarily be categorized as periodic and non- periodic. To investigate the periodic and non-periodic patterns in the data, we analyze four dynamic graph datasets
20

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Rethinking Time Encoding via Learnable Transformation Functions
99.00
94.00
98.00
92.50
97.00
91.00
LoTE Trus.
LoTE: Teass
96.20
FTE Trxes
88.00
FTE Timn.
LeTE
95.50
86.50
2.8
16
32
64
100
16
32
64
100
(a) Wikipedia/TGN
(b) MOOC/TGN
Figure 6. AUC-ROC results comparing different dimensions of the FTE and Spline- Spline-based LeTE on Wikipedia/TGN and MOOC/TGN.
99.30
99.20
99.04
98.96
98.78
98.72
98.52
98.48
L-TE
LeTE
96.26
98.24
LeTE Indac.
LoTE Induo:
PTT Inda::
PTB
98000
(a) AP
(b) AUC-ROC
Figure and AUC-ROC results comparing different dimensions the FTE and Spline-basee based LcTE on Wikipedia/DyGFomee.
using spectral entropy (Shannon, 1948). First, we normalize the time or time differences (since previous dynamic graph representation learning methods typically use time differences as inputs to the time encoding, we include this analysis here as well) for each node with more than five interactions, mapping the values to the range [o, 1]. We then treat each node's interaction times as a signal sequence. The spectral entropy for each node is computed as follows: We begin by applying the Fast Fourier Transform (FFT) to the normalized signal sequences: X (ff) FF where X(f) is the frequency- domain representation of the signal. Next, we caleulate the magnitude of the frequency components M(J) [X())..
The magnitudes are then normalized to form a probability distribution: P(f) = Σ, M(T) Finally, the spectral entropy P(J) entropy value indicates periodicity, while a higher entropy value suggests randomness.
M(S)
computed as: H(P) = - P(J) log P(J), which measures the uniformity of the frequency components. A lower
We present the density plots of the spectral entropy in Figure 8. As shown in the figures, only small portion of the nodes exhibit strong periodicity in their interaction times or time differences, while most nodes show high entropy, indicating non- non-periodic behavior. This suggests that capturing periodic patterns alone is insufficient; it is also important to model non-periodic patterns to enhance the efficiency and expressiveness of the time encoding.
Dessiy Piuc of Spectral Estops of Intrrction))
Danelt Plat of Speeetl Eninay (Time Diflaews betwor Isteaationn)
Whieeaa
Wikjodo
12
Denatt
Specral homopy
Spectral
Figure 8, Density plots of spectral entropy for dynamic graph datasets.
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Rethinking Time Encoding via Learnable Transformation Functions
G.2. Comparative Analysis of Different Variants of LcTE
We conducted set of additional experiments on dynamic graph link prediction tasks, applying different variants of LcTE and comparing their downstream task performance, evaluated by AP and AUC-ROC. The results, presented in Tables 9 and 10, indicate that, in most cases, Combined LcTE achieves the best performance among the three variants of LcTE. This outcome is intuitive, as Combined LeTE leverages the strengths of both Fourier-based LeTE and Spline-based LeTE, enabling it to effectively model diverse time patterns.
Due to differences in the periodicity and non- hon-periodicity of node interactions across datasets, the effectiveness of Fourier- based LeTE and Spline-based LeTE varies. Nonetheless, in most cases, both methods outperform the benchmark. This demonstrates that even when using only Fourier-based LeTE or Spline-based LeTE, they can effectively model different patterns, including periodic, non-periodic and mixed patterns, in the data.
Table Comparing Functional Time Encoding (FTE), Fourier-baser LeTE (F-LeTE). Spline- LeTE (S-LeTE) and Combined LeTE (C-LeTE); Dynamic graph link prediction results in AP. The best results are in bold.
AP
Wikipedia
Reddit
MOOC
LastFM
TE
Transdiactive
Inductive
Transductive
Inductive
Transductive
Inductive
Transductive
Inductive
FTE
96.95 a: 0.24
96.33
98.53 a 0.04
85.34 0.19
72.73 0.11
77.78 0.13
R.LTEE
96.82 + 0.16
96.31 0.13
98.54 + 0.03
97.03 + 0.02
85.25 = 0.29
85.08 0.29
72.31 + 0.30
77.19 + 0.38
TGAT
S-LeTE
97.54 0.06
97.06 + 0.05
98.56 + 0.01
97.05
88.31 = 0.10
88. 0.28
75.68 + 0.55
80.61 + 0.42
C-LeTE
97.82 + 0,09
97.34 = 0,08
98.56 + 0.01
96.99 0.06
88.30 0.05
88.37 0.12
76.22 0.25
81.32 : 0.14
FTE
98.45 0.06
97.83 0.04
98.63 0.06
97.50 0.07
89.15 1,60
89.04 1.17
77.07 3.97
81.45 + 4.29
R-LeTE
98.57 :: 0.09
97,94 = 0.08
98.66 + 0.01
97.39 = 0.07
90.04 = 0.67
89.94
77.58 # 5.22
82,82 + 6.53
TGN
S.LTTE
98.55 0.06
97.98
98.74 + 0.00
0.04
0.20
90.87
82.26 2.27
86,46
C-LeTE
98.78 + 0,07
98.19 0,09
98.74 + 0.01
97.52
91.41 0.55
90.17 0.69
83.64 2.00
87.55 1.88
FTE
96.47 + 0.16
96.22 0. 17
97.53 0.02
94.09 0.07
82.38 0.24
80.60 0.22
67.27 + 2.16
73.53 1.66
R-LeTE
97.83 + 0.05
97.58 10
97.74 # 0.03
94.75 = 0,20
83:40 1.32
81.75 # 1.43
76.08 #: 0.79
80.68 + 0.70
TCL
S-LOTE
1: 0.06
97.05
97.78 a: 0.03
83.87 0.30
69.92 0,46
76.44
C.LeTE
98.19 + 0.04
97.89 0.03
97.75 a 0.09
94.83 0.20
84.24 = 0.10
82,72 0.12
72.76 a: 4.64
78.70 +: 3.67
FTE
99.03 1 0.02
98.59 a 0.03
99.22 0.01
98.84 0.02
87.52 0.49
86.96 0.43
93.00 0.12
94.23 0.09
DyG- Former
F-LeTE
99.04 0.01
98.66 68.66.2. 0.05
99.22 + 0.01
98.85 0.02
87.60 0.26
87.15 0.22
93.06 0.05
94.11 #: 0.09
S.L.TE
99.12. 0.01
98.72 0.03
99.17 0.10
98.78 0.13
88.66 0.20
88.37 0.25
93.50 0.12
94.57 0.15
C-LeTE
99.13 a 0.02
98.73 a: 0.00
99.24 a: 0.01
98.86 a: 0.01
88.70 a: 0.21
88,39 = 0.15
93.64 a: 0.10
94.69 +: 0.12
Table 10. Comparing Functional Time Encoding (FTE), Fourier-based LcTE (F-LeTE), Spline- based LcTE (S-LcTE) and Combined LeTE (C-LeTEE): Dynamic graph link prediction results in AUC-ROC. The best results are in bold.
AUC
Wikipedia
Reddit
MOOC
Las
TE
Transductive
Inductive
Transdutivee
Inductive
Transductive
Inductive
Transductive
Inductive
FTE FLLTE
96.69 1 0,26
95.95 0.33
98.48 1 0.04
96.90 0.07
86.44 0,24
86.04 0.19
70.89 0.10
76.11 i 0.11
TGAT
96.53 +: 0.17
95.94 a: 0.20
98.48 a: 0.03
96.91 0.03
86.36 = 0.29
86.18 :: 0.33
70.53 a 0.18
75.61 +: 0.22
S-LeTE
97.33 + 0.07
96.76 0.06
98.51 0.02
96.96 0.05
89.38 0.14
89.14 + 0.24
73.89 0.51
79.17 + 0.44
C-LeTE
97.63 + 0.11
97.07 + 0.10
98.51 + 0.01
96.88 + 0.05
89.46 = 0.08
89.50 + 0.05
74,24 + 0.28
79.63 + 0.14
FTE
98.37 + 0.07
97.72 + 0.03
98.60 + 0.06
97.39 = 0,07
91.21 = 1.15
91.24 + 0.99
78.47 + 2.94
82.61 +: 3.15
R-LTTE
98. 0.10
97.86 0.09
98. + 0.02
97.27
91.71 = 0.65
91.53 0.33
78.24 4.86
83.16 1 6.28
TGN
L-LTTE
98.47 +: 0.06
97,86 0.06
98.72 0.00
97.55 0.05
92.68 0.37
92.41 0.73
82.48 2.17
86.49 0.62
C-L-TE
98.73 0.07
98,11 0.10
98.72 0.02
97.43 0.11
92.65 0.48
91.27 + 0.85
83.94 1.85
87.74 + 1.75
FTE
95.84 +: 0.18
95.57 0.20
97.42 a 0.02
93.80 0.07
83.12 0.18
81.43 0.19
64,06 1.16
70.84 +: 0.85
TCL
BI&TE
+ 0.07
97.14
97.62 0.03
94,42
0.92
0.98
70.17 0.47
75.86
S-LOTE
96.90 + 0.08
96.62 1.0:+0..1 0.15
97. 68 + 0.03
94.63 0.05
84.50= 0.20
83.02 : 0.23
67.32 0.50
74.37 + 0.51
C-LeTE
97.84 +: 0.06
97.56 1: 0.02
97.6 0.08
94.49 2 0.17
84.73 a 0.13
83.25 = 0.23
69.16 3.21
75.85 :: 2.64
FTE
98.9 + 0.02
98.48 4: 0.03
99. 0.01
98.71 = 0.01
87.91 = 0.58
87.62 0.51
93.05 + 0.10
94.08 + 0.08
DyG- Former
FLOTE
98. 0.02
y8.55
99.16 1 0.01
98.72
88.09 0.16
0.14
93.09 0.03
0.04
S-LeTE
99.04 + 0.01
98.67 0.02
99.08 + 0.13
98.61
89.18 + 0.21
89.16 0.30
93.56 0.06
94,46 + 0.06
C-LeTE
99.04 ::: 0.02
98.65 0.01
99.17 = 0.00
98.74 0.01
89.17 0.20
89.14 0.09
93.65 : 0.07
94.52 : 0.08
G.3. Visualization and Interpretability
As mentioned in Section 3.2, the learned parameters of our proposed method can be used to reconstruct the time embedding feature map or the non-linear transformations. Since the previous time encoding method (FTE) uses fixed non- non-linear transformation functions, we present an example of its feature map and compare it with the Fourier-based LeTE in Figure 9,
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Rethinking Time Encoding via Learnable Transformation Functions
Larger
(8) Functional Time Encoding (FTE)
Langer input range
(b) Fourier-Based TE (Ours)
Figure Example feature map for the FTE Fourier-based LeTE different dimensions (the total dimension the parameters weight are based on learned TGN model on Wikipedia dataset).
pim
0.75
Dim
0.50
1
pām Bim
Time Eritaniding Dimemsion
Transfommed Value
Figure Example of non-linear transformation for Spline- based LeTE different dimensions.
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Rethinking Time Encoding via Learnable Transformation Functions
Crashiee
Malpoayy Ma M
midy
Chmpetonn Caencin
Amma
Synthetic Periodic Data
Synthetic Non-Periodic Data
Synthetic "Mixed" Duta
Figure 11. Capturing periodic, non-periodic and mixed patterns synthetic data.
Coffer Decmemnee Sopereo So
Coopative of of and Revoeraen The Soarro Oreend Guasle
Orgge Pecmmmennt 1il
Oiiggnd Envorrend fro botty ow1)
Figure 12. Capturing periodic, non-periodic and mixed patterns real data.
We compare the local and global mappings of the two time encoding methods. As shown in the figure, our method captures richer and more detailed time patterns in different dimensions of the time encoding for local mappings. By contrast, the FTE method exhibits periodicity in only one dimension. This occurs because the learned frequency parameters in other dimensions are too small to capture sufficient periodicity locally. Similarly, for global mappings, strong periodicity is still observed the feature map of our method, alongside varying degrees of non-periodicity. The FTE continues to exhibit periodicity only one dimension. In other dimensions, the similar frequency parameters result in insufficient periodicity modeling and lack of pon-oorio pattern representation.
We also present the non-linear transformation sketches of Spline-based LeTE in Figure 10. The figure shows that the learned non-linear activation functions vary across dimensions, significantly enhancing the model's expressiveness. Additionally, demonstrates the modeling of both periodic and non-periodic patterns at local and global scales. Since Combined LeTE a combination of Fourier-based LeTE and Spline- based LeTE, is intuitive that Combined LeTE inherits the same interpretability as its components. Furthermore, as our method supports the reconstruction of non-linear activation functions, retains strong interpretability.
G.4. Capturing Periodic, Non-Periodic and Mixed Patterns in Data
Complex periodic and non-periodic patterns often coexist in real-world data, forming mixed time patterns. To demonstrate that our method surpasses previous methods in modeling such patterns, we design mini reconstruction task using both synthetic data and real data from the Wikipedia dataset. Specifically, construct an encoder-decoder model to reconstruct the data. The encoder is either (d-dimensional) our LeTE or the FTE, while the decoder is simple linear layer mapping d-dimensional vector to a 1-dimensional output. The reconstruction objective minimizes the MSE loss, which also quantifies the modeling capability of the time encodings. Additionally, the reconstructed time sequence plots visually indicate the models' ability to fit the data.
To isolate periodic and non-periodic patterns, we first generate synthetic data containing purely periodic signals, purely
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Rethinking Time Encoding via Learnable Transformation Functions
y sin(x)
y sin(x))sin(2x)
log(1 + e*)
y =
1
FTE
445
-12
155
Fourier-based
LcTE
Spline-based LeTE
Target Function
Learned Function
Figure 13. FTE, Fourier-based LeTE and Spline-based LeTE fitting different functions.
non-periodic signals, and mixed signals. These data used to evaluate the performance of different encoders (LeTE FTE). The ground truth and reconstructed sequences are shown in Figure 1.. As illustrated, the FTE method performs reasonably well on periodic data but struggles with non-periodic and mixed data. In contrast, our method consistently outperforms FTE, demonstrating its capability to model both periodic and non non-periodic patterns effectively.
Real-world data often exhibit complex combinations periodic and non-periodic patterns, i.e., mixed patterns. To further evaluate our method, we randomly select nodes' interaction sequences from the Wikipedia dataset and perform the same reconstruction experiments. The results are presented in Figure 12, where the time sequences are smoothed using a ID Gaussian filter for clarity. As shown, the time sequences reconstructed using our LeTE align more closely with the original data compared to those reconstructed using FTE. Additionally, the loss of our LeTE is significantly lower than that of FTE, further validating our method's s ability capture complex periodic and non-periodic patterns in real-world data.
The experimental results show that, regardless of whether the sequence is periodic or non-periodic, our method consistently outperforms better. This is primarily due to the incorporation of learnable non-linear transformations into our time encoding approach.
G.5. Fitting Ability
We conduct a simple toy experiment to further demonstrate that both Fourier-based LeTE and Spline-based LeTE are capable of capturing different patterns. Consequently, Combined LeTE inherits this ability as well. illustrate this, we generate set of training data using 4 different non-linear transformation functions. Two of these functions are periodic: the sine function y sin(x), and more complex periodic function y (1 + sin(x)) sin (2x). The other two functions non-periodic: the Softplus activation function y log(1 + e²") (Misra, 2019), and the Swish activation function y Ite-= (Radford et al., 2019).
We fit the data using simple I-dimensional FTE, Fourier- based LeTE and Spline-based LeTE, evaluating their ability to capture complex patterns, including both periodic and non periodii. The learned non-linear transformation functions are plotted in Figure 13. shown in the figure, both Fourier-based LeTE and Spline-based LeTE successfully capture diverse
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Rethinking Time Encoding via Learnable Transformation Functions
patterns. We also compare our method with FTE. Due to the fixed non-linear transformation functions used in FTE, it fails capture the complex periodic and non-periodic patterns present in the data. These results demonstrate that our proposed LeTE has the capability to model complex patterns in data effectively and is more general than previous time encoding methods.
H. More Explanations and Examples about Interpretability
We choose to use a 4-dimensional Combined LeTE to present our analysis related interpretability of LcTE. The experiments conducted on the Wikipedia and MOOC datasets, with TGN and DyGFormer backhone models. The training process and settings consistent with those used in the main experiments. We will demonstrate the interpretability of our model from the following perspectives:
1. Reconstructing the learned non-linear transformation functions and plotting them to provide a clear and intuitive analysis.
2. Analyzing each dimension understand what information it represents. Specifically, the first two dimensions of the time encoding are Fourier-based, while the last two dimensions are Spline-based.
3. Comparing different datasets under the same backbone model.
4. Comparing different backbone models' LeTE under the same dataset.
5. Comparing the plots of low- vs. high-dimensional LcTE to assess the impact of dimensionality interpretability.
H.1. Reconstructing
As previously discussed, the previous time encoding methods exhibit a degree of interpretability by using fixed sinusoidal functions, which inherently reflect periodic patterns. However, this strong inductive bias also limits their expressiveness and generalization to complex or non- periioiidy
In contrast, our proposed LeTE is fully learnable time encoding, and the learnable non-linear functions can still be reconstructed and visualized from learned parameters, allowing for interpretability analysis through function inspection.
We demonstrate this interpretability using 4-dimensional Combined LeTE, trained on the Wikipedia/TGN, Figure 14 shows the learned transformation functions for each dimension. The first two dimensions are Fourier-based and the last two are Spline-based.
H.2. Analyzing Each Dimension
Fourier-based: The Fourier coefficients explicitly encode frequency components, offering a clear, intuitive view of the captured periodicity. Compared to fixed sinusoidal functions, our learnable Fourier-basee time encoding captures periodic patterns with finer granularity and greater flexibility, enabling the representation of both periodic signals and subtle non-periodicities within specific ranges.
For single dimension, low-frequency components capture long-term trends, while high-frequency components focus on short-term fluctuations. This allows the model to encode both long-term dynamics and short-term variations simultaneously. As an example, we apply this to the Wikipedia dataset, which records editing activities, where nodes represent users or pages, and edges with timestamps capture editing events (frequency magnitude spectrum is shown in Figure 15, note that the inputs are time differences this case).
Specifically, Dim shows strong ligh-frequecyy response. The learned coefficients include cos(32r) +0.29, cos(Az/) -0.16, cos(52/) 0.42. This suggests that Dim 0 is sensitive short-term repetitive edits, i.e., high-frequency editing behavior.
Dim 1 captures low- to mid-freou mid-freeqennc patterns, with large coefficients: sin(1æ/) +0.96, sin(4z)) : +0.61, cos( (4x/) +0.29. These reflect longer-term periodic behaviors. For example, frequency-1 may correspond to daily or weekly editing cycles, while frequency-44 may capture sub-daily repeated interactions. This dimension may reflect user habits or regular community editing patterns. Thus, LeTE's Fourier- dimensions not only retain the periodic interpretability of sine
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Rethinking Time Encoding via Learnable Transformation Functions
Dim 0 (Fourier-based)
0.
-0.6-
Dim 1 (Fourier-busedd)
0-
Dim (Spline-based)
0.0
Dim 3 (Spline-based)
-0.36
-0.38
Time
Figure 14. Plots of the four dimensions of the non-linear transformation functions of a 4-dimensional LcTE trained on Wikipedia/TGN. We further present four functions here, the parameters are learned read from trained model:
fo(x) = -0.0444 cos(1 x6) + 0.0758 sin(1 x²) + 0.0875 cos(2 =/) + 0.0704 sin(2 =6) + 0.0712 cos(3
<b) 0.0327-sinll
+ 0.0040 cos(4 (oo(4-20) mh) 0.0340 sin(4 (6)) + 0.0150 cos(5 -n) 0.0220 sin(5 x6) + 0.0710 cos(1
x1) + 0.05 (in(1.z]]
0.1483
cos(2 + 0.2502 sin(2 (6n(2.1]) + 0.2938 cos(3 0.0878 sin xf) 0.1641 cos
<1 0.0640 sin(4
21) 0.4155 cos(5 21) + 0.0395 sin(5 z() 0.0762,
f1(x) +-.1860 cos(1 =()) 0.0267 sin(1 =66 0.1971 cos(2 =b) 0.0510 sin =²) 0.0225 cos(3
<6) 0.0909 sin(3 =/) 0.0501 cos (4- =/) + 0.1460 sin(4 4. =6) + 0.0952 - ( 0.2974 sin(5 (au(5.))))) x6) 0.1604 cos (1s(1
x1 0.9609 sino0ssnn11110 x1) 0.2323 cos(2- x1) 0.3430
sin(2 xí) 0.0441
(3 - (in1428--in3--.... x1) + 0.29930 1.2930-coo(4 cos(4
<1) + 0.6073 sin(4- 21) + 0.0330 cos(5 z1) - 0.1345 sin(5 x1) - 0.0130,
here, for both fo(x) and f1(2), =² 1.0069 0.0069 and a1 0.0054 0.0108.
f2(x) +0.0013- (x) (support: [-2.20,-1.80]] 0.0017 B (x) (support: [-1.80,-1.40)- 0.0353 B2(x) (support: [-1.40,-1.00]-
0.0321 (1) (support: [-1.00.-0.60]]
0.0455 (z) (support: [-0.60,-0.20])
0.0273 Bs(x) (support: [-0.20, 0.20]) + 0.0211
Ba(z) (support: [0.20, 0.60) 0.0248 Br(z) (support: [0.60, 1.00)) 0.4133 Tanh(=).
fa(=) -0.000s (z( (support: [-2)20.-1.80] 0.0072 (zx (support: [-1.80,-1.40J)
0.0040 B2(x) (support: [-1.40,-1.00))
0.0227 (x) (support: [-1.00,-0.60) - 0.0067 (x) (support: [-0.60.-0.20], + 0.0248 Ba(x) (support: [-0.20, 0.20J) + 0.0165
(z) (support: [0.20, 0.60)) 0.0078 (r)x) (support: [0.60, 1.00J) 0.0100 Tanh(z).
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Rethinking Time Encoding via Learnable Transformation Functions
Erquanyy Mageitake Specner (flend)
Frequsccy Maptiede Soctium
Magstade (1.0)
Migittat
Magaiaae
Migrrera
0.33
025
04-
013
Freguency (k)
Froquescy (k)
(a) Dim 0 FFourier-based)
(b) Dim I (Fourier-based)
Figure 15. Frequency Magnitude Spectrum for the first two dimensions of the non non-lincar transformation functions of LcTE trained on Wikipedia/TGN, These dimensions Fourier-based.
functions but also exhibit richer frequency composition, allowing it to simultancously capture both short-term bursts and long-term rhythms.
In addition, this approach could be extended analyze complex patterns. However, our goal here is present the underlying idea, we will not go deeper here.
Spline-based: The Spline-based functions offer complementary advantages, particularly for non-periodicity.
In our spline-hased dimensions, where we applied basis function (Tanh), if the weight of the basis function is higher, may dominate specific simension-succh as Dim 2 in Figure 14. However, there are other dimensions where splines dominate, such as Dim 3. To further clarify this, we combine the specific Wikipedia dataset and explain:
Dim 2: The output increases monotonically with time difference, indicating lime-decayllike effect the longer the time since last edit, the stronger the encoding response. This may suggest the time encoding has learned that re-activation after long inactivity is significant event in this specific case.
The function exhibits sharp peaks and local bumps, indicating that the model assigns particular importance to certain time intervals. These may correspond to known active editing windows or reaction delays. The sharpness some coefficients suggests the model has captured rare but important temporal phenomena, such as one-off campaigns or anomaly spikes.
Dim 3:
The Spline coefficients inherently capture local temporal features, indicating specific time intervals that the model considers critical or active. Sharp peaks coefficients within these curves suggest the occurrence of sudden events or anomalies. This local characteristic is advantagcous for identifying rare phenomena.
H.3. Different Datasets
We reconstructed and plotted the four non-lineat functions for a 4-dimensional LeTE trained on MOOC/TGN (shown in Figure 16). By comparing these results to those from the Wikipedia (Figure 14), it can be seen that the Dim 0 exhibit lack of periodicity. From the reconstructed equations of Dim 0, the gigher-frequency terms do have coefficients with some magnitude, but they are generally small. For instance, the coefficients of cos(52r) and sin() are relatively small (e.g.. -0.0134 and -0.0136), suggesting that their contribution minimal and insufficient to generate significant fluctuations. As result, the overall function primarily exhibits slow oscillations, making the plot appear be predominantly non-periodic within certain input window.
This observation aligns with the findings in Appendix G. and Figure 8, where the spectral entropy statistics also show that the Wikipedia exhibits stronger periodicity compared to the MOOC.
Thus, by comparing the non-linear functions of LeTE across different datasets, we can indirectly explore the periodic or non-periodic nature of the data present.
28

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Rethinking Time Encoding via Learnable Transformation Functions
Dim 0 (Fourier-based)
0.2
0.0
002
Dim (oourier-basedl
Dim (Spline-hased)
0.25
0.00
Dim 3 (Spline-based)
0.0
-0.2
Time
Figure 16. Plots of the four dimensions of the non-linear transformation functions of a 4-dimensional LcTE trained on MOOC/TGN, We further present the four functions here, the parameters are learned and read from the trained model:
fo(x) -0.0186 cas(1 = - 0.0131 sin(1 =/) 0.0173 cos(2 = + 0.0039 sin(2 =/) + 0.0019 cos(3 x/)) +
1.0013-sin(( uh) + 0.0032 0 56)) 0.0096
0.0134 cos(5 =/) 0.0136 sin mb) + 0.3140 cos(1 21) +
0.5998 sin(1 0.2781 cos(2 x1) + 0.4591 sin(2-a() + 1316 coosocos(33 x() + w..2296-sinnf sin---ssnnn3---
(mm3355) -1} 0.0880 cos(1-20
0.0198 sin(4-2) x1) 0.2621 cos(5 x1) 0.0030 1..0000iin55---. =1) + 0.0016,
[1(x) +0.0816 cos(1 =b) 0.0362 sin(1 =6) 0.0588 cos(2 = 0.0046 sin m²) 0.0199 cos(3
<6) 0.008200-200
0.0121 cas( (4- =/) + 0.0325 - sin(4 (4 =6) + 0.0591 c5s(5
0.0609 sin(5 1.5... x6) 0.4787 cos(1
x + 1.1011 1.-1001--in(---- x1) 0.0467--4---0 cosc-coos22200 x1) 0.4438 sin(2 xí) 0.1300
cos(3 0.2972 sin x1) + +..334 1..344-co6s44
cos
<1) + 0.1396 sin(4- 21) + 0.0593 cos(5 21) 0.0841 sin(5- x1) + 0.1166,
here, for both fo(x) and f1(2), m² 0.9857
0.0971 and a -0.0187 0.0885.
f2(x) +0.0020 Bo(x) (support:
(z) (support: [-1.80, -1140))--00997 B2(x) (support: [-1.40, -1.00D-
0.0916 (x) (support: [-1.00, -0.60)) 0.1039 (2x) (support: [-0.60, -0.20]) 0.344 B5(x) (support: [-0.20, 0.20]) 0.3597
Ba(z) (support: [0.20, 0.60)- 0.3093 Br(z) (support: [0.60, 1.00)) 0.2827 Tanh(x).
fa(=)) 00.001 (x(x) (support: [-2.20, -1.0)) 0.0099 (1999-Bi22 (x) (support: [-1.80, -1.40]+ 00.0664 0.03644BB(z)) (support: [-1.40, -1.00)+
0.0712 (r(r) (support: [-1.00, -0.60J) + 0.0169 (x(x) (support: [-0.60, -0,20]) + 0.2372 (s( (support: [-0.20, 0.20J) + 0.2886
(z) (support: [0.20, 0.60)) 0.2155 (x)x (support: [0.60, 1.00]) 0.0947 Tanh(z).
29

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Rethinking Time Encoding via Learnable Transformation Functions
Dim (Fourier-based)
0-
-1
Dim (Fourier-based)
Dim 2 (Spline-busell)
0-
Dim 3 (Spline-based) based)
0.5
0.0
Time
Figure 17. Plots of the four dimensions of the non-linear transformation functions 4-dimensional LcTE trained Wikipedia/TGNi The y-axes have been scaled to the same level as 18 to facilitate direct comparison. The reconstructed functions are the same as in the Figure 14.
H.4. Different Backbones
We provide plots of the same dataset trained with TGN and DyGFormer, shown in Figure 17 and Figure 18, with the y-axes set to the same level for each backbone to facilitate a direct comparison). As the figures demonstrate, despite using different backbone models, the learned functions exhibit similar trends and shapes for each dimension. This illustrates the stability of our method and makes the interpretability process more reliable.
Of course, there may be some detailed differences between LeTEs trained on different backbones. This is intuitively due the presence of various influencing factors, such the model architecture, the interaction of LeTE with other modules, the optimization process and etc. However, we can validate the idea by inspecting the plot in simplified manner.
H.5. Comparing Lower- and Higher-dimennionall LeTE
We further compare the lower- and higher-dimenssionl LeTE by reconstructing the non-linear functions and plotting them (please refer to and compare Figure 14 and 19). Intuitively, the higher-dimensional representation will provide more information. As seen from the plots, Dim 2 in Figure 19 is dominated by the basis function, partially losing the information captured by Dim Figure 14.
From the perspective of the reconstructed functions, the Fourier-based dimensions, LeTE with only one Fourier-based dimension has single input transformation, I/g, and all frequency components are computed based on this transformation. This means the LeTE encodes on broader time scale (reminder: we used the Wikipedia dataset) and models the time difference variations of editing activities without distinguishing patterns different scales. Since there is only Fourier- based dimension, all frequency components are controlled by the same input transformation, making it harder for the model to interpret editing patterns at different time scales. In contrast, for the LeTE with two Fourier- based dimensions, each dimension has different input transformations (3/ and r/1), ) enabling the model to capture more detailed editing behaviors at different scales. For example, Dim 0 might rely more on I/g (with a larger scaling factor), focusing on short-term fluctuations (high- frggueryy components), while Dim might rely more on s/ (with a smaller scaling factor), focusing
30

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Rethinking Time Encoding via Learnable Transformation Functions
Dim (Fourier-Be based)
Dim (Fourier-based)
0-
Dim 2 (Spline-hare (apline-based
0-
Dim (Spline-basedu based)
0.5
0.0
0.55
Figure 18. Plots of the four dimensions of the non-lincar transformation functions of a 4-dimensional LcTE trained Wikipedia/TGN, The y-axes have been scaled to the same level as 17 to facilitate a direct comparison. We further present the four functions here, the parameters learned and read from the trained model:
fo(=) -00727 cas(1 x6) 0.0704 sin(1 x66 0.1529 cos(2 =b) 0.1720 sin(2 2b) 0.1554 cos(3
26) 0.0320 (in(33 (1)) 0.0385 cos (4- (4)))) - 0.0220 - sin(4 m²)) - 0.0086 cos(5 (2)) 0.1107 sin(5 + 0.0758 oos
21) 0.2909 2900--inn1110- x1) 0.2430--2- cos(2 21) + 0.3166 sin(2 x1) + 0.2097 cos(3 0.0334
10034--inn3---.. x1) 0.3308 cos
0.0664 sin(4 - 21) - 0.4329 cos(5 21) - 0.1351 sin(5 zí) - 0.0009, <1)
(x: 0..2232 cos(1 =²) + 0.1531 sin(1 =6) 0.2963 cos(2 =b) 0.0852 sin(2 mb) 0.0409 cos(3
m6) 0.1263 sin(3 x/) 0.0938 cos(4. (4. 0.1772 sin(4 =/)) + 0.1431 cos(5 (5 24)) 0.3435 sin(5 x6) 0.2795 cos(1
<1 1.0935-06660 sin(1-2f) + 0.1297 cos77ccos222200 21) 0.4377 sin(2 0.0711 cos(3 0.3519- sin x1) + 0.4151 cos(
x1) 0.5552.sim(4
sin(4- x + 0.0696 cos(5 x - 0.1653 sin(5 x() - 0.0245.
here, for both fo( and (z) = 0.9936 æ 0.0016 and 2 0.0009 x 0.0678.
fa(=) = +-.0012- Bo (z) (support: -.000 Bi (z) (support: -14.0, 0.0096 (2) (support: -1.000
0.0071 (2(x) (support: [-1.00, -0.60)) 0.0197 Bxxx)) (support: [-0.60, -0.20]) 0.0075 (2x) (support: [-0.20, 0.20]) 0.0045
Bz(z) (support: [0.20, 0.60)) + 0.0038 (z) (support: [0.60, 1.00)) + 0.4794 Tanh(z).
[s(x) = -0.0008 (x) (support: [-2.20, -1.80] -000))-00007--111 By (x:) (support: [-1.80, -1.40] +0.0))++000000 (x))26-B((() (support: [-1.40, -1.00))
0.0294 (1(x) (support: [-1.00, -0.60J) 0.0009 (a(x) (support: [-0.60, -0.20J) + 0.0179 (x) (support: [-0.20, 0.20J) + 0.0037
Ba(z) (support: [0.20, 0.60)) 0.0031 Br(z) (support: [0.60, 1.00)) 0.0395 Tanh(=).
31

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Rethinking Time Encoding via Learnable Transformation Functions
Dim (Fourier-based)
2
0-
Dim (Spline-based)
0.2
-0.2
Time
Figure 19. Plots of the two dimensions of the non-linear transformation functions of 2-dimensional LcTE trained on Wikipedia/TGN. We further present the four functions here, the parameters are learned and read from the trained model:
fo(x) = -0.9560- cos(1 (x²)) + 0.3821 sin(1 (x")) + 0.2430 cos(2 (z')) 0.3138 sin(2 (x')) 0.4807 cos(3
(z')) 0.1918 sin(3 (z')) 0.6372 cos(1 (z')) 0.2811 sin(1 (x') + 0.1555 c5 (z/) + 0.0840 1..0000sii55--5 (a/)) + 0.0334,
here, x' 0.9963 0.0092.
fi (z) 0.0020 B=()) (support: [-2.20,-1.80)) 0.0022 Bi (z) (support: [-1.80.-1.40D + 0.00970000000 (2009)-2((r) (support: [-1.40,-1.00)]
0.0636 (xx (support: (-1.00,-0.60)) - 0.0556 (x) (support: (-0.60.-0.20)) 0.0123 Ba(x) (support: [-0.20, 0.20)) - 0.0054
(a( (support: [0.20, 0.60)) 0.0011 (x) (support: [0.60, 1.00)) 0.1945 Tanh(x).
more on long-term trends (low-frequency components). Thus, higher dimensions allow the model to handle editing behaviors at different time scales, providing higher interpretability.
Similarly, for the LeTE with only one Spline-based dimension, primarily focuses on adjusting a single level, potentially describing how time affects editing behaviors. However, relying on just one Spline-based dimension may make difficult capture relatively complex time dynamics. For the LeTE with Spline-based dimensions, the weights of the coefficients more distributed, granting the overall LeTE stronger local adjustment capabilities. Moreover, since dimension may be dominated by basis function or Spline functions, higher dimensions naturally have stronger expressive power.
Although higher-dimensional LeTEEs offer stronger performance and better explain the information captured by the model, the interpretability analysis of such higher-dimensional LeTEs becomes more complex and may require a dimension-by dimension analysis.
Limitation and Future Work
In this paper, we introduce LeTE, general time encoding method. Generally, Combined LeTE offers better performance as it leverages the strengths of both Fourier-based LeTE and Spline-based LeTE,, enabling it to effectively capture both the periodicity and non-periodicity of time. However, in practical scenarios, the choice of the hyperparameter or among the three variants may depend on the characteristics the data and the specific task requirements.
We also explore the impact of the time encoding dimension on downstream task performance. Similarly, selecting an appropriate time encoding dimension may vary depending the data and tasks. Notably, observe that even with small dimension, LeTE can achieve acceptable results in downstream tasks.
Additionally, we mention that certain position encoding methods can be considered special cases of our approach. However, as position encoding not the primary focus of this paper, we did not provide formal proofs. We believe that extending the ideas proposed in this paper to models that use position encoding could yield improved results, making this promising direction for future research.
J. Code Implementation
The codes are available at GitHub Repository.
32


Reviews:
[{"review_id": "u4EzHjKG0o", "reviewer": "Reviewer_1gph", "summary": "This paper proposes Learnable Transformation-based Generalized Time Encoding, a new approach to encoding time in machine learning tasks. LeTE generalizes popular functional time encoding strategies and makes the non-linear transformation fully learnable. The authors use techniques from Fourier expansions and spline functions to parameterize these transformations and this flexibility allows for better modeling of periodic, non-periodic, and complex mixed temporal patterns compared to methods that rely on a fixed function. They demonstrate LeTE’s effectiveness on a variety of tasks: event-based image classification, time-series forecasting, dynamic graph link prediction, and a real-world financial risk control application.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback and suggestions. We are happy to further discuss the interpretability of our method real data scenarios. We would also like to clarify that demonstrating interpretability for very high-dim encodings is challenging. We choose to use a 4-dim Combined LeTE to present our analysis. The training process and settings are consistent with those used in the main experiments of our paper. We will demonstrate the interpretability of our model from the following perspectives:\n\n1.\tReconstructing the learned non-linear transformation functions and plotting them to provide an intuitive analysis.\n\n2.\tAnalyzing each dim to interpret what information it represents. \n\n3.\tComparing different datasets under the same backbone.\n\n4.\tComparing different backbones' LeTE under the same dataset.\n\n5.\tComparing the low- vs. high-dim LeTE to assess the impact of dimensionality on interpretability.\n\n**Reconstructing**\n\n As discussed in the paper, the previous TEs exhibit a degree of interpretability by using fixed sine functions, which reflect periodic patterns. However, this strong inductive bias also limits their expressiveness and generalization to complex patterns or non-periodicity. \nIn contrast, the LeTE is fully learnable, and the learnable non-linear functions can still be reconstructed and visualized from learned parameters, allowing for interpretability through function inspection.\n\nWe demonstrate this interpretability using a 4-dim Combined LeTE, trained on the Wikipedia/TGN. Fig. 1 (see [anony. repo.](https://anonymous.4open.science/r/LeTE_ICML25_Rebuttal)) shows the learned functions for each dim. The first 2 dims are Fourier-based, and the last 2 are Spline-based.\n\n**Analysing each dim**\n\n**Fourier-based:**\nThe Fourier coefficients explicitly encode frequency components, offering an intuitive view of the captured patterns. Compared to fixed sine functions, our learnable Fourier-based TE captures periodic patterns with finer granularity and greater flexibility, enabling the representation of both periodicity and subtle non-periodicity within specific ranges.\n\nFor a single dim, low-frequency components capture long-term trends, while high-frequency components focus on short-term fluctuations. As an example: Wikipedia dataset, which records editing activities, where nodes represent users or pages, and edges with timestamps capture editing events (frequency magnitude spectrums are shown in Fig. 2, note the inputs are time differences in this case):\n\nDim 0 shows a strong high-frequency response. The learned coefficients include\n$\\cos(3x'): 0.29, \\cos(4x'): -0.16, \\cos(5x'): -0.42$. This suggests that dim 0 is sensitive to short-term repetitive edits, i.e., high-frequency editing behavior. \n\nDim 1 captures low- to mid-frequency patterns, with large coefficients: $\\sin(1x'): 0.96, \\sin(4x'): 0.61, \\cos(4x'): 0.29$. These reflect longer-term periodic behaviors. For example, frequency-1 may correspond to daily or weekly editing cycles, while frequency-4 may capture sub-daily repeated interactions. This dim may reflect user habits or regular community editing patterns.\n\nThus, Fourier-based dims not only retain the periodic interpretability of sine functions but also exhibit richer frequency composition, allowing it to simultaneously capture both short-term bursts and long-term rhythms. Moreover, this approach could be extended to analyze more complex patterns. As our goal here is to present the underlying idea, we will not go deeper here.\n\n**Spline-based:**\nThe Spline functions offer complementary advantages, particularly for non-periodicity. In Spline-based dims, where we applied a basis function (Tanh), if the weight of the it is higher, it may dominate a specific dim—such as dim 2 in Fig. 1. However, there are other dims where Splines dominate, e.g., dim 3. We combine the specific case on Wikipedia and explain:\n\nDim 2: The output increases monotonically with time difference, indicating a time-decay-like effect — the longer the time since last edit, the stronger the encoding response. This may suggest the TE has learned that re-activation after long inactivity is a significant event in this specific case.\n\nDim 3: The function exhibits sharp peaks and local bumps, indicating that the TE assigns particular importance to certain time intervals. These may correspond to known active editing windows or reaction delays. The sharpness of some coefficients suggests the TE has captured rare but important temporal phenomena, such as one-off campaigns or anomaly spikes.\n\nThe Spline inherently capture local time features, indicating specific time intervals that the model considers critical. Sharp peaks coefficients within the curves suggest the occurrence of sudden events or anomalies. This local characteristic is advantageous for identifying rare phenomena. \n\n**Due to space limitations and similar concerns raised by Reviewer ozTs. Please kindly check the remaining parts in our response to Reviewer ozTs.**", "claims_and_evidence": "The claims are supported by the experiments on various tasks: event-based image classification, time-series forecasting, dynamic graph link prediction, and a real-world financial risk control application.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experiment designs make sounds.", "relation_to_broader_scientific_literature": "Time encoding is very important for time-series analysis, sequence modeling, and graph representation learning.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths\n1. Clear motivation and novelty: The paper addresses a known limitation in existing time-encoding methods: most rely on fixed periodic assumptions (e.g., sine or cosine), making them less effective for mixed or non-periodic time dynamics.\n2. Comprehensive experimental evaluation The experiments span multiple domains:\n\nWeaknesses / Concerns\n\nInterpretability dicussion could be expanded: detailed demonstrations of how domain experts might interpret the learned curves (especially in high-dimensional time encodings) could strengthen the real-world applicability argument. More ablations or visualizations of learned functions in real data scenarios would further highlight interpretability.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Is the learned time encoding functions in real data scenarios interpretable?", "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": "HC6GwluGXF", "reviewer": "Reviewer_ozTs", "summary": "This paper proposes LeTE (Learnable Transformation-based Generalized Time Encoding), a flexible and learnable time encoding framework that generalizes existing methods (e.g., Time2Vec, Functional Time Encoding). By parameterizing nonlinear transformations via Fourier series and B-spline functions, LeTE provides a more expressive representation of temporal information, capable of modeling periodic, non-periodic, and mixed time patterns. Extensive experiments on time series forecasting, dynamic graph representation learning, and real-world applications demonstrate its superior performance and generalizability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback and suggestions. As Reviewer 1gph also raised similar concerns, and due to space limitations, **we have addressed some of these points in our response to Reviewer 1gph. Could you kindly review the first part of our reply there? Below is the continuation of our response following the reply to Reviewer 1gph.**\n\n**Different Datasets**\nWe reconstruct and plot the non-linear functions for a 4-dim LeTE trained on MOOC/TGN (shown in Fig. 3, in [anony. repo.](https://anonymous.4open.science/r/LeTE_ICML25_Rebuttal/Figure%201-6.pdf)). By comparing these results to those from the Wikipedia (Fig. 1), it can be seen that the dim 0 exhibit a lack of periodicity. From the reconstructed equations of dim 0, the higher-frequency terms are generally small. For instance, the coefficients of $\\cos(5x')$ and $\\sin(5x')$ are relatively small (e.g., -0.0134 and -0.0136), suggesting that their contribution is minimal and insufficient to generate significant fluctuations. As a result, the overall function primarily exhibits slow oscillations, making the plot appear to be mainly non-periodic within a certain input window.\n\nThis observation aligns with the findings in our original paper  (Appendix G.1 and Fig. 8 (this refers to figure in the original paper), where the spectral entropy statistics also show that the Wikipedia exhibits stronger periodicity compared to MOOC.\nThus, by comparing the plots of LeTE across different data, we can indirectly explore the periodic or non-periodic nature of the data present.\n\n**Different Backbones**\n\nWe provide plots of the same dataset trained with TGN and DyGFormer, shown in Fig. 4 and Fig. 5, with the y-axes set to the same level for each backbone to facilitate a direct comparison. As the figures show, despite using different backbone models, the learned functions exhibit similar trends and shapes for each dimension. This illustrates the stability of our TE and makes the interpretability process more reliable.\n\nOf course, there may be some detailed differences between LeTEs trained on different models. This is intuitively due to the presence of various influencing factors, such as the model architecture, the interaction of TE with other modules, the optimization process and etc. However, we can validate the idea by inspecting the plot in a simplified manner.\n\n\n**Comparing lower- and higher-dim LeTE:**\n\nWe further compare the lower- and higher-dim LeTE by reconstructing the non-linear functions (please refer to and compare Fig. 1 and 6 in the repo.). Intuitively, the higher-dim representation will provide more information. As seen from the plots, dim 2 in Fig. 6 is dominated by the basis function, partially losing the information captured by dim 3 in Fig. 1.\n\nFrom the perspective of the reconstructed functions, for the Fourier-based dims, the LeTE with only 1 Fourier-based dim has a single input transformation, $x'$, and all frequency components are computed based on this transformation. This means the LeTE encodes on a broader time scale (reminder: we use Wikipedia here) and models the time difference variations of editing activities without distinguishing patterns at different scales. Since there is only 1 dim, it is harder for the TE to interpret editing patterns at different time scales. In contrast, for the LeTE with 2 Fourier-based dims, each dim has different input transformations ($x'_0$ and $x'_1$), enabling the model to capture more detailed editing behaviors at different scales. For example, dim 0 may rely more on $ x'_0$ (with a larger scaling factor), focusing on short-term fluctuations (high-frequency), while dim 1 may rely more on $x'_1$ (with a smaller scaling factor), focusing more on long-term trends (low-frequency). Thus, higher dims allow the model to handle behaviors at different time scales, providing higher interpretability.\n\nSimilarly, for the LeTE with 1 Spline-based dim, it primarily focuses on adjusting a single level, potentially describing how time affects behaviors. However, relying on just 1 dim makes it difficult to capture more complex time dynamics. For the LeTE with 2 Spline-based dims, the weights of the coefficients are more distributed, granting the overall LeTE stronger local adjustment capabilities. Moreover, since a dim may be dominated by basis func or Spline funcs, higher dims naturally have stronger expressive power.\n\nAlthough higher-dim LeTEs offer stronger performance and better explain the information captured by the model, the interpretability analysis of higher-dim LeTEs becomes more complex and may require a dim-by-dim analysis.\n\n**Summary**\n\nWe thank the reviewer for the suggestions on the interpretability of our method. We will consider adding this discussion to the appendix. Additionally, we have prepared code to process the learned LeTE parameters, reconstruct key non-linear transformations, and visualize them. This code will be updated in our publicly available repo after the review process is complete.", "claims_and_evidence": "Most of the key claims in the submission are well supported by empirical evidence and theoretical analysis.\n\nSupported Claims:\n\n1. LeTE is a generalization of existing time encoding methods (e.g., Time2Vec, FTE): This claim is backed by formal derivations and proofs (e.g., Proposition 3.1), showing how specific parameter settings in LeTE reduce to previous methods. The argument is mathematically sound and clearly presented.\n\n2. LeTE is capable of capturing a wider range of time patterns (periodic, non-periodic, mixed) The authors support this through both the construction of learnable nonlinear transformations (Fourier and spline-based), and comprehensive empirical evaluations across tasks that exhibit different temporal dynamics. The wide range of tasks (forecasting, dynamic graphs, financial modeling) provides convincing evidence that LeTE handles complex patterns beyond those captured by fixed-function encodings.\n\n3. LeTE achieves better performance with fewer dimensions (higher dimensional efficiency): This is substantiated via ablation experiments (Section 4.5, Figure 5–7), showing that LeTE with 2/8/16 dimensions can outperform traditional FTE with 100 dimensions—a strong empirical support for the efficiency claim.\n\n4. LeTE is invariant to time rescaling: This is theoretically demonstrated (Proposition 3.2), similar to prior work on Time2Vec and FTE. The formulation and proof are clear and align with expectations for such encoding schemes.\n\nQuestionable Claims:\n\n1. LeTE offers enhanced interpretability: While the authors argue that the learned functions are interpretable (due to their basis in Fourier/spline components), interpretability is only briefly mentioned and weakly demonstrated via some visualization (Appendix G.3). The paper could be strengthened by providing concrete examples of how the learned functions reflect real-world temporal patterns, perhaps via visualization or case studies.", "methods_and_evaluation_criteria": "Yes, both the proposed method and the evaluation protocol are appropriate and well-aligned with the problem of time representation learning in machine learning models.", "theoretical_claims": "The theoretic claims are actually extremely trivial. No need to verify the correctness.", "experimental_designs_or_analyses": "Yes, I have reviewed the experimental design and find it generally sound and reasonable. The authors evaluate their proposed time encoding method on a wide set of tasks that represent standard and widely accepted benchmarks in the field, including time series forecasting, dynamic graph learning, and a real-world classification application.", "relation_to_broader_scientific_literature": "The paper’s contributions build meaningfully on a well-established line of research in time encoding for temporal machine learning tasks, such as time series analysis and temporal graph.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "HxtuOwKWbe", "reviewer": "Reviewer_gPzN", "summary": "The paper proposes a time encoding method that can work as a plug-and-play functionality to capture diverse patterns in the real world. The method is motivated by the observation that the existing time encoding approaches struggle to capture non-periodic and mixed patterns. To capture such complex patterns, the paper proposes to transform timestamps to representations via the combination of Fourier series and Spline functions. By their inherent inductive biases, the Fourier series is more capable of capturing periodic patterns whereas the Spline functions more excel in modeling non-periodic patterns. The paper assesses the effectiveness of the proposed method by replacing the time encoding modules with the proposed one in various time series tasks. The experimental results confirm the superiority of the proposed model.\n\n## update after rebuttal\n\nThe authors have partially addressed my concerns, and I am now leaning toward acceptance. I suggest incorporating the newly conducted experiments and the comparison with prior work (TIDER) in the revised version to better highlight the novelty. Additionally, the current experiments only focus on methods modeling temporal patterns, assessing LeTE’s efficacy in time series forecasting would be more comprehensive by also comparing it with approaches designed to capture spatial correlations, such as iTransformer [2] and Sumba [3].\n\n[2] iTransformer: Inverted transformers are effective for time series forecasting, ICLR 2024.\n\n[3] Structured matrix basis for multivariate time series forecasting with interpretable dynamics, NeurIPS 2024.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Sensitivity to the hyperparameter**\n\nActually, the performance gain can be influenced by the hyperparameter $p$. As we analyze in the experiments on dynamic graphs (please refer to Appendix G.2, “Comparative Analysis of Different Variants of LeTE,” and Table 9 for details), the performance remains robust across different values of $p$. Specifically, as shown in Table 9, regardless of whether $p$ is set to 0, 0.5, or 1, our method almost consistently outperforms the baselines. However, the downstream results do affected by the $p$.\n\nFor the time series forecasting experiments, we actually also choose $p$ only from the set {0, 0.5, 1}, so there is no concern regarding careful tuning of this hyperparameter in practice. We have also uploaded a result table where $p$ is set to 0.5 (Tab. 1 in the [new anony. repo., especially for the rebuttal](https://anonymous.4open.science/r/LeTE_ICML25_Rebuttal/Table%201%20and%205.pdf)). As seen in the newly uploaded tables, our method still achieves high win rates for both MSE and MAE. Intuitively, slight tuning of $p$ within the set {0, 0.5, 1} can lead to even better results.\n\n**Code repository**\n\nWe have updated the [original anony. repo.](https://anonymous.4open.science/r/LeTE), we hope it works now. We also newly added the running logs and the requirements file to the repository. If you still facing the \"The requested file is not found\" problem, you may try to download the repository and check it. Alternatively, you may also download our code from the openreview Supplementary Material, we also uploaded a copy of the same code when we submitted our manuscript.\n\n**Comparison with [1]**\n\nThank you for pointing this out. We acknowledge that [1] (TIDER) also incorporates Fourier series to model temporal patterns in multivariate time-series data. However, our work differs from TIDER in terms of motivation, model design, and application scope.\n\n**Motivation:** TIDER employs Fourier series to model the seasonal component of time-series data as part of a decomposed temporal structure (trend + seasonality + local bias) within a matrix factorization framework, specifically for the task of missing value imputation. \n\nIn contrast, LeTE is a task-agnostic, general-purpose time encoding module, designed to replace previous temporal encodings with fully learnable transformations. Our goal is to provide a unified and flexible time encoding mechanism that can be applied across different temporal modeling scenarios.\n\n**Model Design:** TIDER is specialized for a specific task and evaluated exclusively on imputation. Its architecture is tightly coupled with that objective.\nIn TIDER, Fourier series are used internally to model a latent factor $V_s$, and only for capturing periodicity, within a low-rank matrix factorization design. \n\nIn LeTE, Fourier-based functions are used to encode timestamps into embeddings, which are then used across multiple downstream tasks. The encoding serves as a plug-and-play module and is jointly trained with downstream objectives.\n\n**Application scope:**  Our experiments cover time-series, dynamic graph, event-based tasks, and real-world applications, demonstrating that LeTE is not only expressive but also highly adaptable across domains.\n\n**Moreover,** LeTE introduces a unified framework for time encoding via deep function learning, encompassing not only Fourier-based functions but also spline-based functions, and even hybrid combinations (Combined LeTE). \n\n**In summary,** while both works utilize Fourier series, TIDER focuses on a specific modeling component for a single task, whereas LeTE introduces a general-purpose, extensible, and theoretically grounded time encoding framework that supports a wide range of tasks in temporal modeling.\n\n**Holidays**\n\nFrom the perspective of a long time window, holidays can be regarded as a type of periodic pattern. As we demonstrate that our method can effectively capturing periodic patterns, the patterns caused by holidays can also be captured. Please also refer to Appendix G.4: Capturing Periodic, Non-Periodic, and Mixed Patterns in Data, and Fig. 11 and 12 for more details.\n\nAs a brief recap, we demonstrate that our method enables models to capture periodic, non-periodic, and mixed time patterns. Specifically, the holiday-related periodicity can be simulated by the low-frequency signal shown in Fig. 11 (synthetic periodic data).\n\nWe kindly invite you to review our responses to **Reviewer 1gph** and **Reviewer ozTs**, where we illustrate the interpretability of our proposed method and explain how it effectively captures different temporal patterns.", "claims_and_evidence": "Most of the claims are supported by experiments. In the introduction, the paper argues that the existing time encoding methods cannot capture the complex patterns caused by holidays, but there is no evidence to verify whether the proposed method can capture such patterns. It is better to provide experiments to show this claim.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria seem reasonable to me.", "theoretical_claims": "The proofs seem correct.", "experimental_designs_or_analyses": "In the method part, the authors suggest setting the hyperparameter $p$ to 0.5, but in the experiments, they adjust it across different datasets and methods. The reviewer is curious whether the efficacy of the proposed method is sensitive to this hyperparameter. Because it will significantly limit its applicability if the performance largely hinges on a careful tuning of this hyperparameter in practice. The paper should present the results in Table 1 by setting $p$ to its default value of $0.5$.", "relation_to_broader_scientific_literature": "The proposed method can work as a plug-and-play module for various time series modeling approaches.", "essential_references_not_discussed": "The idea of adopting Fourier series to learn representations for time series has been explored in [1]. The paper should discuss its connection and distinction from the prior work to clarify its novelty. \n\n[1] Multivariate Time-series Imputation with Disentangled Temporal Representations, ICLR 2023.", "other_strengths_and_weaknesses": "Strengths:\n\n* The proposed method is invariant to time rescaling.\n* The paper is well-written and easy to follow.\n\nWeaknesses:\n\n* It is not clear if the performance gain is sensitive to the hyperparameter $p$.\n* The provided code repository is invalid, and this raises a concern regarding reproducibility.\n* The novelty of the proposed method should be further clarified by comparing it with prior work [1]. \n\n[1] Multivariate Time-series Imputation with Disentangled Temporal Representations, ICLR 2023.", "other_comments_or_suggestions": "* The paper should present the results in Table 1 by setting $p$ to its default value of $0.5$.\n* It is better to provide the experiments to support the claim of capturing the complex patterns caused by holidays.\n* Please update the code repository to eliminate the concern regarding reproducibility.", "questions_for_authors": "Please refer to the questions listed above regarding hyperparameter sensitivity, reproducibility, and novelty clarification.", "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": "trjjE2WejZ", "reviewer": "Reviewer_fQEG", "summary": "This paper proposes a learnable time representation framework—referred to as LeTE—that aims to improve upon prior time encoding methods which rely on fixed or narrow inductive biases (e.g., purely sinusoidal functions). The authors introduce two learnable approaches for modeling time: one based on Fourier series expansions and another on B-splines. Additionally, they propose a combined version that leverages both. By making the transformation functions learnable, the approach can, in principle, encompass various existing time encodings (like Time2Vec) as special cases. The authors further claim invariance to time rescaling and better interpretability over prior methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Rescaling**\n\nThank you for the insightful comment. We agree that empirically demonstrating rescaling invariance is important.\n\nTo directly support this, we conducted an additional tiny experiment by applying the Combined LeTE to Wikipedia/TGN, using two different time input scales: t=t/60 (interpreted as minutes), and t=t/3600 (interpreted as hours),\nwhereas previous experiments used Unix timestamps (in seconds).\n\nAs shown in the results of this experiment (please see the [anony. repo. Rescaling](https://anonymous.4open.science/r/LeTE_ICML25_Rebuttal/Rescaling.pdf)), both versions achieve similar performance and outperform the baselines. Minor differences may be attributed to other factors in the training environment.\n\nWhile we provide proofs of rescaling invariance in Appendix C.3, we would also like to clarify that our existing experimental settings indirectly verify this property:\n\nIn Sec. 4.2 (Time-series), we use absolute Unix timestamps, which are large-scale values.\n\nIn Sec. 4.3 (Dynamic Graph), we use relative time differences, typically much smaller in scale.\n\nDespite the difference in time magnitude across these settings, LeTE consistently outperforms the baselines, illustrating strong robustness to changes in time scale. This further supports the claim that LeTE is inherently invariant to time rescaling, thanks to its learnable transformations.\n\n**Interpretability**\n\nPlease refer to our responses to **Reviewer 1gph** and **Reviewer ozTs.** We made a detailed analysis of the learned curves and demonstrate how these curves can be interpreted in practice. \n\n**How TEs Are Used**\n\nIn practice, the learned LeTE are used in one of two ways: Added to feature embeddings (in time series models): $\\mathbf{x} = \\text{TokenEncode}(x) + \\text{LeTE}(t)$; or concated with node and edge features (in dynamic graphs): $\\mathbf{x} = [\\text{Node Features} | \\text{Edge Features} | \\text{LeTE}(t)]$.\n\nIn this way, LeTE provides temporal signals to the model, enabling it to modulate attention weights or node interactions based on temporal context. \nBy contrast, prior time encoding methods either: use hand-crafted encodings, or use fixed sine functions, limiting their ability to represent complex time patterns (periodicity, non-periodicity, and mixed).\n\nMoreover, LeTE leverages learnable non-linear transformations, including both Fourier-based and spline-based, allowing it to learn time patterns directly from data in a flexible, data-driven manner and capture a richer time patterns. As shown in our experiments (Sec. 4), replacing prior TEs with LeTE consistently improves performance across a diverse set of downstream tasks, demonstrating that the learned embeddings are not only more expressive, but also more generalizable.\n\n**Scaling weight**\n\nThe reason we added the learnable scaling weight $s_i$ after LayerNorm can be concluded as follows.\n\nActually, we first consider to add this learnable scaling weight is due to we conducted some experiments and compare that the performances of adding or without these scaling weights, we found that the experiments with the scaling leads to better performance.\n\nUpon further reflection, we believe the performance gain can be attributed to the following reasons:\n\nWe apply LayerNorm to each dimension of the transformed signal to stabilize optimization and ensure comparable scales across dims. However, this normalization step removes the original scale information that might have been encoded by the learned function coefficients. The scaling factor $s_i$ reintroduces flexible amplitude control after normalization.\n\nIn practice, adding the scaling leads to slightly better performance, as it allows each dim to adjust its impact during learning. Without this factor, some dims may become under- or over-represented.\n\n**Relationship with KAN**\n\nYou are correct that the use of B-splines in LeTE is conceptually connected to the ideas from the KAT.\n\nThe motivation behind LeTE arises from the limitation observed in existing time encoding methods: they typically employ fixed non-linear functions, which constrain their ability to model diverse time patterns. To overcome this, we adopt a deep function learning perspective, introducing the learnable transformations — either through Fourier series or B-spline. This enables LeTE to flexibly encode complex mixed time patterns.\n\nWhile this design philosophy aligns with the spirit of KANs, there are some differences:\n\n1. LeTE is designed specifically to address limitations in time encoding, serving as a lightweight, plug-in module for downstream tasks. In contrast, KANs are proposed as general network architectures.\n\n2. LeTE includes not only spline-based functions but also introduces Fourier-based one, particularly suited for modeling periodic patterns. However, KAN primarily relies on splines.\n\n3.  KANs are typically layered architectures, while LeTE acts as a plug-and-play TE that maps time to vector embeddings, which are then fed into larger models.", "claims_and_evidence": "**Time Rescaling Invariance**\nThe authors assert that the proposed method is invariant to time rescaling (e.g., changes in units from days to hours). While they provide a theoretical argument, the empirical validation of this property is less explicit in their experiments.\n\n**Enhanced Interpretability**\nThe paper claims better interpretability over previous methods by allowing direct reconstruction of the learned transformation functions. However, the evidence provided (largely visualizations in the appendix) may not conclusively show that these representations are more interpretable than, for example, a straightforward sinusoidal basis. Additional qualitative or quantitative assessments would strengthen this claim.", "methods_and_evaluation_criteria": "Yes. The paper’s chosen methods and evaluation criteria appear well-matched to the goal of improving time encodings and testing them in realistic contexts.", "theoretical_claims": "I checked the correctness of their equations in the main text.", "experimental_designs_or_analyses": "Overall, the experimental designs are broadly sound, using appropriate metrics and mainstream baselines for time-series and dynamic graph tasks.", "relation_to_broader_scientific_literature": "Overall, the paper’s key contributions fit naturally into—and extend—existing streams of research on time-encoding and functional approximation within neural models, offering a flexible drop-in alternative that maintains or improves upon the advantages of earlier fixed-basis (sinusoidal) approaches.", "essential_references_not_discussed": "Not to my knowledge.", "other_strengths_and_weaknesses": "- Strength\n\n**Comprehensive Literature Review**\nThe authors provide a clear, concise overview of earlier time encoding methods, illustrating how LeTE builds upon and generalizes them.\n\n**Well-Organized Composition**\nThe paper’s structure—with clear figures, tables, and references—helps the reader grasp the proposed approach and its variants (Fourier-based, B-spline-based, combined).\n\n**Generalization to Prior Methods**\nBy demonstrating that prior time-encoding approaches (like Time2Vec) are particular cases of LeTE, the authors show strong potential for “plug-and-play” deployment. This could be attractive for practitioners seeking simple drop-in enhancements.\n\n- Weakness\n\n**Overstated Claims**\nWhile the paper provides theoretical proofs for invariance and interpretability, the experimental demonstrations of these claims are not as strong. For instance, readers would benefit from direct empirical evidence or metrics that validate rescaling invariance.\n\n**Clarity on How Time Embeddings Are Used**\nIt would help to explain more concretely how these learned embeddings tie into the final predictions, and clarify why this learnable embedding is superior to the previous ones.\n\n**Interpretability Remains Nuanced**\nAlthough LeTE can reconstruct learned transformations, “interpretability” is not necessarily obvious. Additional evidence (beyond raw visualizations) would make the case more convincing.", "other_comments_or_suggestions": "Please see weakness.", "questions_for_authors": "**Nature of the Learned Curve**\nFrom Figure 2, it looks like the time embedding is effectively learning some function of t through various basis functions. Beyond the direct representation of time, do you see these learned curves capturing other hidden phenomena (e.g., seasonalities, abrupt events)? How should readers interpret or use these curves in practice?\n\n**Role of the Scaling Factor [s]**\nYou introduce a learnable scaling factor after the layer normalization in each dimension. Given that you already learn coefficients within the B-spline or Fourier expansions, can you clarify why this additional scaling is necessary?\n\n**Connection to Kolmogorov–Arnold Theorem**\nThe use of B-splines and references to function superposition remind me of current lines of work involving the Kolmogorov–Arnold Theorem (KAN). Could you elaborate on whether LeTE is conceptually related to KAN-based approaches?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes LeTE, a new time encoding method for ML models. This method parameterizes nonlinear transformations via Fourier series and B-spline functions, providing a more expressive representation of temporal information, and thus is capable of modeling complex temporal patterns. The method is shown to be effective in experiments on time series forecasting, dynamic graph representation learning, and other real-world applications. 

Reviewers give generally positive feedback to this paper, with concerns being resolved by the rebuttal.

I recommend to accept this paper, but strongly encourage the authors to include the key points of their rebuttal (e.g., new results, arguments about novelty) in the camera-ready.
Decision: Accept (poster)