Optimal Transport for Time Series Imputation
Authors: Hao Wang
zhengnan li
Haoxuan Li
Xu Chen
Mingming Gong
BinChen
Zhichao Chen
Venue: ICLR 2025 Poster (2025)
Keywords: Time series
Imputation
Primary area: learning on time series and dynamical systems

Abstract: Missing data imputation through distribution alignment has demonstrated advantages for non-temporal datasets but exhibits suboptimal performance in time-series applications. The primary obstacle is crafting a discrepancy measure that simultaneously (1) captures temporal patterns—accounting for periodicity and temporal dependencies inherent in time-series—and (2) accommodates non-stationarity, ensuring robustness amidst multiple coexisting temporal patterns. In response to these challenges, we introduce the Proximal Spectrum Wasserstein (PSW) discrepancy, a novel discrepancy tailored for comparing two \textit{sets} of time-series based on optimal transport. It incorporates a pairwise spectral distance to encapsulate temporal patterns, and a selective matching regularization to accommodate non-stationarity. Subsequently, we develop the PSW for Imputation (PSW-I) framework, which iteratively refines imputation results by minimizing the PSW discrepancy. Extensive experiments demonstrate that PSW-I effectively accommodates temporal patterns and non-stationarity, outperforming prevailing time-series imputation methods. Code is available at https://github.com/FMLYD/PSW-I.

Published as a conference paper at ICLR 2025
OPTIMAL TRANSPORT FOR TIME-SERIES IMPUTATION
Hao Wang Zhengnan Li²
Haoxuan Li Xu Chen'
Mingming Gong
Bin Chen
Zhichao Chen
1 ZZhejiang University *Communication University of China
SPeking University
'Remmin University of China 'The University of Melbourne
Mohaaed bin Zayed University of Artificial Intelligence University of Chinese Academy of Sciences
Ho Ho-waze No-ward@outt/ look. com fmlyd@ cuc. edu en
justuschencaeézjuu edu, cn
ABSTRACT
Missing data imputation through distribution alignment has demonstrated advan- tages for non-temporal datasets but exhibits suboptimal performance in time-series applications. The primary obstacle is crafting a discrepancy measure that simulta- neously (1) captures temporal patterns- accounting for periodicity and temporal dependencies inherent in time-series and (2) accommodates -on-stationartt, en- suring robustness amidst multiple coexisting temporal patterns. In response to these challenges, we introduce the Proximal Spectrum Wasserstein (PSW) discrepancy, a novel discrepancy tailored for comparing two sets of time-series based on optimal transport. It incorporates pairwise spectral distance to encapsulate temporal patterns, and a selective matching regularization to accommodate non-stationarity. Subsequently, develop the PSW for Imputation (PSW-I) framework, which iter- atively refines imputation results by minimizing the PSW discrepancy. Extensive experiments demonstrate that PSW-I effectively accommodates temporal patterns and non-stationarity, outperforming prevailing time-series imputation methods. Code is available at https: / github............ com/EMLYD/PSK.
INTRODUCTION
The incompleteness of time-series data is widespread issue in many fundamental fields. For example, in healthcare (Prosperi et al., 2020), users utilize monitoring devices to track their health data, but these records may be incomplete due occasional device failures disconnections. Similarly, in manufacturing and process engineering (Wang et al., 2024a), automation equipment often includes sensors that monitor operational statistics for safety and efficiency, but these records can also be affected by human errors or sensor malfunctions. Such incompleteness damages data integrity, which critical for accurate and reliable analytics (Qiu et al., 2024; 2025; Liu et al., 2024), highlighting the importance of effective Time-Sere Seee-Series Imputation (TSI) techniques.
Deep learning methods have attracted substantial attention TSI due to their ability to model nonlinearities and temporal dependencies. These methods can be classified into two categories: predictive methods, which estimate deterministic values for the missing entries within the time- series (Cao et al., 2018; Du et al., 2023), and generative methods, which generate missing values conditionally given observed ones (Luo et al., 2018). Despite their effectiveness, they are primarily challenged by model selection amidst incomplete data (Jarrett et al., 2022), and require masking some observed entries during training to generate labels (Chen et al., 2024), which can limit performance especially given high missing ratios.
To counteract the defects with deep imputation methods. alignment baannent-bbsedd methods have emerged as alternative. These methods eliminate the need for masking observed entries and training parametric models on incomplete data, offering advantages in sample efficiency and implementation simplicity. While alignment-based methods have proven be effective for imputing missing data non-temporal datasets (Muzellec et al., 2020; Zhao et al., 2023; Wang et al., 2024a;b), their application to TSI is challenging and remains largely unexplored. Our experiments (see Section 4.2) indicate that directly
"Corresponding author.

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Published as a conference paper at ICLR 2025
applying existing alignmenn-basee methods to temporal data yields poor performance. Therefore, how to adapt distribution aligmment methods to TSI remains a challenging and open problem.
In distribution alignment, the choice of discrepancy measure matters, which should accommodate the dataset characteristics (Courty et al., 2017; Wang et al., 2023; Liu et al., 2022b). Typically, time-scries uniquely characterized by temporal patterns, such periodicities and temporal dependencies, and often exhibit non-stationary fluctuations. Motivated by this, the key to accommodate the alignment- based imputation methods TSI is devising discrepancy measure that captures temporal patterns while accommodating non-stationarity in time-series.
To this end, propose the Proximal Spectrum Wusserstein (PSW) discrepancy, discrepancy tailored to compare sets of time-series based on optimal transport. Specifically, PSW integrates Pairwise Spectral Distance (PSD), which transforms time-series into the frequency domain and then calculates the pair-wise absolute difference. By comparing time-series in the frequency domain, the underlying temporal dependencies and patterns are captured. Moreover, PSW incorporates Selective Matching Regularization (SMR), which relaxes the hard matching constraints of traditional optimal transport to enable flexible mass matching between distributions. This relaxation enhances robustness to non-stationarity. Building upon PSW, we propose the PSW for Imputation (PSW-I) framework, which iteratively refines the imputed missing values by minimizing the PSW discrepancy. Extensive experiments demonstrate that PSW-I effectively captures temporal patterns and accommodates non-stationarity and significantly outperforms existing time-series imputation methods.
Contributions. The key contributions of this study are summarized follows:
We propose the PSW discrepancy, which innovatively extends optimal transport to compare distributions of time-series by encapsulating temporal patterns and accommodating non-stationarity.
We develop PSW-I, the first alignment- method for TSI. It eliminates the need for masking observed entries during training and the complexities of training parametric models on incomplete data, enhancing sample efficiency and ease of operation.
We conduct comprehensive experiments on publicly available real-worl datasets to validate the effectiveness of PSW-I, demonstrating its superiority over existing TSI methods.
2
PRELIMINARIES
2.1 PROBLEM FORMULATION
Suppose (id) NxD is the ideally complete time-series with N chronologicallly-ordeee obser- vations and D features. The missing entries are indicated by a binary matrix M E {0, NDD where M,,d is set
if the corresponding entry X (id) is missing, and 0 otherwise. Con- sequently, the observed dataset (obs)) can be obtained via the Hadamard product: ((oss) = (id) o (1 (1-M) + nan @ M. The goal TSI is to construct an imputed data matrix X (imp) E NxD based on the observed entries in X (ohs), such that (imp) x (id)
2.2 OPTIMAL TRANSPORT
Optimal Transport (OT) is a mathematical tool that quantifies the discrepancy between two probability distributions by finding the least-cost plan to transform one distribution into the other. Originally proposed by Monge (1781), the formulation involved finding an optimal mapping between two continuous distributions. However, this original formulation posed challenges related to the existence and uniqueness of solutions. Addressing these issues, Kantorovich (2006) proposed a more computa- tionally feasible formulation in Definition I, a convex programming problem solvable via the simplex algorithm (Disser and Skutella, 2019).
Definition 2.1. Consider empirical distributions a = Q1: and = 31.m- each with n and m samples, respectively; seek a feasible plan T TR k transport
/ at the minimum possible cost:
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(a) Diserepancy ETThl and Electricity.
(b) 8 temporal patches ETTh1 and their spectra.
Figure 1: Case study the discrepancies calculated in the time and frequency domains.
3) = min (D,T),
Tell(a,
T + + T Tym a,
(1)
(), = T T + . + T bj.j -,,1... m. m,
Tíj 2 0, ,,,,, D. = 1, I.
where W(o 3) denotes the minimum transport cost, known as the standard Wasserstein discrepancy;
D E _ represents the pairwise distances calculated as D ||a, B3 21 and b- [b...., bm] are the masses of samples in a and 3, respectively.
a anl
3 METHODOLOGGY
3.1 MOTIVATION
Distribution alignment has proven effective for imputing missing data in non-temporal datasets (Muzellec et al., 2020; Zhao al., 2023), offering advantages in sample efficiency and implementation simplicity. These methods operate by iteratively sampling subsets of the incomplete dataset and updating missing entries to minimize distributional discrepancies between these subsets. This ensures that the imputed values maintain statistical properties consistent with the entire dataset, grounded in the assumption that different subsets from the same dataset share the same distribution.
However, applying distribution alignment to TSI poses significant challenges. Our experiments (see Section 4.2) indicate that existing alignment-based methods perform poorly on temporal data, In the field of domain adaptation (Courty et al., 2017; Liu et al., 2022b), it is well-recognized that distribution alignment heavily depends on the choice of discrepancy measure,
the effectiveness which must be tailored to the specific properties of the data and the task. Therefore, refine the discrepancy measure to accommodate the unique characteristics of time-series data, for enhancing TSI performance. Importantly, there several questions that need to be answered. Do existing discrepancy measures accommodate the characteristics, such as the temporal patterns and non-stationarity, in time-series data? How design discrepancy measure for comparing distributions of time-series? Does designed discrepancy improve imputation performance?
aim to
3.2 PAIRWISE SPECTRUM DISTANCE FOR TEMPORAL PATTERN EXCAPSULATION
Time-series data are characterized by temporal patterns, which reflects correlations between different time steps and provide rich semantic information essential for comparing time-series. The canonical Wasserstein discrepancy (W) fails to capture these temporal patterns because the pairwise distance is computed on step-wise basis, treating the observations at each step independently and disregarding temporal correlations. A simple modification might involve using patch-wise distance: employing a sliding window to generate temporal patches of size T and subsequently computing distances between patches. However, this method still treats different steps within the patch individually, failing to measure dissimilarity between patches in a way that encapsulates their temporal patterns.
To address this limitation, we propose the Pairwise Spectrum Distance (PSD), which uses the Discrete Fourier Transform (DFT) to convert time-domain data into the frequency domain. The DFT decomposes each temporal patch into its spectral components (Wu et al., 2025; Wang al., 2025),
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pord
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(a) Ideal casc.
(b) Exemplar case 1.
(c) Exemplar case 2.
Figure 2: The impact of non- non-staiin stationarity and the resilience of PSW to it. The toy dataset includes two batches of samples, each distinguished by color. Two modes co-exist due to non-stationarity, differentiated by the vertical positioning of the samples.
where each component corresponds to specific temporal patterns in the data. By comparing patches in the frequency domain, PSD effectively captures and compares the underlying temporal patterns. Building upon PSD, we propose the Spectrum-enhanced Wasserstein distance in Definition 3.1. Definition 3.1 (Spectrum-enhanced Wasserstein Distance). The distance between two distributions a,8 E IR Bx of temporal patches is defined as W(FF) (a. 8) minree)(...)) (D(F), T) where the patch size, D(F) is the pairwise distance matrix with elements computed using the Pairwise Spectrum Distance: D (F) = [|F((a)) F(8 and denotes the DFT operator.
i.j
Case Study. Fig. 1 (a) compares W and W(F). Specifically, W(FF consistently decreases with increasing batch sizes and achieves comparable performance of W with smaller batch sizes. For instance, the value of W)FF) with a batch size of 128 approximates the value W with a batch size of 1024. This efficiency arises because spectrum captures the consistent patterns across patches, especially in periodic data where spectral amplitudes are stable despite significant changes in the time domain Fig. 1(b). These findings demonstrate the advantage of (V(F) for time-series data, as it better captures temporal patterns.
3.3 SELECTIVE MATCHING REGULARIZATION FOR NON-STATIONARITY ROBUSTNESS
Time-serics data often exhibit non non-stationn son--saatinnaity. characterized by time-varying patterns and sudden fluctuations. For instance, the Electricity dataset shows significant differences in consumption patterns between weekdays, weekends, and holidays; similarly, the Weather dataset displays distinct climato- logical patterns across different seasons. This non-stationarity produces multiple coexisting patterns or regimes within the data, complicating the accurate calculation of distributional discrepancies.
The canonical Wasserstein discrepancy (Definition 2.1) struggles in the presence of non-stationarity. As illustrated in Fig. 2, it may incorrectly pair patches from different modes, leading to inaccurate distributional discrepancies and misleading imputation updates. This issue arises due to the matching constraints that require matching of all masses of all samples (Wang et al., 2023). Consequently, assuming that patch from new mode, denoted as 6g, added to a, the matching constraints force match between óz and patches in B, distorting the matching process and yielding an imprecise estimate of the discrepancy. This vulnerability formalized in Lemma 3.2, which shows that the Wasserstein distance W increases the added mode deviates further from the typical elements of B. Lemma 3.2. Suppose that à Çō2 +1 -))x is a distribution perturbed by a Dirac mode with relative mass C E (0, 1). For sample y" in the support of B, Fatras et al. (2021) demonstrate:
W(ā, 3) (1 ()W((, + (D(2.))) (D(
f gd3
where D (2.y") is the deviation of 6e, g is the optimal dual potential of W(a
To enhance robustness to non-stationarity, it is plausible to relax the marginal matching constraints, allowing for matching flexible mass of each patch. Inspired by weak transport principles (Chizat
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Figure 3: The workflow of PSW-I with Pmiiss 0.3. The batch size (B) is set to 2 and the patch size (T) is set to 4. The number of features (D) is omitted for clarity.
al. 2018; Séjouréé et al., 2019), we introduce the Proximal Spectral Wasserstein (PSW) discrepancy, which replaces the hard marginal constraints with Selective Matching Regularization (SMR). This approach removes the requirement to match all mass between distributions, thus accommodating the coexistence of multiple modes and enhancing robustness non-stationarity.
Definition 3.3 (Proximal Spectral Wasserstein Discrepancy). The PSW discrepancy seeks a transport plan T R that transports the distribution ar to B at minimal cost, defined as:
(a,
min (D (D) T (TImIIA)) + DKI.TTTT 1.1/Am))) T20
where ((F) is the pairwise distance matrix computed using PSD; K is the matching strength; An 1a/n and Am 1m/m /m are uniform simplex vectors; 7 denotes the PSW discrepancy.
Case Study. To illustrate the limitations the existing Wasserstein discrepancy and the robustness of PSW non-stationarity, conduct case study shown in Fig. 2. Ideally, in the absence of non- stationarity, Wasserstein discrepancy based on OT accurately matches patches within the same mode and correctly estimate the distributional discrepancy. However, in the presence of non-stationarity, we investigate two exemplary cases:
When the proportions of different modes vary across batches, standard OT incorrectly matches patches from different modes due to its force to match all mass. This leads to a biased Wasserstein discrepancy. In contrast, P employs selective matching strategy, focusing on typical patches and effectively avoiding false matches across modes given small matching strength x S 10.
Example 1:
Example When a newly emerging mode exists non-overlapping area between two modes, standard OT forces matching of the outlier mode with others, leading to inappropriate pairings and interfering with the matching of other patches. Conversely, P resists the interference as decreases. When is reduced to 5, the outlier mode is effectively excluded from the matching process.
Our approach differs from the entropic unbalanced optimal transport (Séjourné al., 2019; Fatras et al., 2021), which also relaxes matching constraints through regularization, by omitting entropic regularization from formulation. This omission is crucial, as entropic regularization has been shown to hinder missing value imputation (Chen et al., 2024). Notably, without entropic regularization, the OT associated with PSW cannot solved using the Sinkhorn algorithm. Instead, we employ the majorization minimization majorization-mmiiiizztooon algorithm (Chapel al., 2021) to solve the OT problem. In Section 4.4, we present a rigorous comparative study that highlights the advantages of PSW over traditional entropic unbalanced optimal transport methods in the context of TSI.
3.4 PSW-I: PSW FOR TIME-SERIES IMPUTATION
While PSW effectively compares and balances distributions of temporal patches, does not directly perform time-series imputation. To fill this gap, propose the PSW for Imputation (PSW-I)
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framework, which iteratively minimizes the PSW discrepancy between batches of patches to refine the imputation of missing values. The core steps of PSW-I are outlined in Fig. and explained below.
Initialization. The incomplete dataset X is initialized by filling each missing entry with the average of its nearest observed steps, producing an initial imputation matrix X The imputed values are treated as learnable parameters, and their gradients are tracked subsequent steps.
Forward Pass. Two batches of temporal patches, denoted as E RB and 3 E RB are sampled from the current imputed dataset X with batch size B. The PSW discrepancy is then computed between these two batches according to Algorithm 1.
Backward Pass. The gradients of the PSW discrepancy P with respect to a and B are calculated as follows using automatic differentiation:
op =
Tu
T . sign(e i= 1.
0a;
==1
0P
Σ Tr T W sign(es.)) j j= 1,2,...,
0ßj
where ((F) denotes the DFT matrix eij (a; W() E R.T. These gradients are used to update the imputed values in and 3 via gradient descent with update rate 1). Only the imputed values are updated during this process. PSW-I iteratively executes the forward and backward passes until hitting the carly-stopping criteria on the validation dataset.
Theoretical Justification. We demonstrate that the PSW discrepancy is a valid discrepancy measure and satisfies the properties of metric under mild conditions (Theorem C.I). Furthermore, we prove that PSW is robust to non-stationarity in data (Theorem C.2). Due to the convexity of the PSW discrepancy with respect to inputs a and 3, the convergence of PSW-1 is guaranteed with error bounds (Theorem C.3). Detailed proofs are provided in Appendix C.
4 EMPIRICAL INVESTIGATION
4.1 EXPERIMENTAL SETUP
Experiments are performed public time-series datasets (Wu et al., 2021; Liu et al., 2024), including ETT, Electricity, Traffic, Weather, PEMS03, Illness, and Exchange. To simulate point-wise missingness (Du et al., 2024), a binary mask matrix is generated by sampling Bernoulli random variable with predetermined mean for missing ratios. Additional missing mechanisms and their associated results presented in Appendix D.
Datasets:
Baselines: PSW-I is compared against representative TSI methods: (1) the predictive TSI methods (DLinear (Zeng et al., 2023), FreTS (Yi et al., 2023), TimesNet (Wu al., 2023), iTTansformer (Liu et al., 2024), PatchTST (Nie et al., 2023), Transformer( connsfrrner(Vaswaii Vaswani et al., 2017), SAITS (Du et al., 2023) and SCINet (Liu et al., 2022a)), (2) the generative TSI methods (CSDI (Tashiro et al., 2021)). Additionally, the performance of distribution alignment methods tailored for non-temporal data (TDM (Zhao et al., 2023) and Sinkhorn (Muzellec et al., 2020)) is evaluated for comparison.
Implementation details: To ensure consistency in experimental conditions, the batch size B is fixed at 256. The Adam optimizer, known for its adaptive update rate and effective convergence, is employed for training, with an update rate 7) = 0.01. We exclude 5% of the indices from the training data to form the validation set. The key hyperparameters involved in PSW-I are tuned to minimize the MSE on the validation set. The patch size tuned within {24, 36, 48); the matching strength is tuned within {1, 10, 100, 1000). The experiments are conducted on platform with two Intel(R) XR) Platinum 8383C CPUs @ 2.70GHz and a NVIDIA GeForce RTX 4090 GPU. Performance is evaluated using modified mean absolute error (MAE) and mean squared error (MSE) following (Zhao et al., 2023; Jarrett et al., 2022), with a focus on imputation errors over missing entries. We set Tmax 200 and Lmax 1, 000 to ensure convergence, applying early stopping on the validation dataset with patience of 10.
'The definition of the DFT matrix is presented in Definition C.4, and the gradient derivation is detailed in Theorem C.5. Notably, our gradient formulation omits the gradients from the optimal transport plan T with respect and ( enhance the efficiency and stability of the calculation process.

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Table 1: Imputation performance in terms of MSE and MAR 10 datasets.
ETTh1
ETTb2
ETTml
ETTm2
Electricity
Traffic
Weather
Dliness
Exchange
PEMS03
Darasets
Metries
MSE MAE
MSE MAE
MSE MAE
MSE MAE|
MSE MAE
MSE MAE
MSE MAE
MSE MAE
MSE
MAE
MSE MAB
Transfonner
0.222 0.322
0.221 0.312 0.06 0.16 0.041 0.134
0.12 0.228
0.216 0.214
0.214|0.195
0.195 0.132
0.24 0.3 0.224 0.186
0.081 0.184
DLincar
0.144 9.267
0.108 0.231 0.103 0.221 0.084 0.201
0.137 0.271
0.251 0.272
0,274 0.185
0.21 0.273
0.261 0.216
0.112 0.261
TimesNet
0.253 0.353 0.133 0,263 0.061
0.173 0.068 0.186
029 0.254
0.201 0.243
0.28 0.189 0.231 0.277 0.319 0.264
0.076 0.19
FreTS
0.184 0.312 0.147 0,259 0.055
0.159 0.039 0.135
0.155 0.285
0.234 0.27 0.178 0.12 0.278 0.325 0.228 0.189
0.109 0.252
PatcTTS
0.171 0.297 126 0.258 0.05
0.149 0.03
0. 0.262
0.235
0.238
0.247 0.167 0.605 0.505 0.237 0.197
0.065 0.179
SCINet
0.149 0.275 0.128 0,248 0.067 0.176 0.064 0.179
0.125 0.239
0.29
0.314
0.198 0.136 0.617 0.473 0.2998 0.247
0.106 0.249
ilransformer
0.163 0.281 0.101
0.211 0.056
0.156 0.034 0.125
0.128 0.251
0.252
0.269
0.188
0.127 0.316 0.31
(0.068
0.056
0.08 0.203
SATTS
0.216 0.305 0.183
0,256
0.956
0.154 0.042 0.129
0.114
0.216
0.224
0.207
0.132
0.089
0.167
0.216
1.005
0.833
0,083
0.189
CSDI
0.151 0.269
0.00%
001
0.177 00.158 0.113
0.533
0.269
0..30
0.324
0.158
0.107
0.384
0.1
0.103
0.1115 0.17
Sinkhorn
0.877 0.65
0.800
0.603
0.934
0.698
0.8933
0.67
0.9986
0.821
1.012
0.729
0.793
0.672
0..449
0.382
0.723
0.599
0,968
0.855
TDM
0.982 0.742
0.971 0.999
0.752 0..988
0.742
0.999
0.795
0.698
00.665
0.689
0.841
0.607
0.9338
0.776
0.925 0.828
PSW-I (Ours) 0.126 0.231
|0.046 0.142 00.047 0.131 (0.02 0.094 00.6600006 0.20s 0919 0.199 0.107 0.072/0.066 0.1 (0.031 0.026 (0.049 0.149
Kindly Note: Each entry represents the average results at four missing ratios: 0.1, 0.3, 0.5, and 0.7. The best and second-best results are bold and underlined, respectively.
Table 2: Ablation study results.
Flectricity
ETTh
Model
SMR
PSD
MSE
MSB
MAE
PSW-1
X
0.116 004
0.227 2.0 00s
0.096go.002
0.204_0..00
PSW-1I
X
/
0.0801 to #0.003
0,196 40. 004
0.085 200001
0.208 0.208 1.004
PSW-111
V
0.115
0,218 005
0.09 20.000
0. 194 20.006
PSW-I
0.075 10.75..003
0.180 004
0.077 40.022
0,188 0..04
Kindly Note: The best results are bold. 1he" marks the results that signilicantly outperform PSW-1", with p-value < 0.05 over paired sample Litss.
4.2 OVERALL PERFORMANCE
Table compares the performance of PSW-I with baseline methods, averaged over four missing ratios: 0.1, 0.3, 0.5, and 0.7. Key observations from the results are as follows:
Effectiveness of Existing TSI Methods. Existing TSI methods demonstrate promising perfor- mance. Notably, fundamental time-series models such as iTTrasformmr and SCINet exhibit very competitive results, achieving the best performance among baselines in out of 20 cases. These models effectively capture temporal patterns present in the data and leverage them for imputation tasks. Meanwhile, methods specifically crafted for TSI, such as SAITS and CSDI, also achieve comparable performance, each attaining the best performance among baselines in 4 out of 20 cases. Albeit with suboptimal structure for capturing temporal patterns, the specialized mechanisms for TSI, such as the conditional generation strategy in CSDI, effectively raise the performance comparable to advanced fundamental time-series models.
Limitations of Alignment- Based Methods in TSI. Alignment- based methods, including Sinkhorn and TDM, demonstrate strong performance on non-temporal data (Zhao al., 2023) but fall short in TSI tasks. Specifically, their performance in Table 1 is significantly inferior to that of prevailing TSI methods. This discrepancy can be attributed their inability to capture temporal patterns inherent in time-series data and vulnerability to non-stationarity due to their inherent i.i.d. assumption (Muzellec et al., 2020), resulting in suboptimal imputation results.
Superiority of the Proposed PSW-I Method. PSW-I retains the advantages of alignment-based methods: it does not require masking observed entries during training or training parametric models on incomplete data. This distinguishes it from prevailing TSI methods and positions it as a promising alternative. Moreover, PSW-I counteracts the limitation of alignment- based methods to capture temporal patterns and accommodate non-stationariiy. Overall, PSW-I achieves the best performance across all 10 datasets, often surpassing the best baseline by significant margins (e.g.. on PEMS03), showcasing its efficacy in real-world applications.
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(a) Varying patch length results on Illness
(b) Varying patch length results on ETTh1
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(d) Varying matching strength results on ETTh1
Figure 4: Varying patch length and matching strength results with missing ratios 0.1, 0.3, 0.5 and 0.7.
Table 3: Varying pairwise distance results.
Table 4: Varying discrepancy results.
Electricity
Electricity
Distances
MSB
AMSE
MRE
AMRE
MAE
AMAB
Discrepancies
MSE
AMSE
MRE
AMRE
MAB
AMAE
PSW-T
0.080
0.233
0.193
OT
0.082
0.239
PSW-A
0.079
1.3%
0.231
0.9%8.
0,191
1.0%
EMD
0.080
2.5%
0.225
6,2%%
0,184
6.5% J
P.W.P
1.3%%
1.7%%.
0.189
2.1%64
6.5%
5.8%
5.4%
PSW
0.075
6.7%
0.218
6.9%
0,180
7,2%
Qurs 0.075
9.3%
0.218
9.6%
0.180
8.9%L]
6T
ETThl
Distances
MAE
AMAE
Discrepancies
MSE
AMSE
AMAE
PSW-T
D
0.268
0.202
OT
0.084
0.273
0,206
PSW-A
0,081
1,25.
0.265
1.19.
0.200
1.0%-]
EMD
0.079
6.3%
0.264
3,4%
0.202
2.0%
PSW-P
0.081
1,2%6]
0.263
1.9%
0.198
2.0%
UOT
0.081
3.7%
0.263
3.8%
0.198
4.0%
PSW
0.077
6.5%
0.250
7.2%
0.188
7.4%&
Ours
0.077
9.1%
0.250
9.2%
0.188
9.6%
4.3
ABLATIVE ANALYSIS
Table 2 presents an ablation study dissecting the contributions of the Pairwise Spectrum Distance (PSD) and Selective Matching Regularization (SMR), the two key components of the proposed PSW-I framework. The baseline model without PSD and SMR computes the Sinkhorn discrepancy using the Euclidean distance between patches. While this naive model performs suboptimally. it still outperforms the standard Sinkhorn imputation method (as shown in Table 1) because the patch-wise distance somewhat captures more temporal patterns than the step-wise distance.
Incorporating PSD and SMR significantly improves imputation performance by effectively encap- sulating temporal patterns through spectral representation, and accommodating non-stationarity by matching flexible set of samples. The PSW-I framework, which integrates both PSD and SMR, achieves the best performance, demonstrating the effectiveness of combining these two components.
4.4 GENERALITY ANALYSIS
In this section, we explore some alternative implementations to the key components of PSW-I to justify its rationale and advantages. Table 3 and 4 present the results, where A denotes the relative performance reduction. The primary observations are summarized as follows.
Effects of the Discrepancy Measure. Our approach accommodates the non-stationarity by replacing the matching constraints in OT with soft regularizer (SMR), leading to significant performance improvements over standard OT Table 3. Notably, the UOT method in domain adaptation (Fatras et al., 2021; Séjourné et al., 2019) also relaxes matching constraints and can be adapted to TSI. similarly enhancing imputation performance compared to standard OT. However, method differs from the standard UOT formulation by omitting entropic regularization, which proves to be beneficial to missing value imputation (see the discussion by Chen et al. (2024)). The results comparing OT (with entropic regularization) versus EMD (without entropic regularization)
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(a) Varying batch size results on Illness.
(b) Varying batch size results on ETThI.
0.30
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0.5
0.1
0.5
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(c) Varying update rate results on Illness
(d) Varying update rate results on ETTh1
Figure Varying batch size and update rate results with missing ratios 0.1, 0.3, 0.5 and 0.7.
are provided in Table By avoiding entropic regularization, our method acquires rectified transport plans that are more effective for imputation tasks.
Effects of the Distance Metric. The proposed Pairwise Spectrum Distance (PSD) computes the patch-wise distance by measuring the difference of their spectrum. Compared to calculating the patch-wise distance in the time domain (PSW-T in Table 3), using distance metrics that consider amplitude characteristies (PSW-A) phase characteristies (PSW-P) individually leads to performance improvements, as they capture distinct aspects of temporal patterns. Specifically, PSW- A focuses on the strength of frequency components, while PSW-P captures the timing information inherent in the phase. Combining both amplitude and phase characteristics in PSW provides a comprehensive encapsulation of temporal patterns and achieves the best performance.
4.5 PARAMETER SENSITIVITY ANALYSIS
In this section, we examine the impact of critical hyperparameters on the performance of PSW-I. The results are presented in Fig. and 5. The primary observations are summarized as follows:
The patch length (T) determines the scale of temporal patterns that PSD captures. When T is reduced to I, the model effectively degrades to naive alignment- models (Muzellee et al., 2020) that overlook temporal patterns. Increasing T to 24 leads to substantial performance improvements, underscoring the importance of temporal patterns for TSI. However, further increasing T beyond 24 deteriorates performance, which could be attributed the curse of dimensionality in the Wasserstein discrepancy, reducing its discriminability in high-dimensional spaces (Chizat et al., 2020).
The matching strength (xx) controls the flexibility of matching mass in SMR. As x decreases, the robustness to non- non-stationarity property is enhanced. The consistent performance gains underscore the importance accommodating non-stationarity for TSI and the efficacy of SMR.
The batch size (B) determines the scale of the optimization problem when calculating PSW discrepancy. Overall, model performance is not highly sensitive to it in cases with low missing ratios. However, in cases with high missing ratios, a smaller batch size beneficial, as it enables more fine-grained comparisons between distributions and can improve sample efficiency. Therefore, choosing smaller batch size is advantageous for both accuracy and efficiency in such scenarios.
The learning rate (1)) controls the model convergence. As increases, the imputation error initially decreases and then increases, indicating the presence of an optimal value. learning rate of 0.01 yields the best overall results, effectively balancing convergence stability and speed.
5 CONCLUSION
This study introduces the PSW-I approach, offering a fresh perspective on OT for time-series imputation. The core innovation is the PSW discrepancy: instrumental and versatile discrepancy

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measure for comparing distributions of time-series. Based the proposed PSW discrepancy, further derive a novel time-series imputation approach, termed PSW-I, which significantly enhances imputation performance various real-world time-series datasets.
Limitations and Future Work. This work employs the DFT to encapsulate temporal patterns, which primarily captures global frequency components and may not effectively represent local transient pattern.. Future work could explore advanced spectral methods, such as wavelet transforms, to enhance temporal resolution. Additionally, due the curse of dimensionality, there is bottleneck when increasing the patch size to improve performance. Enhancing the scalability of PSW-I to exceedingly long-term patterns remains a challenging yet promising direction for future research.
ACKNOWLEDGEMENT
This work was supported by National Natural Science Foundation of China (623B2002). National Science and Technology Major Project of China (20222D0120005), ARC DE210101624 and ARC DP240102088. The first author extends special gratitude to Prof. Degui Yang of Central South University. for his exceptional signal processing lectures and generous research guidance during M.T.E.M. studies.
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A RELATED WORKS
A.1 MISSING VALUE IMPUTATION FOR TABULAR DATA
In this section, we review the predominant methodologies for missing data imputation in tabular datasets, which differ from time-series data due to the absence of temporal patterns and dependencies. Broadly, these methods can be categorized into four paradigms, each offering unique strengths and limitations (Jarrett et al., 2022).
The simple direct paradigm involves basic statistical methods such as mean, median, and mode imputation to replace missing values. These techniques are fast and easy to implement but often fall short in capturing the complex dependencies present in weal-world datasets (Malarvizhi and Thanamani, 2012; Mazumder et al., 2010), As a result, this approach frequently leads to oversimplified and inadequate imputations that may not perform well in practice.
The round-robin paradigm (Stekhoven and Bühlmann, 2012; Royston and White, 2011), originally developed from the Iterative Conditional Expectation (ICE) algorithm (Royston and White, 10, im- putes missing values for each feature by utilizing other observable features as predictors. Building on ICE, modern approaches incorporate advanced parametric models, including neural networks (Mattei and Frellsen, 2019; Kyono et al., 2021), Bayesian models (Royston and White, 20.1), and random forests (Stekhoven and Bühlmann, 2012), to better handle intricate missing patterns. Furthermore, diverse training strategies such as multiple imputation (Royston and White, 2011), ensemble learn- ing (Stekhoven and Bühhmann, 2012), and multitask learning (Mattei and Frellsen, 2019) have been explored to improve performance across different contexts. Although this paradigm enhances flexibility and imputation accuracy, the necessity of careful model selection poses a challenge since the complete data is unavailable for validation (Jarrett et al., 2022).
The generative paradigm approaches imputation as a conditional generation problem, leveraging advanced neural architectures and generative models such as Generative Adversarial Networks (GANs) (Yoon et al., 2018; Sun et al., 2024; Li et al., 2024) and diffusion models (Tashiro et al., 2021; Xu et al., 2024; Ouyang et al., 2023; Chen et al., 2024) to estimate the underlying data distribution and generate plausible values for missing entries. This approach can capture complex relationships and dependencies, leading high-quality imputations, especially when sufficient data is available. However, the generative paradigm inherits the challenges of generative models, including the instability of adversarial training and the significant data requirements of diffusion models. These issues are further exacerbated by the high missing ratios and complex missing patterns often encountered in industrial settings, limiting their practical utility.
A more recent innovation in this field is the distribution matching paradigm (Zhao et al., 2023; Muzellec et al., 2020; Wang et al., 2024b;a). This paradigm samples subsets of dataset and update the missing entries to minimize the discrepancy between the sampled subsets. The underlying assumption that, under the i.i.d. principle, any two batches of data should follow the same distribution, allowing for effective imputation by reducing distributional divergence. Our research extends this methodology by addressing its vulnerability to outliers and enhancing its capability to account for temporal patterns, thus adapting it for time-series imputation tasks.
A.2 MISSING VALUE IMPUTATION FOR TIME-SERIES DATA
Deep learning-basee based imputation methods have attracted substantial attention in TSI due to their ability to model complex nonlinearities and temporal dependencies in time-series. The literature can be broadly classified into two categories: predictive methods and generative methods (Du, 2023; Du et al., 2024; Wang et al., 2024c).
The predictive methods focus on estimating deterministic values for the missing entries within the time-series. These methods leverage various neural network architectures to capture nonlinearities and temporal dependencies, such as RNNs (e.g., DeepAR (Salinas et al., 2020), S4 (Gu et al., 2021), I), GRU-D (Che et al., 2018), BRITS (Cao et al., 2018)), CNNs (e.g., TemesNet (Wu et al., 2023)), and Transformers (c.g., iTransformer (Liu et al., 2024), SAITS (Du et al., 2023)). For example, GRU- D (Che et al., 2018) introduces decay mechanism that effectively addresses irregularities arising from missing values, while TimesNet (Wu et al., 2023) transforms time-series data into an image-like like representation using spectral analysis, allowing the application of vision models for imputation tasks.
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Despite their ability to provide effective imputation, predictive methods are challenged by model selection amidst incomplete data. Additionally, these models typically require masking of some observed entries during training to generate labels, which reduce sample efficiency, reducing performance especially when the proportion of missing values is high.
The generative methods aim to generate missing values conditionally given observed ones (Chen et al., 2023; Luo et al., 2018; Gao et al., 2024). Notable generative approaches include variational autoencoders (VABs), generative adversarial networks (GANs), and diffusion models. For instance, Fortuin et al. (2020) proposed GP-,AE, a model that integrates a Gaussian process prior with a variational autoencoder to model incomplete time-series data. Similarly, Miao et (2021) introduced US-GAN, which improves the masking matrix to facilitate the generator during imputation. Tashiro al. (2021) proposed CSDI, a diffusion model that employs a conditioned training strategy to guide the imputation of missing entries. While generative methods offer more probabilistic framework and can capture the inherent uncertainty in the imputation process, they are often more computationally intensive and complex to train, which can present scalability issues. Similar to predictive methods, generative models also require masking observed entries to construct training labels, leading to reduced sample efficiency, particularly in cases with high proportion of missing data.
B BACKGROUND ON DISCRETE OPTIMAL TRANSPORT
This section introduces the foundational concepts and algorithms necessary for computing OT between discrete measures. Consider a scenario involving n warehouses and m factories, where the i-th warehouse holds a, units of material and the j-th factory requires by units of material (Peyré and Cuturi, 2019). The objective is to establish a mapping from warehouses to factories that: (1) completely allocates all warehouse materials, (2) fulfills all factory demands, and (3) restricts the transportation from each warehouse to at most one factory. Each potential mapping is evaluated based on global cost, which aggregates the local costs incurred from transporting a unit of material from warehouse to factory j.
Definition B.1. The Monge problem for discrete measures, where a = Em== a,0x, and 3 _ bydx,, seeks a mapping T: {xx - { that optimally redistributes the mass from o to 3. Specifically, for each j, must hold that b, -E a, and Tya = B. The goal is to minimize the transportution cost, represented by c(1, v), leading to the following formulation
Edx,, T(x)))
min T:T
The original Monge formulation does not guarantee the existence or uniqueness solutions (Peyré and Cuturi, 2019). Therefore, Kantorovich (Kantorovich, 2006) extended this framework by relaxing one-to-one mapping constraint, allowing transportation from a single warehouse to multiple linear programming problem.
factories, and reformulated the problem
Definition B.2. The Kantorovich problem for discrete measures and B defines a cost-mininization task over feasible transport plans म R R The objective is find plan that minimizes the overall transport cost:
W(a, B) == min (D
xEII(a.,))
Ma, := =
IR " m1m a,
T =b},
where W(a, B) represents the Wasserstein discrepancy between a and 3; D denotes the distance matrix computed using the squared Euclidean metric (Courty al., 2017), and a. b are vectors describing the mass distribution in a and B, respectively.
The subsequent research in discrete OT primarily follows two trajectories. The first aims to reduce the computational complexity of solving OT problems. While exact solutions can be obtained through linear programming algorithms, these come with cubic complexity in relation to the number of samples (Bonneel et al., 2011). To address this, various approximate algorithms for acceleration have been developed, such as the Sinkhorn and sliced OT algorithm (Altschuler et al., 2017) with quadratic and linear complexity, respectively. The second line of research focuses modifying the transport
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Algorithm 1 The imputation workflow of PSW-I.
Input: X (nbs); the incomplete data; M: the mask matrix.
Parameter: 6: the matching strength; e the entropic regularization strength; maxi max number of iterations to solve PSW; max: the max number of epochs; B: the batch size; the update rate. Output: (Imp): the imputed dataset.
I: tt- 0.
2: X' + Pre-Impute(X (ohs)) (ohbs)
3: while I. T max do
4:
or 3, M ,MM <- Sample (M;
5:
(FF) + F(a),
((F)) t F(3)
D ||a
-Bj - 1si,j B
6:
7:
P - Algorithml (D(F); K. kmax).
8:
a'
P M.
9:
8' + 3 VVpP OM. M.
10:
X t Updato(a',,''x8
11:
t ++11.
12:
if X - -X ||p < 1e then
break.
14:
(imp) t- X
problem suit specific applications. Examples include the weak transport problem in domain adaptation (Chizat et al., 2018), the Schrödinger bridge problem in generative modeling (Marino and Gerolin, 2020), the Gromov problem in graph matching (Xu al., 2019), the unbalanced transport problem in causal inference (Wang et al., 2023), and the adversarial formulation in GANs (Yoon al., 2018; Spinelli al., 2020; Li et al., 2024).
C THEORETICAL ANALYSIS
C.I THEORETICAL JUSTIFICATIONS
Theorem C.1 (Metric Properties). PSW defines valid divergence the space of probability measures over time series. For any probubility measures 0, R.T PSW satisfies:
(i) neg-tiiat P(a, 3) 2 0,
(ii) Identity of Indiscernibles: P(a,, 3) = 0 if and only if = 3,
(ii) Symmetry: P(a,3) a(,,a))
Moreover, when 0, PSW satisfies the triangle inequality and thus constitutes metric.
Proof. The proof proceeds by verifying each of the four metric properties for the PSW discrepancy.
Non-Negatinn :on--eegatiity: The PSW discrepancy is defined via an optimal transport formulation incorporating the spectral distance metric depectral, the transport strategy, and a selective matching regularizer. All these components are non-negative. Consequently, P(a, -the infimum of sum of non-negative terms-satisfee Pa,, 2 0.
3, the optimal coupling 7 the identity coupling, and for i,j such that %i 0. The selective mass regularizer contributes zero minimal constant that can be canceled based on its formulation. Thus, P(a, 3) = O. Conversely, P(a, B) 0, the spectral distance metric depectral (æt, y) must vanish almost surely under the optimal coupling 7. Assuming the DFT is injective for sufficiently rich and non-degenerate time series, we have y almost every leading to B.
Identity of Indiscernibles: If
depectral (a,, (3)
Symmetry: The spectral distance metric satisfies depeetra (2, y) = dspectral (y. 2). Therefore, swap- ping and does not change P(a, B). Moreover, transposing does not change the value of
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selective mass regularizer, which immediately follows from the common assumption that the sample masses are uniformly distributed each distribution, ensuring P(a, 3) = P(3,a).
Triangle Inequality (when 0): Consider three probability distributions a, B, and &. Let Ta,8 E 7(a, 3) and 78.8 E () be the optimal couplings that minimize P(a, B) and P(3, E). respectively. Utilizing the concept of transitivity in optimal transport, define the coupling Ya, by
ea, 2) / Ta, (ae.yyyaaa (),,, dy.
Next, we compare the PSW discrepancy between and with the discrepancy based on Ya.Ei
P(a,E) & / dapoctratls; z) ( z).
Substituting the expression for Ya,g, we have:
P(a,E) = dapectral z) (fr.stes e 2) dy dr dz.
Applying Fubini's theorem to interchange the order of integration, the above expression becomes:
E)a,E) - (fdapectral 2)rsp, (v, 2) dz) Ta,e(a, y) dy dz.
By the triangle inequality property of the spectral distance metric depectral, we have: dspectral 2) S dspectral (2. y) + dspeccrally 2). Substituting this into the integral, we obtain:
P(a, < / (dapectral v) + dapectral 2)) T0, v), (7,,,,1,,2 (v,, dy dr dz.
This expression can be separated into two distinct integrals:
P(a,E) dspectrrll (I,, Va, v) dy dx + f dapeccral 2)118. 2) dy dz. Recognizing that the first integral corresponds to P(a, B) and the second P),,E), we have: g)a,5) P(a, P(S,))
Thus, P satisfies the triangle inequality when 0.
Theorem C.2 (Robustness of PSW Outlier modes). Suppose that Çō2 + (1 ()xx is distribution disturbed by a Dirac mode at with relative mass ( E (0, 1). We have:
Pd (ā, (1 (c)p"(a
e -
where d(z)= = (D (z, (), Am) is the average distance of z and samples in B.
Proof. This theorem builds upon the foundation of Lemma 1.1 Fatras al. (2021) and extends its applicability to our proximal spectrum wasserstein discrepancy. Let T be the OT plan associated with ps (ax 3) in (2). For the disturbed distribution ã, suppose the transport plan is T (1- T
{6g x 3, where & is parameter to be optimized. The marginals of T are Ti = (1 - <)T; + C65
and T = (1 - <)T; + çØ3. The KL. divergence of them to a and 3 satisfies
DKL (Tilla) < D) (T (Ti + (DKL(65,11,,
DRL. (T (8) & (1 - DK. T 11 () + (6x(1)))))))
which immediately follows from the joint convexity KL divergence. Therefore, we have:
pd (a.9)
()( 5 (1 (1-)) (DEF), T
RD (Ti (a) + RD
e(0)
+c (F Am)
+ DKI,(062111 + DKI (133113)
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where d(2) = (D(()) (z, 3), Am) can be viewed the average from the distance of 2 and samples in 3. According to the first order condition, the term e(o) achieves the minimum when 2(x) Substituting it into the above inequality, we have:
p" (a, 9) S (1 c)P" (a, B) 2x - -dd(())/2m)
Theorem C.3 (Error Bounds of PSW-I). Suppose 0x (axz (xk) is the imputation values the k-th iteration, and 0 = (a", 3 be the optimum imputation values that minimizes P( ((a,8). Assume that P satisfies the following regularity conditions:
1. Convexity: P(O) is convex with respect to (0), which naturally holds since it is an composition of affine functiins and I-norms.
Smoothness: P(0) has Lipschitz continuous gradients with Lipschitz constant i.e. [VoP)(k) - VeP(Oe+1)] S L 0xx1 0xl1;
Initialization Proximity: The initial distributions satisfy 1.0 - 0*1l S e some constant
Under these conditions, for all iterations K 2 O, the PSW-I framework ensures:
Plak, Bx) - P(a", 8 20y
Proof. Given that P(a, () is convex with respect to (a, (3 and has I--Lipschitz continuous gradients, can follow the standard steps for deriving the error bound for optimizations.
Suppose 0g = (ak,, 3x) is the imputation values at the k-th iteration, and 0" (a", ,88) be the optimum imputation values that minimizes P(a,, B). Consider the iterative update rules of the PSW-I framework using gradient descent:
86+1 = 0x - - VoP(()))
For L-LLischiiz convex function P, we have
P(0++1) P(O)) + VOP(ON)(0 - 0%) + " ||0 0xl1
L
2
= -(()) 1/ (.P(b)11 + Laj
2 [VP((O))1
= P(O)) 71 - -n
2
S P(A)) IVOP(OWI1
S P(0') + VOP(Os) - 0r)
V
P(P()1)
= P(0') + 2n 10 1-1
1
10
1-1
VoP(BL) - - IVo
17
2n
1
P(0") + 10
1-1
10
2n)
24
P(8")
(11@g
111
11
1*
2n
(3)
where the first line is a well-known property of Lipschitz convex functions; the second line incorpo- rates the update rules of the gradient descent, the fourth line holds as we set S 1. which makes
- 2 < Ø-
Summing up the inequality above with l: 0, 1,.,,.., K:
K
P(Ox) KKP(')) < 2n (1lb 1-)1 - 110k - -8"11)) s
110 101
2n
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According to the 4-th line of (3), we have P(0x+1) - P(0k) ......... P(86). Therefore, (P(0k) Σ P))+)) holds, which immediately follows by
1
1 6² 211
KP(Ok) - KP(00) 2 P(Os) KP(0*) S
_ 100 011
21
Therefore, have:
1
P(Ok) -PO0) - 24yR
2
Definition C.4 (DFT), Let [xo, (1. 21...., ;,, denote a T-length sequence, the DFT of is T-length sequence with the k-th component defined as:
w o.k w (P) LA
T-1
j2nkt/TT = [X0, XT-I
X
t=
w(F) n-1
where j is the imaginary unit, ((F))
= e e-jnktt/T
On the basis, the DFT sequence x(P)
1.k
(x .... x11 can calculated as:
(
W
W
0.0
0.1
OTT-I (F)
W
(F)
w
(F)
w
x (P) .oo....
1,0
1,
1,T-1
(F()
=x
(FF) T-1.0
((F)
(F)
T-111
T-1,T-
where (F) defined as the DFT matrix. Theorem C.5 (Gradient calculation). can be expressed as
The gradients of the PSW discrepancy with respect to a and
0P
Σ
T ((().
sign(er.))
0o; j=1
OP
n
Σ Tr, sign(e.,)
03,
where the grudient of to and 8 is Iruncated for accelerating and stabilizing the update process.
Proof. After acquiring the optimal transport matrix-T-by solving the optimization problem in (2),
the PSW is calculated
P(a, 3)
: B)=
(D(F), (D(F), T + (DKL (T1m An) DKL (T (m))
which depends and through the distance matrix D by
|| - 11(a:l 13)a=-3) W(
where (F) the DFT matrix Definition C.4. Using the chain rule, we have:
OP
OP
8D
a
(P)
Σ
T
Σ
T (F). sign(e.,))
Da:
day
da;
j=1
j=1
1=1
0P
Op
n
0D 1d
Σ
T
Σ Th (FF) sign(e
03,
D
F
03;
09;
where e,, (a; ( E RT, T is the sequence length. Notably, the derivatives d' day and dT/d33, should be considered ideally. However, obtained through an iterative numerical optimization procedure that is not always differentiable. Even if the optimization procedure was
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1.2
0.29
1.0
+
100
Time (ms))
100
Time (ms)
0.285
Time
0.8
-
500
0.28
500
0.6
0.27
0.260
100
500
84
128
256
512
8
18
32
64
1
10
50
Batch size
Patch length
Match strength
Figure 6: Running time of calculating PSW under diverse settings. Each color indicates distinct maximum of iterations. The colored lines indicate the mean values 100 trials, with the shadowed areas representing 99.9% confidence intervals.
differentiable, calculating d'T d'//da; /a, and d' /d3; would be computationally intensive due the need to compute gradients across up to 1,000 iteration steps. Even worse, large values in iteration could cause excessively large gradients and lead to gradient explosion. Therefore, we choose update and B through the pathway p (F) a, B. This method stabilizes the optimization procedure and effectively guides the imputation updates.
0D!F)
(D and 33, as follows:
Finally, let Gigi be the k-th element of eij, we detail the derivation of Da,
aD (P)
aD
deijjk
Gi.j. do;
Jes
Σ sign(er,,.) da;
Σ sign(e, e ).jkk)
(FF i,k
[W!PP) ...., 5m- (siig((es,,.))) sigu(ei.fu-11))
(
WF sign((,.))
(D
05g
Ci.jk Oa;
dei.j.
Σ sign(e
0B,
Σ
sign(ei.j. W(FF (FW(F)
,0'... .... w<F)
in (og.))) sign(Ci.ja-1)
&
(
si,
D ADDITIONAL EXPERIMENTS
D.I RUNNING TIME ANALYSIS
We examine the computational demands of the proposed PSW-I method, focusing on the time required for each update, specifically the execution time of Algorithm to calculate the PSW distance. Results are detailed Fig. and delineated below.
Increasing B from 64 to 512, while keeping T - 8, leads to predictable increase in running time. This is due to the enlargement the transport matrix (T), which escalates the complexity of the optimization problem. Nevertheless, the rumning time remains manageable, with durations 1.2 seconds even at the largest batch size of 512. Therefore, while larger batch sizes do increase computational load, they do not significantly impede practical application.
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01
0.5
011
0.20
+
03
0.20
6.95
MAE
23
0.15
0.15
0.10
0.001
000
0.01
0.02
6.05
16
32
64
128
250
12
40
96
0.5
0.5
07
0.05
0.05
MSE
0.05
MSE
0.05
0.001
000
0.02
0.03
16
10
30
50
70
100
(a) Varying hyperparameter performance ETTh2 dataset.
0.43
0.5
0.45
0.63
0.54
0.45
0.36
0:45
0.27
0.96
0.27
1
0.36
0.27
0.38
0.15
0.18
0.001
000
0.02
0.03
0.18
128
10
30
50
70
100
0.45
0.1
0.5
0.7
0.45
0.27
0.58
0.36
0.7
0.27
0.00
0.011
0.005
0.01
0.02
32
64
128
256
12
24
48
72
50
100
K
(b) Varying hyperparameter performance on Electricity dataset.
Figure 7: Hyperparameter sensitivity results on ETTh2 and Electricity datasets, with missing ratios
being 0.1, 0.3, 0.5 and 0.7.
Holding B = 128, we observe a direct, but modest, positive correlation between (T) and running time. Nevertheless, an increase in T does not alter the size of the transport matrix which is primarily influenced by B. Therefore, the impact of T on the computation cost relatively minor.
An interesting observation is the effect of varying k. As к decreases, the running time increases. This trend likely arises from the complexity of achieving a more concrete transport strategy, which requires more iterations to reach plateau.
D.2 FULL RESULTS
Table 5 provides a comprehensive comparison of PSW-I's performance against baseline methods, with results for missing ratios of 0.1, 0.3, 0.5, and 0.7 listed separately. Additionally, the standard deviation calculated from five random seeds is presented in Table 6.
D.3 PARAMETER SENSITIVITY ANALYSIS
We examine the sensitivity of hyperparameters on two additional Mansets-Electricitg and ETTh2 STTTh--spppemmenting the results presented in the main text, as shown in Fig. 7.
D.4 DOWNSTREAM TASK PERFORMANCE
We evaluate the prediction performance of the iTransformer trained on the datasets imputed using PSW-I and baseline methods. As shown Table 7, all methods demonstrate comparable performance; however, our PSW-I method achieves the best results, particularly on the ETTh2 dataset. This highlights the effectiveness of our approach in downstream tasks.
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Table 5: Full comparison results with missing ratios: 0.3, 0.5, 0.7. The length of history window set to 96 for all baselines. Avg. indicates the results averaged over missing ratios.
Datasets
CT1bl
BrTh2
CT1ml
DTTml
Elecriiity
Traffic
Weather
liness
Excharge
PEISSI
Motrics
MSE
MAE|
MSE
MAE|
MSE
MAE|
MSE
MAE
MSE
MAB|
|
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Transfomecr
0.223
0.164
0.119
0.017
0.085
0.101
0.209
0.158
0.184
0.161
0.109
0.135
0.212
0.122
0.102
0.06
0.158
0.3
0.116
0.344
0.157
0.285
0.042
0.13
0.033
0.129
0.108
0.215
0.190
0.192
0.19
0.129
0.316
0.251
0.137
0.113
0.063
0.159
0.
0.349
0.359
0.25
0.36
0.009
0.173
0.17 121
0.12
0.228
0.257
0.219
0.122
0.129
0.293
0.316
0.251
0.215
0.079
0.179
0:425
0.313
0.219
0.05
0.199
0.149
0.26
0.239
0.342
0.236
0.16
0.316
0.362
0.373
0,313
0.121
0.239
Arrg
0.222
0.302
0.2211
0.16
0.041
0.134
0.32
0.238
0.216
0.214
0.195
0.132
0.34
0.3
0.224
0.186
0.008
0.184
0.1
0.083
0.213
0.073
0.067
0.152
0.053
0.163
0.101
0.237
0.244
0.269
0.251
0.17
0.121
0.18
0.225
0.189
0.108
0.259
DL DLiocar
0.117
0.342
0.081
0.303
0.083
0.201
0.07
0.188
0.125
0.262
0.253
0.277
0.276
0.187
0:149
0.343
0.246
0.301
0.113
0.267
0.5
0.148
0.272
0.117
0.24
0.11
0.229
0.089
0.206
0.142
0.214
0.209
0.268
0.271
0.182
0.284
0.333
0.252
0.208
0.114
0.259
0.225
0.248
0.31
0.257
0.273
0.296
0.2
0.286
0.346
0.323
0.267
0.114
0.258
Arg.
0.144
0.367
00.008
0.351
0.103
0.221
0.054
0.201
0.137
0.271
0.251
0.212
0.274
0.185
0.21
0.273
0.261
0,216
00.112
0.261
TimessNot
0.1
0.173
0.201
2.084
0.314
0.002
0.127
0.031
0.135
1.11
0.236
0.159
0.217
0.251
0.17
0.084
0.185
0.238
0.2
0.057
0.168
0.147
0:200
0.175
0.225
0.252
0.31
0.266
0.5
0.063
0.297
0.126
0.26
0.001
0.179
0.012
0.197
0.124
01249
0.205
0.246
0.272
0.183
0.378
0.304
0.943
0.288
0,377
0.194
0.7
0.523
0.532
0.168
0.295
0.11
0.238
0.114
0.243
0.38
0.300
0.266
0.283
0.346
0.233
0.252
0.306
0.401
0,332
0.109
0.23
Aug.
0:253
0.358
0.138
0.263
0.061
0.173
0.068
0.186
0.120
01254
|
0.201
0.243
0.28
0.189
0.331
0.202
0.319
0.364
0.006
0.19
0.1
0.127
0.256
0.18
0.271
0.053
0.16
0.041
0.143
0.125
0.27
0.23
0.275
0.168
0.114
0.002
0.156
0.233
0.196
0.126
0.283
ReTTS
0.145
0.282
0.001
0.311
0.041
0.141
0014
0.176
0.133
0766
0,229
0.27
0.169
0.114
0.196
0.204
0.231
0.181
0.007
0.239
0.358
0.141
0,261
0055
0.149
0.077
0.134
0.168
0.292
0.236
0.27
0.174
0.117
0.295
0.349
0.256
0:111
0.115
0.259
0.7
0.255
0.381
0.176
0.294
0.074
0.187
0.042
0.139
0.185
0.315
0.261
0.267
0.3
0.135
0.533
0.472
0.201
0.166
0.087
0.227
Avg.
00114
0.312
0.147
0.340
0.085
0.15
0.019
0.195
0.155
0.285
0.234
0.37
0.178
0.12
0:378
0.935
| 0.238
0.189
|
0,253
PatchTST
0.3
0.118
0.133
0.347
0.264
0.102
0.116
0.251
0.337
0.044
0.044
0.152
0.137
0.020
0.004
0.124
0.106
0.099
0.116
0:234
01243
0.228
0.228
0.228
0.233
0.241
0.232
0.157
0.164
0.115
0.321
0:217
0.443
01299
0.196
0:252
0.361
0.004
0.066
0165
0.164
0.153
0.384
0.117
0.360
0.846
0.139
0.000
0.116
0.196
0.200
02235
0.24
01182
0.122
1.045
0.609
0.204
0.160
0.091
0.234
0.275
0.156
0.127
0.203
0.33
0,253
0.349
0.332
0.224
0.942
0.063
0.25
0.206
0.069
0.163
Arg.
0.388
0.118
0.138
0,200
0,235
0.218
0.347
0.167
0.605
0.506
0.252
0.197
00,005
0.179
0.085
0.209
0.084
0.009
0.136
0.029
0.121
0.084
0.198
0.297
0.329
0.178
0.12
0.135
0.136
0.261
0.219
0.099
SCINet
0.121
0.349
3.085
0.20
0053
0.16
0.577
0.173
0.129
0247
0.40
0.205
0.206
0.139
0:335
0.438
0.281
0:311
0.1
0.241
0.5
0.109
0.3
0.176
0.785
0.02
0.175
0077
0.306
0.130
0.25
0.301
0.578
0.178
0.12
1.128
0.7
0.28
0.231
0.115
0.253
0.168
0.216
0.259
0.309
0.304
0.229
0.165
0.87
0.617
0.571
0.307
0.252
Arg.
0.140
0.275
0.128
0.345
0.067
0.176
0.064
0.179
0.125
0.250
0.29
0.314
0.198
0.136
0.617
0.473
0.296
0,247
0.106
0.249
iTansfornce
0.1
0.101
0.219
0.069
0.175
0.044
0.141
0.028
0.113
0.084
0.209
0.178
0.23
0.131
0.089
0.131
0.215
0J042
0.036
0.058
0.171
0.234
0.221
0.349
0.162
0.109
0.375
0.317
0057
0.048
0.07
95
0.068
0.289
0.085
0.21
0.04
0115
0.09
0.113
0.146
01267
0.771
0.219
0.208
0.14
0.156
0.335
8038
0.045
0.006
0.216
0.7
0,244
0.353
0.155
0.263
0.082
0191
0057
0.167
0.177
0.295
0.34
0.319
0.252
0.17
0.403
0.376
0.116
0.095
0.103
0.233
Aug.
0.163
0.281
0.101
0.211
0.006
0.156
0.044
0.125
0.128
0.251
0.252
0.269
0.188
0.127
0.316
0.31
0.068
0.066
0.08
0.203
0.1
0.119
0.226
0.109
0.191
0.027
0.109
0.001
0.085
0.1
0.202
0.217
0.207
0.000
0.067
0.075
0.156
1.003
0.543
0.078
0.191
SAITS
8.3
0.173
0.276
0.151
0.225
0.04
013
0.29
0.109
0.106
0.266
022
0.204
0.111
0.089
0.122
0178
1.000
0.623
0.083
0.19
0.188
0064
0109
0.57
0.156
0,115
01216
0.225
0.205
0.138
0.033
0.245
0.733
1.007
0.31
0:084
0.188
0.7
0.343
0.395
0.287
0.341
0.044
0.207
0.061
0.164
0.137
0.237
0.234
0.21
0.159
0.107
0.223
0.261
1.006
0,833
0,085
0.167
Arg.
0.316
0.905
0.183
0.356
0.006
0.154
0.042
0.120
0.114
0.216
|
0.224
0.207
|
0.132
0.089
0.167
0.216
| 1.005
0.833
00003
0.180
0.008
0.233
0.087
0.25
0.043
0.14
0.263
0.074
0.174
0.188
0.278
0.316
0.000
0.067
0.092
0.156
0.049
0.41
10:45
0.141
CSDI
0.193
0.346
0.092
0.788
0.011
0.15
0049
0.076
0.5
11285
1.287
0.288
(110)
0.068
0:475
01417
0:009
0.301
0.161
0.066
0.269
0.126
02295
0.323
0.542
0.21
0.141
0.305
0.444
0.125
0.139
0:081
0.148
0.213
0.312
0.117
0.273
0.252
0.277
0.136
0.174
0.855
0.359
0.342
0.351
0.234
0.152
0:351
0.457
0.143
0.161
0:061
0.167
Aug.
0.761
0.101
0.177
0.158
0,113
0.533
0.300
0.306
0.304
0.158
0.107
0.156
0.1
0.003
0.:11
0.1
0.753
0.385
0.712
0.885
0.67
0.846
0.642
0.974
0.816
0.982
0.729
0.677
0.562
0:422
0.200
0.641
0,530
0.340
0.843
Sinkhor
0.5
0.878
0.645
0.361
0.525
09911
0.689
0.814
0.66
0.844
0.819
1.012
0.729
0.718
0.683
0.579
0.358
0.64
0:561
0.86
0881
0.5
0.91
0.609
0.82
0.614
0.95
0.708
0.905
0.678
0.992
01823
1.05
0.729
0.819
0.722
0.557
0.395
0.24
0.611
0.976
0.96
0.962
0.997
0.825
0.729
0.939
0.751
0.64
0.476
0.826
0.684
0.992
Arg.
0.877
0.68
0.807
0.601
0.994
0.698
0.891
0.67
0.986
0.81
1.012
0.739
0.799
0.672
0.540
0.362
0.723
0.500
0.966
0.855
0.1
0.941
0.722
0.932
0.71
0.994
0.748
0.971
0.731
0.808
0.775
0.913
0.68
0.992
0.583
0.763
0.352
0.886
0.745
0.87
0.8
TDM
1.005
0.956
0.251
0985
01942
0.93
0.29
0.933
0.896
0.98
0.661
0:387
0.601
0.206
022
0.915
01823
0.5
0.993
0.749
0.978
0.738
1,002
0.754
0.995
0.746
0.944
01801
0,961
0.704
0.833
0.712
0.862
0.622
0.957
1129
0.946
0.858
0.7
0,989
0.749
1.017
0.751
1.004
0.757
1.001
0.749
0.955
03822
0.976
0.715
0.905
0.798
0.552
0.651
0.981
0,812
0.969
03851
Avg.
0.982
0.742
0.901
0.731
0,999
0.752
0.988
0,942
0.919
10.99
0.948
0.698
(1865
0.689
0.341
0.600
| 0.000
0.776
| 0.905
01824
PWS-((OurS)
0.1
8,079
0.188
8.135
0.124
0.004
0.112
R.016
0.083
0.073
1L177
0.183
0.175
0.092
0.062
01,099
0.086
0.022
0,044
0.142
0.3
0.105
0.213
0.041
0.133
0.84
0.12
0.018
0.065
0.889
11.9%
0.173
0.186
0.098
0.066
0.856
0.100
0.023
0,046
11.14
0.508
0.204
0.107
0.072
0.17
0.114
0.032
0.049
077
0.194
0.259
8.059
0.065
0.065
0.1.5
0.029
0.11
0.148
0.239
0.26
0.237
0.131
0.088
0.114
0.17
8L039
0.833
0.056
0.159
Avg.
0.126
6,231
6.016
0.343
0,047
0.131
0.031
0.094
0.106
0.208
0.197
0.199
0.107
0.073
0.067
0.132
[U031
0.606
0.049
0.149
23

---

Published as a conference paper at ICLR 2025
Table 6: The standard deviation of Table 5,
Datreets
FTTh
RTTM
FTTml
RTTm?
Flearisty
Traffic
Wasther
Biacas
Fachunge
PFMS03
Metriss
MSE
MAE
MAB
MSB
MAE|
MSB
MAB
MSE
MAB
MSB
MNB
MSB
MAB
Transformoe
0.005
0.009
0.009
0.01
0.008
0.0
0.004
0.004
0.004
0.3
0,003
0.005
0.002
0.003
0.003
0.008
0.007
0.005
0.003
0.001
0.008
0.004
0.005
0.005
0.005
0.008
0.0
0,002
0.001
0.0
0.5
0.007
0.005
0.007
0.008
0.009
0.035
0.008
0.005
0.006
0.003
0.0
0.029
0.009
0.003
0.005
0.007
0.009
0.003
0.007
0.003
0.001
0.002
0.006
0.006
Awg.
0.004
0,004
0,005
0.005
0.007
0.007
0.006
0.005
0.004
0.002
0.005
0,006
0.005
0.005
0.004
0.005
0.004
0.002
0,005
0.002
0.1
0,000
0.007
0,002
0.008
0.01
0.003
0.007
0.009
0.00
0.001
0,006
0.006
0.001
0.001
0.006
0.001
0.008
0.004
0.002
D DLinew
Q3
0.005
0.008
0.005
0.004
0.001
0.007
0.003
0.009
0.01
0.002
0.007
0.005
0.005
0.000
0.007
0.006
0.005
0.013
0.5
0.008
0.003
0.004
0.007
0.01
0.001
0.005
0.002
0.004
00003
0.004
0.004
0.009
0.005
0.001
0.006
0.01
0.003
0.008
0.002
0,000
0.002
0.007
0.007
0.005
0.01
0.000
0.000
0.004
0.006
0.001
0.005
0.005
9.007
0,006
0.007
0.000
lavg.
0,007
0.004
0.004
0.006
0.009
0.004
0.003
0.005
0.007
0.000
0,005
0.005
0.005
0.004
0.004
0.004
00003
0,006
0,006
0.004
Tinessot
0.1
0,000
0.006
0,009
0.006
0:001
0.009
0.005
0.006
0.001
11009
1.004
0.01
0.005
0.002
0,000
0.007
0.000
0,006
0,005
0.009
0.004
0.007
0.005
0.007
0.006
0.006
0.002
0.01
0.005
0.009
0.007
0.000
0.001
0.004
0.002
0.008
0.005
0.005
0.002
0.0
0.001
0.01
0.003
0.004
0.007
0.002
0.003
0.001
0.005
0.000
0.000
0.007
0.005
0.007
0,000
0.01
0,002
0,005
0,005
0.002
0.002
0.005
0.002
0.005
0,002
0.001
0.004
0,002
0.01
0.003
0.000
0.004
0,001
0.004
AArg.
0,007
0.005
0.006
0,005
0,004
0.005
0.003
0.006
0.003
0.006
0.005
0.004
0.005
0.003
0.007
0.006
0.007
0,005
0,003
0.005
[..1
0,007
0.005
0.004
0,007
0,005
0.005
0.000
0.009
0,005
0.005
0,002
0.005
0.001
0.01
0.001
0.002
0.000
0,006
0,002
0.000
FroT'S
0.3
0.002
0.008
0.007
0.003
0.005
0.039
0.004
0.005
0.0
0.01
0.001
0.008
0.007
0.003
0.003
0.001
0.007
0.001
0.001
0.003
0.008
0.004
0.001
0.008
0.001
0.003
0.007
0.002
0.002
0.009
0.005
0.003
0.006
0.006
0.009
0.008
0.007
0.006
0.002
0.001
0.008
0.005
0.007
0.008
0.006
0.0
0.006
0.002
0.009
0.009
0.007
Avgg.
0,005
0.005
0.005
0.005
0.005
0.005
0.002
0.008
0.002
0.007
0.007
0.007
0.003
0.002
0.004
0.005
0,005
0.004
PatchTST
0.1
0.3
0,008
0.006
0,001
0.007
0.005
0.007
0.0
0.006
0.001
0.002
0.006
0.000
01000
0.001
0.009
0.001
0.000
0.0
0.007
0.0
0.004
0.01
0.000
0.006
0.00
0.01
0.006
0.004
0.009
0,005
0.004
0.009
0.5
0.003
0.008
0.003
0.003
0.005
0.004
0.006
0.001
0.001
0.001
0.004
0.007
0.001
0.005
0.002
0.006
0.000
0.007
0.008
0.003
0.003
0.004
0.003
0.004
0.006
0.009
|Avg.
0.006
0.006
0.004
0.005
0.003
0.004
0.003
0.004
0.005
0.003
0.005
0.005
0.003
0.004
0.006
0.002
0.007
0.006
0.005
0.1
0.004
0.
0.009
0.004
0.002
0.0
0.000
0.01
0.005
0.00
0.0
0.004
[.007
0.005
0.008
0.066
0.001
0.004
0,000
0.001
SCINet
0.008
0.007
0.009
0.006
0.0
0.002
0,000
0.006
0.01
0.002
0.005
0.002
0.000
0:007
0.006
0,001
0.005
0.004
0.005
0.001
0.004
0.001
0.0
0.006
0.008
0.006
0
0.009
0.001
0.005
0.006
0.001
0.003
0.004
0.007
0.001
0.006
0.001
0.007
0.005
0.009
0.00
0.004
0.001
0.001
0.000
0.009
0.0
0.005
0.003
0.006
0.000
0.006
0.01
0.009
0,0002
0.005
0,006
0.005
0.004
0.005
0.006
0.004
0,003
01.005
0,000
0,005
0.004
0.003
0,005
0.005
00004
0,005
0,000
0.006
Trrnsfocuee
0.1
01004
0.008
0.005
0.009
0:001
0.008
0.005
0.001
0.006
0.0
0.007
0.001
0.000
0.0
0,004
01000
010006
0.003
0,008
0.00
0,005
0,006
0.007
0.009
0.003
0.001
0.008
0.005
0.01
0.000
0.007
0.006
0.009
0.003
0.007
0.009
0.004
0.004
0.003
0.002
0.009
0.01
0.004
0.007
0,000
0.006
0.005
0.002
0.003
0.001
0.000
0:007
0.004
0.005
0.7
0:004
0.007
0.007
0.001
0:003
0.008
0.001
0.0044
0.007
0.01
0.005
0.0
0.005
0.003
0.004
01006
0006
0.005
0.01
0.008
Avg..
0,005
0.008
0.005
0,004
0,003
0.006
0.006
0.005
0.006
0.007
|
0.006
0.002
0.005
0.003
0.005
0.004
| 0.006
0,005
|
0,005
0.004
0.1
0,000
0.008
0.008
0,007
0,005
0.007
0.007
0.001
0,003
0.000
0.001
0.006
0.005
0.005
0.001
0.004
0.000
0,009
0,009
0.001
SAITS
0.3
0,007
0.009
0.005
0.009
0:007
0.006
0.092
0.007
0.01
1.005
0.004
0.006
0.005
0.003
0.007
0.007
00007
0.006
0.005
0.005
0.005
0.001
0.009
0.009
0.004
0.005
0.044
0.004
0.002
0.005
0.003
0.001
0.004
0,002
0.002
0.009
0.009
0.001
0.000
0.001
0.009
0.001
0.007
0.003
0.000
0.01
0.0
0,005
0.00
0.007
0.007
0.005
0,008
0:005
0.006
0.005
0.005
0.006
0.006
0,003
0.005
0.004
0.007
0.004
0.005
0.005
0,005
|
0,006
0.004
0,009
0,003
0.005
0.001
0.001
0.006
0.005
0.005
0.005
0.005
0.01
0.000
0:004
0.005
0.005
CSDI
03
0,002
0.01
0.005
0.007
0.005
0.002
0.007
0.003
0.001
0.0
0,003
0.004
0.009
0.004
0.002
0.004
0.002
0.009
0.001
0.009
0.005
0.001
0.011
100
0:009
0.009
0.00
0.008
0.001
0.001
0.005
0.008
0.002
0.004
0.004
0.006
0:0
0.008
0.002
0.0
0.004
0,001
0.006
0,005
0.006
0.001
0.006
0,004
0.009
0.002
0.005
0.000
0.005
0.001
0.000
Avg..
0.003
0.005
0.005
0.004
0.003
0.001
0.005
0.005
0.005
0.005
0.005
0.007
0.004
0.006
0.002
0.006
Sinkhoon
0.3
0.1
0,002
0.001
0.008
0.007
0.009
0.001
0003
0.005
0,002
0.002
0
0.006
0.004
0.07
0.003
0.006
0.003
0.002
0.009
0.006
0.002
0.009
0.001
0
0.006
0.001
0.005
0.01
0.01
0.0
0.002
0.01
0.000
0,009
0:001
0.008
0.009
0.001
0,002
0.5
01008
0.007
8002
001
0.005
0.003
8.00
0.008
0.009
0000
0.005
0.002
0.00
0.008
0.008
01006
00005
0.001
0.005
0.004
0.009
0.008
0.001
0.002
0.009
0.009
0.005
0.004
0.000
0.002
Avve_
0,003
0.006
0.005
0.005
0.004
0.005
0.004
0.005
0.003
0.003
0.005
0.003
0.005
0.005
0.006
0.005
0.007
0.004
0.007
0.002
0.1
0,003
0.006
0.007
0.009
0.0
0.007
0.004
0.003
0.006
0.001
0.002
0.002
0.008
0.001
0.002
0.007
00004
0.001
0,005
0,002
1DM
0.005
0.01
0.000
0.003
0.004
0.0
0.0
0.000
0.000
0.086
0.001
0.006
0.002
0004
0.008
0.007
0.009
0.003
0.005
0.0
0.004
0.005
0.007
0.008
0.004
0.002
0.005
0.007
0.07
9006
0.006
0.003
0.00
0.008
0.008
0.007
0.007
0.003
0.006
0.008
0.0
0.01
0.001
0.002
0.006
0.004
0.008
0.007
0.004
0.001
0.009
0.0
0.001
Arg.
0.004
0.005
0.006
0.007
0.003
0.006
0.005
0.003
0.008
0.004
0.004
0.004
0.005
0.004
0.006
0.007
0.003
0.004
0.003
0.003
(SSS)((Our))
0.1
0,008
0.004
0.007
0,002
0,001
0.000
0.003
0.003
0.008
0.005
0.007
0.005
0.005
0.01
0.005
0.005
0.0
0,002
0,008
0.009
0.3
0.004
0.008
0,005
0.002
0,009
0.009
0.004
0.009
0.002
0.001
0,602
0,004
0.004
0.303
0.007
0.004
0.001
0,005
0.009
0.0
0.003
0.007
0.002
0.002
0.003
0.0
0.009
0.006
0.007
0.005
0.005
0.003
0.006
0.004
0.006
0.001
0.001
0.003
0.009
8.006
0.002
0.002
0.000
0.007
0.005
0.002
0.001
0.007
0.006
0.009
0.004
0.009
0.00
Avrg
0.005
0.007
0.005
0.003
0.003
0.007
0.006
0.005
0.004
0.004
0.003
0.004
0.004
0.005
0.006
0.005
0.006
0.004
0.007
0.005
24

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Published as a conference paper at ICLR 2025
Table 7: Downstream task performance with iTransformee as the prediction model.
ETTh1
ETTb2
Exchange
Weather
Datasets
Transformer
0,389
0.406
0.299
0.385
0.093
0.215
0.177
0.218
DLincar
0,389
0.406
0.299
0.386
0.091
0.212
0.178
0.219
TimesNet
0.390
0.407
0.302
0.388
0.098
0.222
0.179
0.218
SCINet
0.389
0.406
0.296
0.383
0.098
0.222
0.179
0.219
ilransformer
0.389
0.406
0.293
0.381
0.093
0.215
0.179
0.219
0.381
0.176
CSDI
0.389
0.405
0.297
0.384
0.098
0.222
0.176
0,216
PSW-I(0urs)
0.387
0.401
0.279
0.371
0.091
0.212
0.175
0.215
25


Reviews:
[{"review_id": "gMp66D0j18", "reviewer": "Reviewer_fN5T", "summary": "Providing methods for time series imputations while respecting the temporal dependence.", "questions": "- What if the time series components are observed at different temporal frequencies (say days vs. hours, or hours vs. mins)?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- Very good empirical investigation.\n\n- Clear presentation.", "weaknesses": "NA", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lsiRafzbJ8", "reviewer": "Reviewer_8uNw", "summary": "The paper proposes a time series imputation method based on optimal transport. The key idea is the combination of a frequency-based Wassertein discrepancy  and selective matching regularization. Theoretical justification is also provided. The experimental results show the imputation accuracy outperforms many sota methods.", "questions": "see above.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. An interesting and well-motivated design of the spectral-enhanced Wasserstein distance (WD)\n2. A theoretical justified design of proximal spectral WD to account for non-stationarity.\n3. seemingly excellent performance in real-world benchmark datasets.", "weaknesses": "1. The biggest issue from my end is the lack of standard deviation. From Table 1, the error of the proposed method seems really good, but i am not informed if these results are averaged over multiple train/test runs or just one run. To avoid cherry picking, the authors are encouraged to highlight how these numbers were obtained, what the training/test splits were, and what hyperparameter selection/cross-validation process was involved, etc.  Similar expectations apply to table 3 and 4. \n\n2. Lack of convergence discussion/analysis. From Fig. 3 and Section 3.4, the imputation procedure seems to repeatedly sample patches from the time series, compute PSW, and use the gradient of the PSW to update the imputation. There seems a lack of convergence guarantees or discussions about this procedure. Can authors provide at least some discussion on this? \n\n3. Data noise issue. Real time series data often includes noises. That means, just computing the distance after DFT in lin154 might be affected by data noises. Have the authors consider any methods or trade-offs, such as low-pass filters, in your SWD definition, to improve the robustness and/or counteract the noise effect?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3v00I1T431", "reviewer": "Reviewer_4Wtk", "summary": "The paper proposes an optimal transport (OT) based time-series imputation method. The authors claim that naive application of OT does not work for time-series data. The proposed method consider applying OT in the frequency domain of the original data, called pairwise spectrum distance (PSD). Further, to deal with multiple modes, proximal spectral Wasserstein (PSW) distance is also proposed, in which mass constraint is removed to make transportation more flexible.", "questions": "- In 'Contributions', the authors mention that PSW-I eliminates the need for masking, but PSW-I seems to use masking (eg as shown in Fig3). What does this description mean?\n\n- What does Lemma 3.2 indicate? How do you know 'deviates more from the typical elements of \\beta'? Further, how do you know the PSW discrepancy avoid the problem indicated by this lemma? In Theorem C.2, the perturbation in PSW discrepancy is shown. Is it possible to compare this with Lemma 3.2? Even if it is possible, what does it mean? Is it clear how large (or small) perturbation caused by outliers ultimately affect the final imputation?\n\n- Although D_KL is used in (2), T 1_m and T^T 1_n is not a probability distribution (because the normalization (sum equals 1) constraint is removed). How are they normalized?\n\n- What is the definition of the 'DFT matrix' in the gradient? How does the gradient of (2) become \\Delta_alpha_i P and \\Delta_beta_j P? The second term in (2) disappear? The optimality condition wrt T is considered in this gradient calculation?  \n\n- After (1), D is squared distance, while in definition 3.1 D is distance without square. Which is correct?\n\nMinor issues:\n\n- In the end of p3: Fig. 4 should be Fig 1? In Fig 1(a), the left is W^(F)?\n\n- The first word of Sec3.2: 'time-series' -> 'Time-series'", "limitations": "", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Using OT to impute time-series data is an interesting approach.\n\n- Empirical evaluation shows high performance.", "weaknesses": "- Some technical justification is vague. Clearer descriptions would be desired.\n\n- Introduction is a bit too abstract about the proposed method. It describes what problem is solved in the paper, but does not describe the basic technical idea how it is achieved.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)