LSCD: Lomb--Scargle Conditioned Diffusion for Time series Imputation
Authors: Elizabeth Fons
Alejandro Sztrajman
Yousef El-Laham
Luciana Ferrer
Svitlana Vyetrenko
Manuela Veloso
Venue: ICML 2025 poster (2025)
Keywords: time series
diffusion models
frequency spectrum
Primary area: deep_learning->sequential_models_time_series
TL;DR: We propose Lomb–Scargle Conditioned Diffusion (LSCD), a diffusion-based time series imputation method that leverages a differentiable Lomb–Scargle periodogram to handle irregular sampling and preserve spectral consistency

Abstract: Time series with missing or irregularly sampled data are a persistent challenge in machine learning. Many methods operate on the frequency-domain, relying on the Fast Fourier Transform (FFT) which assumes uniform sampling, therefore requiring prior interpolation that can distort the spectra. To address this limitation, we introduce a differentiable Lomb--Scargle layer that enables a reliable computation of the power spectrum of irregularly sampled data.
We integrate this layer into a novel score-based diffusion model (LSCD) for time series imputation conditioned on the entire signal spectrum. 
Experiments on synthetic and real-world benchmarks demonstrate that our method recovers missing data more accurately than purely time-domain baselines, while simultaneously producing consistent frequency estimates. Crucially, our method can be easily integrated into learning frameworks, enabling broader adoption of spectral guidance in machine learning approaches involving incomplete or irregular data.

LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
Elizabeth Fons"
Alejandro Sztrajman" *2 Yousef El-Laham Luciana Ferrer 34 Svitlana Vyetrenko
Manuela Veloso"
Abstract
computation. However, such pre-processing can distort the underlying data distribution, leading spurious attenu- ated frequency estimates.
Time series with missing or irregularly sampled data are a persistent challenge in machine leamn ing. Many methods operate on the frequency- domain, relying on the Fast Fourier Transform (FFT) which assumes uniform sampling, therefore requiring prior interpolation that can distort the spectra. To address this limitation, we introduce differentiable I Lomb-Scargle layer that enables reliable computation of the power spectrum of irregularly sampled data. We integrate this layer into novel score-based diffusion model (LSCD) for time series imputation conditioned on the en- tire signal spectrum. Experiments on synthetic and real-world benchmarks demonstrate that our method recovers missing data more accurately than purely time-domain baselines, while simulta- neously producing consistent frequency estimates. Crucially, our method can be casily integrated into learning frameworks, enabling broader adop- tion of spectral guidance in machine learning ap- proaches involving incomplete or irregular data.
By contrast, the Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982) can estimate power spectra without assuming uniform sampling. does not require an imputation step to compute frequency components, making it far more reliable in the presence of irregular or missing data. Despite its advantages, Lomb Scargle remains under-explored in the broader machine learning community.
In this paper, we introduce a Lomb-Scargle Conditioned Dif- fusion (LSCD) approach for time series imputation. Inspired by score-bascd diffuusion methods (Song & Ermon, 2019), we develop a diffusion model that operates in the time do- main, but is conditioned on Lomb-Scargle-based spectral representation. By leveraging the full Lomb-Scarge peri- odogram an additional input, our model effectively cap- tures underlying frequency structures and imposes greater consistency between the final time-domain imputation and the original spectral content.
We evaluate our approach on both synthetic and real datasets, introducing varying degrees of missingness (up to 90%). We show that our conditioning on the Lomb-Scargle spectrum consistently improves time-series imputation metrics such as MAE and CRPS compared to purely time-domain baselines. Crucially, the method produces PSD estimates closer to the ground truth, demonstrating improved consistency of frequency recovery.
1. Introduction
Time series data often exhibit missing observations ir- regular sampling intervals, creating challenges for machine learning tasks such as imputation, forecasting, classification, and anomaly detection (Yang et al., 2024b). Many methods operate in the frequency domain (Alaa et al., 2021; Yang et al., 2024a; Wu et al., 2023; Crabbé et al., 2024) using the Fast Fourier Transform (FFT), which requires data to be uniformly sampled on a regular grid. To address this requirement, these methods typically handle missing val- ues by interpolating zero-filling the gaps prior to FFT
Our main contributions are:
1. We introduce a diffusiin-based time-series impu- tation framework that directly conditions on the Lomb-Scargle spectrum, thereby eliminating the need for interpolation zero-filling in the frequency do- main. To further enhance performance with high rates of missing data, we propose consistency loss that refines the alignment between the time-domain signal and its Lomb-Scargle representation.
Equal contribution 'J.P. Morgan AI Research University of Cambridge Uniiversity of Buenos Aires 'CONICET. Correspon- dence to: Elizabeth Fons <elizabeth. @ jpmorgan.com>>.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
We provide a differentiable implementation of Lomb- Scargle and show how it can be seamlessly integrated

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LSCD: Lomb-Scargle Conditioned Diffusson for Time series Imputation
into learning methods to handle missing or irregularly sampled data.
et al., 2023) introduces optimizable auxiliary variables in the dynamics for improved accuracy. Another approach considers diffusion process over function space to handle irregularity in time-series (Biloš et al., 2023), connected with neural processes (Garnelo et al., 2018). Naturally, ex- tensions based on neural SDEs have also been proposed, which incorporate a diffusion term in the latent dynam- ics (Kidger et al., 2020; El-Laham et al., 2025; Oh et al., 2024). Given that most neural ODE and SDE models re- quire computationally expensive numerical solvers at both training and inference time, efforts have focused on develop- ing efficient alternatives. Continuous recurrent units (CRUs) (Schirmer et al., 2022) model hidden states using linear SDEs solvable analytically with a Kalman filter, offering efficient compute times. Neural flows (Biloš et al., 2021) learn the solution path of the ODE directly, bypassing the need for a solver. Attention-based methods like multi-time attention networks (MANs) (Shukla & Marlin, 2021) use specialized embedding modules to map irregular and sparse observations to continuous time representations. Graph- based methods such as GraFITi (Yalavarthi et al., 2024) convert irregularly sampled time series into sparse structure graphs, reformulating forecasting as edge weight prediction using graph neural networks (GNNs). Transformable patch- ing graph neural networks (T-PATCHGNN) (Zhang et al., 2024) transform each univariate irregular time series into transformable patches with uniform temporal resolution.
3. We provide comprehensive empirical results showing improved imputation performance and accurate fre- quency recovery on both synthetic and real datasets.
Beyond time series imputation, we hope our work encour- ages broader use of Lomb-Scargle- techniques for irregular data within machine learning pipelines.
2. Related Work
Time series imputation has been extensively studied across statistical, deep learning, and spectral approaches. Early statistical methods often rely on assumptions of local con- tinuity or cross-variable dependencies. Simple heuristics, such as mean or median imputation (Fung, 2006; Batista & Monard, 2002), are straightforward but fail to capture complex temporal dynamics. More advanced techniques, including Expectation- Sopeetation-MMxxmizatiinn (EM) (Ghahramani & Jordan, 1993; Nelwamondo et al., 2007), auto-regressive models, and state-space approaches (Durbin & Koopman, 2012; Walter.O et al., 2013), incorporate structured depen- dencies but can struggle with high rates of missingness and irregular sampling. Gaussian Processes (GPs) (Fortuin et al., 2020) provide uncertainty-awwn -wware imputation yet face scala- bility hurdles and often assume smooth temporal trends.
Spectral methods have been explored to capture periodic be- haviors in time series. TimesNet (Wu et al., 2023) and other Fourier-based models (Alaa et al., 2021; Crabbé et al., 2024) leverage the Discrete Fourier Transform (DFT) to extract frequency components, improving long-term forecasting and imputation. Additionally, Fons et al. (2022; 2024) intro- duced a Fourier loss to better preserve frequency structure in learning, aligning time-domain reconstructions with their spectral representations. However, like other FFT-based methods, these approaches still require uniform sampling or pre-interpolation and thus can distort frequency estimates under high missingness or irregular sampling (VanderPlas, 2018). By contrast, the Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982) computes frequency spectra directly from unevenly sampled data, sidestepping the need for in- terpolation. Despite its advantages, underutilized in machine learning, with most of its applica- tions still confined to astronomy and signal processing.
Deep learning methods have significantly advanced time series imputation by learning long-range dependencies and complex temporal patterns. RNN-based models such as BRITS (Cao et al., 2018) and GRU-D (Che et al., 2016) incorporate explicit modeling of missingness, while Transformer-basee methods like SAITS (Du al., 2023) can better capture long-range interactions. Generative archi- tectures, including GANs (Miao et al., 2021) and VAEs (For- tuin et al., 2020), allow sampling-base based imputation under high uncertainty. More recently, diffusion models (Tashiro et al., 2021) have shown promising results by iteratively refining noisy samples, but they remain confined time- domain representations and typically overlook the spectral properties of irregularly sampled series (Yang et al., 2024b).
Several approaches have been developed to handle irreg- ular time-series data without requiring pre-defined fixed- length windows. MADS (Bamford et al., 2023) proposes an auto-decoding framework built upon implicit neural repre- sentations. Latent ODEs (Rubanova et al., 2019) define a neural ODE in latent space, solved using numerical methods like the Euler method, naturally accommodating irregularly sampled observations for standard time-series tasks such as imputation and generation. Extensions of latent ODEs, such as the GRU-ODE- GRU-ODE-Bayes model (Brouwer et al., 2019), handle sporadic observations with jumps, while the neural aatinuous-discret state space model (NCDSSM) (Ansari
remains
While prior work has explored both time-domain diffusion- based imputation (Tashiro et al., 2021) and frequency-based representations (Wu et al., 2023; Yang et al., 2024a), our ap- proach is the first to integrate a differentiable Lomb Scargle estimator into the diffusion process. By conditioning time- domain diffusion on Lomb Somb-Scargle-deriver spectral infor- mation, we avoid spectral distortions introduced by uniform resampling. This enables more faithful recovery of underly-
2

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LSCD: Lomb-Scargle Conditioned Diffussin for Time series Imputation
ing frequency structures, ensuring that imputed time-series data align with their true spectral content.
3.2. Lomb-Scargle Periodogram
Let S { St} denote set of time indices and x = [ ] R' denote the observed values defining potentially irregular time series. We define frequency grid w = (ui.....w)) of interest, where each W/ = 2 denotes the corresponding angular frequency (in radians) of the jth frequency component fz. For each frequency component w, Lomb-Scargle Scargle approximates the power spectral density (PSD) P(w) by fitting sinusoids directly to the observed data, circumventing the need to fill in missing points:
3. Background and Motivation
In this section, we present the mathematical foundations of Lomb-Scargle (Lomb, 1976; Scargle, 1982) and demon- strate its effectiveness as robust alternative traditional FFT-based frequency analysis when handling missing data. Additionally, in Sec. 3.3, summarize the conditional diffusion framework for time series imputation introduced by Tashiro et al. (2021).
P(w) = (En [,,, -2 coofw).1) E, cos (wos)
(E, Ex, - Σ; sin [
3.1. FFT vs Lomb-Scargle
Time series data are frequently subject to missing or irregularly spaced observations, caused by sensor fail- ures, asynchronous measurements, or domain-specific con- straints (Yang et al., 2024b). Although the Fast Fourier Transform (FFT) a powerful tool for frequency analysis, implicitly assumes uniform sampling. When data points missing, common approaches include using interpola- tion or zero-filling to address the missing observations. Both strategies can distort the true frequency content, leading to spurious or attenuated peaks (VanderPlas, 2018). This is especially problematic at high missingness levels, where interpolation becomes highly unreliable.
with ā ž= - Ts. denotes the sample mean of the ob- served values, and Φ; - 8; - T, where T is a time shift introduced to ensure the periodogram is invariant to time translations defined as:
Σ sin sisnniwss (2ws.)
Lan
2w
cos((wss)
This shift aligns the cosine and sine terms optimally with the observed data, ensuring robust spectral estimates. Unlike the FFT, no uniform grid or zero-filling is required, making Lomb-Scann Scargle particularly effective for irregularly sampled time series. We provide a basic review on the derivation of the Lomb-Scargle periodogram in Appendix A.
Fully observed TS
TS with missing data
False Alarm Probability. It can be shown that under the assumption of additive standard Gaussian noise, P(w) fol- lows x² distribution with 2 degrees of freedom. This fact can be exploited to approximate the false alarm probabil- ity (FAP) for each frequency component Wk. denoted by PFA (wx), using the following expression:
1.5
Donsity
1.0-
LS
FFT
15
FFT
0.5
0.0
Frequency
Frequency
PFA(W) = 1- exp (-))
(I)
Figure 1: Density of leading frequencies for FFT and Lomb- Scargle (LS) on a synthetic sine dataset. (Left) Fully ob- served time series. (Right) Time series with 75% missing data. The interpolation required by FFT significantly dis- torts the spectral distribution, whereas LS better preserves the original frequency structure.
where Joif denotes an effective number of independent fre- quencies, which can be estimated using heuristics in practice. The FAP can be used to detect and filter out spurious fre- quency components in the periodogram. Using the FAP, we can define the weighting function as:
w(wx) = PFA(W)) +
(2)
This is illustrated in Figure 1, where we show a comparison of FFT vs Lomb- Scargle applied to a synthetic dataset of sine waves with known frequenices. Under fully observed conditions, both FFT and Lomb Lomb-Scargle reliably recover the ground-truth leading frequencies. However, once we artifi- cially remove data (75% missing at random), the FFT with linear interpolation yields shifted or spurious peaks, whereas Lomb- Lomb-Scargle remains well-aligned with the ground truth frequencies.
where e is a small constant to prevent division by zero. This weighting scheme ensures that frequencies with lower FAP (i.c., more significant frequencies) have a greater impact on the loss.
3.3. Conditional Diff'usio for Time Series Imputation
We summarize the conditional diffusion framework for time series imputation (Tashiro et al., 2021). Let us consider
3

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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
Neise
Predicted Neise
Imputed Time Series X
MS +
Conditional Mask 1I1 20
Imputalicn Targel Xg
Natey Imputation Target
m
Denuising Network
Lomb-Scargle
Pesitional Encoder
Observed Condition X
Lomb-Scargle Layer
Spectrum Encoder
(X, s]
Lomb--Saangee Spectrum (S()))
(XS)) (S188)))
Prod Specirum (XS))
Figure 2: Diagram of our Lomb-Scargee Conditioned Diffusion (LSCD) approach for time series imputation.
N multivariate time series, each denoted by X,M,S), where X E k corresponds to the time series values with features over L time steps, and s = fs...... } contains the timestamps. We assume that some entries of X are missing, defined by the observation mask M E
In practice (Ho et al., 2020; Song et al., 2021), is param- eterized via a denoising network ea (x x t0 xxxx)
3e
no (x{* t x
c
at
V
a
(0, 1) 1) Our goal is to model and then sample from
1-a1
Br
t >> 1
1-ae
the conditional distribution px(x xf"). where x5" M ④ X corresponds to the conditional observations, and to = (1- M) O X to the imputation targets, i.e., the
B1
1=1
The parameters are learned by matching the predicted noise 6ø to the true noise 6. We minimize
Training.
missing entries that need to be imputed, with @ denoting element-wise multiplication.
(()) B[Il co(x , 13]
Forward Process. To learn this conditional distribution using score-based diffusion model, we adopt a forward process that gradually adds noise to the target portion x Specifically, for diffusion steps i. 1,2,
where x²" is sampled from the forward process. In prac- tice, during training a conditional mask m E (0, Kj is introduced to artificially split the observed values into x = m C X and x = (M - m²)) @ X, in order to train the conditional denoising function. At inference time (imputation),
qxx]0 x(=))== = =N NN Bixte 11 81)
generate samples of x/e by initializing x with random Gaussian noise and iteratively applying the learned reverse transitions, while conditioning on the observed data xfi
where {B:} is a noise schedule. Under suitable definitions of at = II B²), we have the closed-form
4. Methodology: Spectrum-Condttionel Diffusion
x = /ou x + are e~ NOO.)
The reverse process, parameterized by 8, attempts to denoise xte step by step back xf". while being conditioned on x6 Hence define
Reverse Process.
We now present our Lomb-Scargle Conditioned Diffusion (LSCD) framework for time series imputation, illustrated in Fig. 2. Building on the conditional diffusion approach in Sec. 3.3, we introduce two crucial forms of frequency-domain supervision. At each denoising step, Lomb Searbb--rrrebbbrrre based spectral repre- sentation is learned and provided as conditional information. Furthermore, after the main diffusion training, we incor- porate a spectral alignment term that encourages the final imputed time series to match the Lomb-Scamt spectrum extracted from the observed data.
Comb-Scargle layer that supplies
po(2 ............ ,..., x6²) = p(x I po (xte x x
where
Pol pa(211 10 x00 _xxx21;000xx =

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LSCD: Lomb-Scargle Conditioned Difffusion for Time series Imputation
4.1. Diffusion with Lomb-Scargle Conditioning
Random Missing
Subsequence
Block Missing
Ronse 10
Rovere 3D
Given partially observed time series Xo defined by {X, M, S}, we divide the observed values in two parts, according to a randomly generated conditional mask m
Seop
Step
Figure 3: Visualization of the three missingness mechanisms used in our study.
00 X
= (M- m )
thus defining an imputation target x5 (i.e. artificially gener- ated missing entries) and an observation condition x6 of ob- served values. Following Sec. 3.2, we define [S(xx") E R' as the Lomb-Scargle periodogram of x0" evaluated over a frequency grid w = {wi...... ...................... The spectrum is filtered via False Alarm Probability (FAP), as detailed in Eq. 1, and we additionally apply a log transform and normalization.
with X Ro m where Xo is the fully reconstructed time series (i.e. observed plus imputed parts), obtained by following the backward denoising process from = T to = 0. The pipeline to compute Lscons is illustrated in dashed lines in Fig. 2. This term penalizes discrepancies in the Lomb Lomb-Scargle periodograms, ensuring that the learned reconstruction not only fits the observed data in the time domain, but also preserves the essential frequency structure.
Forward Process. We follow the usual diffusion setup, wherein at each diffusion step = I...., T the imputation target x is perturbed according to
la Voxxt
La + -1-00 Et,
E, ~O,I).
5. Experiments
with {ar} a noise schedule.
In this section, we evaluate LSCD on both synthetic and real-world datasets. We focus on time series imputation under varying degrees of missingness (e.g.. 10%, 50%, 90% missing data). We also include ablation studies to isolate the impact of key components such Lomb-Scargle condition- ing, the spectral encoder Expecs and the spectral consistency loss Lscons. We outline the datasets, baselines, metrics, and ablation protocols below.
Backward (Denoising) Process. A denoising network co(x"", x (S(x)")) is trained to invert the forward noising, predicting the noise 6, given both the noisy time- domain data xt" and the fixed spectral representation LS (x,"). The final sampling procedure iterates the re- verse transitions x + _ conditioned on xi and (x("), until an imputed x is obtained.
5.1. Experimental Setup
4.2. Attention-Basee Spectrum Encoder
5.1.1. DATASETS
Our approach leverages all frequency components from the spectrum as a conditioning signal for diffusion. To that end, we incorporate a spectral encoder:
Synthetic Sine Waves. We create controlled dataset of sine waves with known frequencies (f1, f2....) and various amplitudes. To make the setting more realistic, the dataset is initially generated with 10% missing data. We evaluate our method using three missingness mechanisms: MCAR, sequence missing, and block missing. MCAR introduces missing values randomly, independent of the data itself, serving baseline. Sequence missing creates contiguous gaps in time, simulating sensor downtimes or transmission failures. Block missing removes entire sections of data, mimicking large-scale outages. These mechanisms test the model's ability to handle both short-term and long-term dependencies. Each missingness mechanism applied at three levels (10%, 50% and 100%) on top of the initial missing data to assess the robustness of our method under increasing data loss. Figure 3 visualizes these missingness patterns, illustrating their real-world relevance, from random noise to structured data loss. Additional details on dataset generation and missingness parameters can be found in Appendix D.
2,5 Expec (CS(xx)))))))
composed of two multi-head self-attention layers, designed to encode inter-frequenc and inter-feature dependencies. The resulting embedded representation of the spectrum 2s then incorporated into each step of the denoising process. This allows the network to exploit rich frequency-domat cues while reconstructing the time-domain signal. We refer the reader to Appendix E.2 for additional details on the encoder and denoising network architectures.
4.3. Spectral Consistency Loss
To reinforce alignment between the final imputed time series and the observed spectrum, we perform a final refinement phase during training, incorporating a spectral consistency loss:
LSCam = [|())xx]) LS(2))) I|?
5

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LSCD: Lomb-Scargle Conditioned Difffusion for Time series Imputation
Real datasets We conduct experiments on two real- world datasets with missing data. The first dataset, Phy- sioNet (Silva et al., 2012), comprises 4,000 health measure- ments from ICU patients, covering 35 features. Following the standard pre-processing procedure (Cao et al., 2018; Tashiro et al., 2021), each measurement is processed hourly, resulting in time series with 48 time steps. The prepro- cessed dataset contains approximately 80% missing values. As there is no ground truth for missing data, we randomly select 10%, 50%, and 90% of the observed values to serve as ground truth for the test set. The second dataset consists of PM2.5 air quality measurements collected from 36 sta- tions in Beijing over a 12-month period (Yi et al., 2016). Following previous work (Cao et al., 2018; Tashiro et al., 2021), we construct time series with 36 consecutive time steps. This dataset exhibits approximately 13% missing data, with non-random missingness pattemss and an artificial ground truth containing structured missingness.
5.3. Results on Synthetic Sine Dataset
Table 1 presents the imputation performance of various methods on the Sines dataset, across different missingness mechanisms and levels. Our proposed method consistently achieves the best overall performance, particularly excelling in spectral reconstruction (S-MAB). For MCAR data, our signisicaniii
method significantly outperforms CSDI for all levels of missingness, showcasing the effectiveness of our approach in handling random data loss. Notably, CSDI consistently generates results that have good agreement in S-MAF, par- ticularly in sequence and block missing cases, indicating that its probabilistic modeling helps retain spectral proper- ties. SAITS demonstrates competitive performance in time- domain imputation (MAE) for some cases (e.g., sequence missing at 10%). but consistently exhibits poor spectral re- construction (S-MAE). This highlights that SAITS, while effective in the time domain, fails to capture the underlying frequency structure of the data.
5.1.2. BASELINES
Figure 4 evaluates the spectral consistency of different mod- cls on the Sines dataset with 50% missing data at random. The top row compares the distribution of leading frequen- cies between the ground truth (GT) and predicted (Pred) time series. Our model closely follows the GT distribu- tion, while SAITS and TimesNet exhibit broader spreads, and TimesNet shows more pronouneed deviations, partic- ularly around peaks at higher frequencies. This suggests that our model better preserves the dominant spectral char- acteristics. The bottom row presents the PSD difference PSD - (SD(P)e)) where the black line represents the mean difference, and the shaded area indicates one stan- dard deviation across samples. We observe that our model maintains difference centered around zero with minimal variance, indicating accurate spectral reconstruction. In con- trast, SAITS and CSDI exhibit larger variability, suggesting inconsistent frequency estimation, while TimesNet shows a systematic bias, particularly in higher frequency ranges. These results further exhibit the effectiveness of our method in preserving the spectral properties of time series, ensuring more stable and reliable frequency reconstruction.
We compare our model against different imputation meth- ods, including classical statistic-based methods Mean and Lerp, deep learning based BRITS (Cao et al., 2018), GP- VAE (Fortuin al., 2020), US-GAN (Miao et al., 2021), TimesNet (Wu et al., 2023), SAITS (Du et al., 2023),
CSDI (Tashiro et al., 2021) and foundation model for time series, ModernTCN (Donghao & Xue, 2024).
5.2. Evaluation Metrics
We evaluate the performance of our models using a com- bination of time-domain and frequency-domen domain metrics to comprehensively assess both the accuracy of imputed val- ues and the preservation of the underlying spectrum. In the time domain, we utilize Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) to quantify the ac- curacy of the imputed values, providing measures of both average absolute deviation and squared error sensitivity to larger discrepancies. To assess the fidelity of our models in capturing the spectral properties of the time series, we calculate the Spectral Mean Absolute Error (S-MAE) be- tween the estimated Power Spectral Density (PSD) of the imputed signal and the ground-trutt PSD. The spectrum is computed using both observed (conditional) and imputed values, while missing values in the original time series are masked to ensure a fair comparison. To focus on relative differences in spectral shape, we normalize both PSDs so their total power sums to one before computing S-MAE as the mean absolute difference between them. This met- ric quantifies how well the model preserves the frequency characteristics of the time series.
5.4. Results on Real Data
To assess the performance of our spectrum-conditiioned diffusion approach, we conduct experiments on two diverse real-world time-series datasets-Phyye PM2.5-unde multiple missing-data scenarios. Table 2 summarizes the results, where we compare against a wide range of basclines. We repeat experiments five times and report the averaged result. We first consider Physionet with three different missing rates: 10%, 50%, and 90%. Across all these scenarios, our method consistently achieves the best besr-best time-domain results (MAE and RMSE) while also exhibiting competitive S-MAE scores. Notably,
and
6

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LSCD: Lomb- Conditioned Diffusion for Time series Imputation
Table 1: Imputation performance of time series reconstruction and frequency spectrum over sines datasets.
TYPE
METRIC
MEAN
LERP
BRITS
GPVAE
US-GAN
TIMESNET
CSDI
SAITS
MODERNTCN
OURS
POINT
0.1
MAE
1.380
1.305
0.943
1.399
0.933
1.220
1.336
0.885
0.973
0.765
RMSE
1.947
1.991
1.657
1,986
1.636
1.803
1,889
1.569
1.727
1.453
S-MAE
0.081
0.081
0.052
0.082
0.053
0.069
0.008
0.043
0.049
0.003
0.5
MAE
1.373
1.449
1.095
1.383
1.152
1,481
1.359
1.041
1.129
0.975
RMSE
1.930
2.070
1.759
1,950
1.845
2.017
1.922
1.699
1.817
1.658
S-MAE
0.264
0.324
0.170
0.266
0.191
0,239
0.027
0.159
0.173
0.014
0.9
MAE
1.375
1.586
1.320
1.377
1.369
1.579
1.361
1.292
1.360
1.271
RMSE
1.935
2.142
1.899
1,938
1.970
2.143
1.925
1.878
1.963
1,870
S-MAR
0.439
0.572
0.383
0.439
0.407
0.462
0.044
0.375
0.406
0.036
SEQ
0.1
MAE
1.353
1.542
1.330
1,355
1.384
1.391
1.413
1.323
1.329
1.359
RMSE
1.905
2.092
1.915
1.908
1.995
1.959
1.988
1.890
1.931
1.962
S-MAE
0.055
0.075
0.056
0.054
0.061
0.062
0.006
0.055
0.056
0.005
0.5
MAE
1.374
1.564
1.347
1.376
1.393
1.467
1.378
1.342
1.354
1.316
RMSE
1.934
2.115
1.928
1.936
1.999
2.038
1.943
1.917
1.960
1.913
S-MAE
0.271
0.369
0.269
0.271
0.297
0.321
0.028
0.268
0,277
0.026
0.9
MAE
1.386
1.573
1.362
1.388
1,403
1,489
1.372
1.352
1.375
1.313
RMSE
1.946
2.127
1.941
1.949
2.007
2.062
1.943
1.929
1.982
1.913
S-MAR
0.288
0.389
0.286
0.288
0.305
0.338
0.029
0.283
0.292
0.027
BLOCK
0.1
MAE
1.306
1.507
1.255
1.309
1.334
1.379
1.304
1.268
1,275
1.259
RMSE
1.807
2.014
1.786
1.811
1.885
1,898
1.804
1.785
1,825
1.774
S-MAR
0.105
0.146
0.100
0. 104
0.116
0.124
0.011
0.103
0.106
0.010
0.5
MAE
1.306
1.505
1.279
1.308
1.333
1,451
1.314
1.285
1.309
1.269
RMSE
1.815
2.014
1.806
1.817
1.881
1.978
1.835
1.804
1,852
1.810
S-MAE
0.287
0.383
0.278
0.286
0.306
0.344
0.029
0.285
0.296
0.027
0.9
MAE
1.339
1.523
1.319
1,340
1.359
1,506
1,329
1.320
1,356
1.320
RMSE
1.868
2.052
1.862
1,869
1.927
2.054
1.874
1,859
1.916
1.870
S-MAE
0.359
0.473
0.351
0.358
0.376
0.439
0.036
0.358
0.374
0.035
Pred
Pred
0.6-
Decatty
Decdity
Deasity
0.4
02:
0.2
0.0
0.0
X352
Count
Lead Prequency
Laa Prequency
Laad Prequency
Land Prequency
1.00
3.28
3.50
1.50
(9D0Pra))
PSDICT
1SD0T:
0.75
6.75
0.75
0.75
p4p/00 (3bitee)
Focqucayy
Peaquein
(a) Ours
(b) SAITS
(c) ModertTCN
(d) TimesNet
Figure 4: (Top) Distribution of leading frequency the Sines dataset (Missing Point 50%). (Bottom) Difference between the power spectral densities (PSD) of the ground truth and predictions, with the black line representing the mean difference and the shaded area indicating one standard deviation. Our approach shows the best performance.
at the lower 10% missing rate, our approach yields the low- est MAE and RMSE, improving upon the already strong baseline CSDI. For the spectral metric (S-MAF), we again outperform all alternatives, demonstrating more accurate frequency-dome domain reconstruction. At 50% and 90% miss- ing rates, approach maintains the lowest scores in all
metrics, with S-MAE sometimes on par with CSDI. These results suggest that explicit conditioning on frequency com- ponents helps maintain signal characteristics even under high data sparsity. In the PM2.5 pollution dataset, our model reports significant gains in MAE and RMSE compared to the baselines. Additionally, also observe lower S-MAE

---

LSCD: Lomb-Scargh Conditioned Diffusion for Time series Imputation
Table 2: Imputation performance of time series reconstruction and frequency spectrum over real datasets.
DATASET
MISS
METRIC
MEAN
LERP
BRITS
GPVAE
US-GAN
TIMESNET
CSDI
SAITS
MODERNTCN
OURS
PHYSIONET
10%
MAE
0.714
0.372
0.278
0,469
0.323
0.375
0.219
0.232
0.351
0.211
RMSE
1.035
0.708
0.693
0.783
0.662
0.690
0.545
0.583
0.697
0.494
S-MAR
0.032
0.020
0.016
0.020
0.022
0.013
0.014
0.012
50%
MAE
0.711
0.417
0.385
0.521
0.453
0.307
0.315
0,440
0.303
RMSE
1.091
0.840
0.833
0.907
0.852
0.840
0.672
0.735
0.803
0.664
S S-MAE
0.111
0.087
0.064
0.083
0.076
0.076
0.052
0.055
0.071
0.052
90%
MAE
0.710
0.565
0.560
0,642
0.670
0,642
0,481
0.565
0,647
0.479
RMSE
1.097
0.993
0.975
1.038
1.060
1.031
0.834
0.971
1.026
0.832
S-MAE
0. 0.148
0.189
0,104
0.124
0.125
0.131
0,093
0.108
0.137
0.093
PM25
10%
MAR
50.685
15.363
16.519
23.941
32.999
22,685
9.670
15.424
24.089
9.069
RMSE
66.558
27.658
26.775
40.586
48.951
39.336
19.093
30.558
40.052
17.914
S-MAE
0.135
0.039
0.039
0.060
0.080
0.056
0.023
0.034
0.059
0.022
1.4
1.4
2.5
CT
GT
1.2
Pred
Pred
Pred
2.0
Pred
1.0-
1.0-
0.5
0.2-
Count
1.3
Count
1.5
0.c-
1.0
0.2
0.2-
0.2
0.5
Data value
Data value
Data value
Data value
3.5
GT
GT
2.5
Pred
255
Pred
2.-
Pred
2.0
Pred
8.54
2.8
Coaant
Count
3.5
1.8
1.0
1.8
6.5
0.-
Data valuue
Daca Value
Data valuue
Data value
(a) Ours
(b) CSDI
(c) SAITS
(d) BRITS
Figure 5: Distribution of imputed values across different models on real datasets. (Top) Physionet dataset with 10% missing
data; (Bottom) PM25 dataset.
than competing methods, indicating stronger preservation of the signal's 's spectral structure. Finally, Figure 5 shows comparison between the imputed values and the observed ones for Physionet (top) and PM25 (bottom). In Physionet, CSDI and our method capture the general distribution shape, but LSCD has a slight advantage for larger values, while SAITS and BRITS show a concentration in lower values, without covering the tails of the distribution. For PM25, both CSDI and our method show similar performance, but method covers more accurately larger values. These periments underscore the advantage of using Lomb-Scargle- based spectrum conditioning within a diffusion framework. By circumventing the need to fill in missing points before spectral analysis, our approach better preserves the time series' inherent frequency characteristics. As evidenced by strong results on both Physionet and PM2.5, frequency- aware conditioning leads not only to lower time-domain emrors but also to more accurate frequency reconstructions.
5.5. Ablation Studies
To validate the contribution of each component in our LSCD framework, we conduct ablation studies on the Physionet and PM25 datasets, progressively removing key elements. Results are summarized in Table 3. The full model achieves the best performance across all settings, demonstrating the effectiveness of Lomb-Scargle conditioning (LS), spectral encoder (Espes), and spectral consistency loss (Lscoms). Re- moving LsCons marginally increases MAE, highlighting its role aligning imputations with spectral properties. Further removing Expec degrades performance more significantly, emphasizing the importance of encoding inter-frequency dependencies. Removing LS conditioning entirely causes the largest degradation, underscoring the necessity of direct spectral supervision. Notably, S-MAE remains stable at high missing rates. These results confirm that all compo- nents contribute accurate time-domain imputation while preserving spectral fidelity.
8

---

LSCD: Lomb- Conditioned Diffusion for Time series Imputation
Table 3: Ablation of LSCD in Physionet and PM2.5.
METHOD
10% MIsS.
50%
Miss.
90%
MISs.
PM25
MAE
S-MAE
MAE
S-MAE
MAE
S-MAE
MAE
S-MAE
OURS
0.211
0.012
0.303
0.052
0.479
0.093
9.069
0.0219
W/o LsCons
0.213
0.012
0.303
0.052
0.480
0.093
9.085
0,0223
W/O LsConss Eove
0.218
0.013
0.304
0.052
0,480
0.093
9.334
0.0222
W/o LsCons. Earee LS
0.219
0.013
0.307
0.052
0.481
0.093
9.669
0.0239
6. Limitations
age further exploration of spectral-guided learning methods.
Lomb-Scargee natively supports irregularly sampled data. However, our LSCD architecture for time series imputation works with grid-based data and thus our differentiable imple- mentation of Lomb-Scargle is adapted for this type of input. While bosre-baed diffusion models such as CSDI (Tashiro et al., 2021) and LSCD rely on a fixed time grid, they are able to operate on irregularly sampled time series, provided that the interpolation time points are known at training time. In contrast, continuous-time methods only require knowl- edge of the interpolation time points at inference time.
Impact Statement
Our work focuses on improving time series imputation. Po- tential applications include healthcare (e.g., filling gaps in patient vitals), climate modeling (handling sparse sensor readings), and finance (dealing with stock transactions). By restoring continuity and enhancing frequency fidelity, enable more reliable downstream analyses, which can lead to better decision-making in critical domains.
A Acknowledgments
7. Conclusions
This paper was prepared for informational purposes in part by the Artificial Intelligence Research group of JPMorgan Chase & Co. and its affiliates ("JP Morgan") and is not product of the Research Department of JP Morgan. JP Morgan makes no representation and warranty whatsoever and disclaims all liability. for the completeness, accuracy or reliability of the information contained herein. This docu- ment is not intended as investment research or investment advice, recommendation, offer or solicitation for the purchase or sale of any security, financial instrument, fi- nancial product or service, or to be used in any way for evaluating the merits of participating in any transaction, and shall not constitute a solicitation under any jurisdiction or to any person, such solicitation under such jurisdiction or to such person would be unlawful.
We presented Seabb-e Conditioned Diffission (LSCD), a novel approach for time series imputation that leverages a diffusion model conditioned on the Lomb-Scan Saabll peri- odogram. Our method directly incorporates spectral infor- mation within the diffusion framework, ensuring that both time-domain and frequency-doman characteristics of the data are preserved.
Our experimental evaluation on synthetic and real-world datasets demonstrated that LSCD consistently outperforms existing imputation methods in both time-domain accuracy (MAE, RMSE) and spectral consistency (S-MAE). Notably, our method maintains superior performance even under ex- treme missingness (up to 90%), demonstrating its robustness in practical scenarios.
Our approach employs a spectral encoder to process the Lomb-Scargle spectrum and a spectral consistency loss to reinforce alignment between imputed signals and their fre- quency representations. Our ablation studies confirmed that each of these components contributes significantly to the overall effectiveness of our method. The spectral encoder captures inter-frequency and inter-feature dependencies, en- hancing the model's 's ability to utilize spectral cues, while the spectral consistency loss ensures that frequency components are faithfully reconstructed in the imputation.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
A. Lomb-Scargle Discussion
The Lomb-Scargle periodogram provides a spectral representation of a signal. It is typically used to understand the power spectrum of an irregularly sampled time series sample, typically in the context astronomy data. Consider a univariate signal x ( Ity), which is assumed be irregularly sampled at times .1...., ig. For given frequency f. the idea behind the Lomb-Scargle periodogram is to fit x to sinusoid using least-squares. Equivalently, assume that at each time L,, the signal can be modeled as a noisy sinusoid:
= A fl + ④) + +t..
where e, NN(0,, o²) ) assumed be white Gaussian noise. Using trigonometric identity, the model can equivalently be represented as:
Ft an ft, 0g f2m.) Ct.
(4)
where the amplitude and phase @ can be related to the parameters a and a2 as follows:
A = V af + az
(5)
022
Φ = tan
(6)
a1
The model formulation in (4) is convenient, because it can be expressed as a linear model with additive Gaussian noise:
I'i
(os(22ftt1)
sin(2
Q1
:
-,,
02
cos(2nf(x)
where e ~ o(0,0²IN) The maximum likelihood estimator of θ is equivalent to the least-squares solution and is given by:
= H HT
8
The above equation yields solution for the estimators ã and ã. which can readily be converted into the original parameters A and @ as:
En tt,, cos(2 ((t (0 -b))]
EnN Ti, f ((--n(x)) ())) -5))
(9)
Σm=1 cos? (2T (In - x))
Σm=1 sin (TTTTT2// - +))
ΣN sin sin(44 fta)
tan
(10)
4m
DA cos( (o)(4A)
Note here that @ is not random, since it does not depend on the observed signal. Since are working with sinusoid, A the amplitude estimator À can be converted to a measure of power as P(f) x 2 to provide the relative strength of the frequency f:
P(f) 1 (IN= Lie cos cos (1,, - +2))
(IN The Σn sin -(( +))
+))) +))))
(11)
Σx cos? (2 ft,, -()) Φ))
Repeating this exercise for collection of candidate frequencies f1.... ......... fK (and yielding corresponding power estimators P(fi)...., P(fx)), P(J) follows x² distribution with 2 degrees of freedom, allowing for ease of access to the false alarm probabilities of the power for each frequency.
get obtain a periodogram of the signal. can be shown that under Gaussian noise (a² = 1),
B. Theoretical Discussions
In this section, we provide a high-level theoretical discussion of our work.
12

---

LSCD: Lomb-Scargle Conditioned Diffusson for Time series Imputation
B.1. Conditional Entropy of Reverse Process
First, we leverage result that was presented in (Yang et al., 2024a) related to the conditional entropy of the reverse process of diffusion model. Specifically, the theoretical result presented by the authors shows that the conditional entropy of the diffusion reverse process given the frequency representation of time series is strictly less than the conditional entropy of the reverse diffusion process without the frequency information:
x to
ta X CH, CD) < H X X
( (12)
where X denotes the diffusion process of the target at time t, X denotes the time-series observation condition, and CH and CP denote the feegurryy and Nominant-frequeet condition, respectively. Here, H.(-I-) denotes the conditional entropy, which for two random quantities X given Y is defined as:
TH (X f p)x y) logp(x)y))))))
(13)
where y) denotes the joint distribution of X and Y, while y(x denotes the conditional distribution of X given Y. The essence behind the theoretical result shown in (12) is that incorporating additional conditional information reduces the entropy of the reverse process. This result also applies conditioning on additional information, such as the encoded Lomb-Scargle representation Zy = Expec( (CS (CS()))) and thus we can directly deduce that:
H X ta
X
X Zs) < H(X |
(14)
The proof of this result is straightforward and follows the same logic as the proof presented in Appendix A of (Yang et al., 2024a).
B.2. Potential Bias of Sinusoidal Assumption of Lomb-Scargle
Recall from Appendix A, each frequency component of the Lomb-Scargle periodogram can be derived using least-squares, where the underlying time-series signal is assumed to be a sinusoid with zero-mean and additive noise. From the maximum likelihood point of view, the noise is Gaussian with variance a². One important aspect of this work to discuss is the potential bias of utilizing the Lomb-Scargle periodogram in the imputation model, since the periodogram is based on the assumption that the underlying data is sinusoidal. In this work, the Lomb-Scargle periodogram is incorporated in two aspects: (1) as a conditioning variable in the imputation model; and (2) as part of the spectral consistency loss LsCons used to fine-tune our model in the final stage of training to enforce alignment between the spectrum of the observed data and that of the generative model.
In the first aspect, the Lomb-Scargle periodogram is encoded using a transformer module and incorporated as a conditioning vector throughout the reverse process of the diffusion model. This conditioning provides spectral information about the target time series, guiding the generation process toward samples with similar frequency characteristics. Unlike conditioning on independent variables, the periodogram contains meaningful information about the spectral properties of the time series being generated, making it an informative conditioning signal that should improve generation quality. The sinusoid assumption of the time-series utilized in the derivation of the Lomb-Scargle periodogram does not introduce bias into the model, but rather guides the diffusion process. In the scenario in which the periodogram is not useful, the model would learn to ignore the conditional information. In our work, we have observed that utilizing this information does improve the generative quality.
In the second aspect, the spectral fine-tuning introduces regularization of our model from the ecore-matching optimum obtained in the first stage of training, creating a trade-off between distributional accuracy (captured by the initial training) and spectral fidelity. To be precise, in the first part of training, the standard score-matching objective is optimized, that is,
[()) = E[ile - (x,",, 1 13),
(15)
and in the second part of training, the spectral consistency loss is minimized, where:
Lscous (0) = E [x]))(x;)) - [
(16)
While the score-matching objective learns to approximate the full data distribution, it may not adequately capture the frequency representation of the time series data. The spectral consistency loss addresses this limitation by explicitly
13

---

LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
enforcing that generated samples maintain spectral characteristics consistent with the observed data, which believe is especially useful for time-series with high degrees of missingness. One can roughly interpret training algorithm minimizing single regularized loss with two terms:
Lrgg (0) A1 ( + (
(17)
where A > 0 and A2 > 0 are scaling constants. As can be seen in (17), our training algorithm will introduce bias, since the spectral consistency loss will deviate the parameters from optimal solution, which minimizes the score moore-ntchhngg objective.
C. Computation times
We present analysis of the computational speed of our method (LSCD). In Tables 4 and 5 show a comparison of training and inference computation times respectively, for the LSCD and CSDI models, evaluated on PhysioNet and PM2.5 datasets. All computations in this analysis were performed using a g5 2xlarge AWS instance (AMD EPYC 7R32 CPU, with an Nvidia 10G 24 GB GPU). As shown in the tables, the percentage increase in computation time is approx. 9% for training and 13% for inference. However, the training LSCD includes a final fine-tuning stage using the spectral consistency loss LsCons from Section 4.3, which takes 288.7 s/ep for PhysioNet and 430.3 s/ep ep for PM2.5, due requiring running the inference pipeline part of the computation. Considering this step together with the full training process, LCSI) resulted in an additional 43% training time for PhysioNet and 45% for PM2.5.
Table 4: Training time per epoch for CSDI and LSCD. The last column shows the relative increase for LSCD. Measurements averaged over 10 epochs.
CSDI (s/epoch)
LSCD (slepoch)
Time (%)
PhysioNet
10.30
11.18
8.5%
PM2.5
14.82
16.23
9.5%
Table 5: Inference time per batch for CSDI and LSCD (batch size = 16). The last column shows the relative increase for LSCD, Measurements were averaged over batches.
CSDI (s/batch)
LSCD (s/batch)
ATme (%)
PhysioNet
88.19 69.36
99.18 78.58
13.3%
PM2.5
12.5%
D. Datasets
Here introduce in detail the datasets used in our evaluation. We evaluate method on both synthetic and real-world datasets with varying degrees of missingness to assess its robustness in imputation tasks. We use publicly available real data from two domains, healthcare and climate, and we generate a synthetic dataset where we have control over the ground truth frequencies. A summary of the datasets characteristics shown in Table
Table 6: Summary of datasets used for evaluation.
DATASET
SAMPLES
FEATURES
TIME STEPS
MISSINGNESS TYPE
MISSING %
LINK
SYNTHETIC SINES
2000
5
100
MCAR. SEQUENCE, BLOCK
10% 80% (NATURAL)
N/A
PHYSIONET
4000
35
48
NATURALLY OCCURRING MCAR
[[INK]
PM2.5 AIR QUALITY
5633
36
36
NON-RANDOM ARTIFICIAL
13% (NATURAL)
[LINK]
PhysioNet. This dataset contains multivariate physiological measurements from ICU patients (Silva et al., 2012). We use the preprocessed version from Cao et al. (2018); Tashiro et al. (2021), consisting of 4,000 patient records with 35 variables measured hourly over 48 time steps. Approximately 80% of values are naturally missing. For evaluation, we hold out 10%, 50%, and 90% of the observed values as ground truth and assess imputation quality on these missing entries.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
PM2.5 Air Quality Data. This dataset contains hourly PM2.5 pollution measurements from 36 monitoring stations
in Beijing 12-month period (Yi al., 2016). Following prior work (Cao et al., 2018; Tashiro et al., 2021), extract sequences of length 36. The dataset exhibits approximately 13% naturally occurring missing values, with additional structured missingness artificially introduced for evaluation. The dataset presents real-worke challenges such as non-random missing patterns and periodic fluctuations in pollution levels.
D.1. Synthetic Sines dataset
We generate a synthetic time series dataset for evaluating the proposed method. The dataset is designed to simulate multivariate time series with distinct frequency and amplitude characteristics across different channels. The process is described below.
D.1.1. TIME SERIES GENERATION
Let L represent the number of timesteps, K the number of channels, and I the maximum time horizon. For each sample, multivariate time series X RI*K is generated using sum of sinusoidal components with additive Gaussian noise. Specifically, the time series for each channel k is generated as follows:
fx(1)) Σ i sin(2n + du.i) Eks
where aR Ju,i, and dx.;; are the amplitude, frequency, and phase of the i-th sinusoidal component of channel k, respectively. The term EN represents Gaussian noise with zero mean and standard deviation σ₁. The number of sinusoidal components Ni. as well as the parameters ag.ir fkiis and Φk,i. vary across channels to simulate heterogeneity.
D.1.2. FREQUENCY AND AMPLITUDE SAMPLING
For each channel k the amplitudes ax,; are fixed for all samples within the dataset, while the frequencies fu,s are drawn from Beta distribution, defined as:
Sk.i Beta(a= 2 2) wg (11
wkk
where μ* and we represent the mean and width of the frequency range for channel Sampling from a Beta distribution ensures that frequencies concentrated around μ4g but still allow variability, making the generated time series more realistic. This approach follows Fourier Flows (Alaa et al., 2021), which leverages Beta distributions for frequency modeling in generative time series tasks. Fixing the amplitudes dx.i for each frequency simplifies the interpretation of each sinusoidal component's contribution and reduces the complexity of parameter selection while retaining diversity in frequency combinations.
1.1.3. DATASET CHARACTERISTICS
The dataset contains 2000 samples, each with 100 timesteps spanning a time horizon of T 10.0 units. The dataset consists of K 5 channels with distinct amplitude and frequency configurations, summarized in Table 7.
Table 7: Characteristics of the generated dataset for each channel.
Channel
Components (N₂)
Mean Frequencies (1²)
Frequency Widths (ww)
Amplitudes (ax,i)
[1.0]
[1.0]
[1.0]
2
[1.0, 2.0]
[1.0, 1.5]
[0.5, 1.0]
3
[1.0, 2.0, 3.0]
[1.0, 1.0, 1.5]
[0.5, 1.0, 1.5]
4
[0.5, 1.0, 2.0]
[1.0, 1.0, 1.0, 2.0]
[0.8, 1.2, 1.5, 2.0]
[0.5, 1.0, 2.0, 4.0)
[0.5, 1.0, 1.0, 1.5, 2.0]
[1.0, 1.5, 2.0, 2.5, 3.0]
1.4. MISSINGNESS SIMULATION
binary mask M e (0, 10 generated to simulate missingness in the dataset. To make the setting more realistic, the dataset initially generated with 10% missing data, where values are randomly masked. This ensures that even before
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
applying specific missingness mechanisms, the dataset already contains some level of real-world uncertainty.
We evaluate three missingness mechanisms:
MCAR (Missing Completely Random): Values are masked independently of the underlying data distribution. This serves as a baseline missingness pattern.
Sequence Missing: Contiguous time intervals are masked to simulate sensor downtimes or transmission failures.
Large regions of the dataset, spanning both time steps and feature dimensions, masked to mimic large-scale data outages. The actual missing rate of block missingness is difficult to strictly control due to the overlap between blocks. Instead, "factor" is used to adjust the missing rate approximately.
Block Missing:
Recent studies time series imputation (Du et al., 2024; Mitra al., 2023) highlight that block and sequence missingness patterns better represent woal-world scenarios compared to MAR MNAR mechanisms. These structured patterns closely align with real applications such sensor failures, data transmission losses, and large-scale system outages. Unlike MCAR, which assumes uniform randomness, structured missingness provides a more challenging and realistic evaluation of imputation models. Furthermore, methods that perform well under sequence and block missingness conditions tend to generalize better across diverse real-world datasets.
We use the PyGrinder' library to generate missingness patterns in our dataset, ensuring standardized and reproducible approach missing data simulation.
Each missingness mechanism is applied at three levels (10% 50%, and 90%) assess the robustness of our method under increasing data loss. The specific hyperparameters used for each mechanism and the resulting missing rates, computed from the actual data, are summarized in Table 8.
Table 8: Missingness parameters and actual missing rates for different mechanisms.
Mechanism
Hyperparameters
Target Rate
Actual Rate
MCAR
0.1
10%
10.0%
MCAR
0.5
50%
50.0%
MCAR
= 0.9
90%
90.0%
Sequence Missing
p = 0.1, seq_een 50
10%
10.0%
Sequence Missing
p = 0.5, seq 50
50%
50.0%
Sequence Missing
p = 0.9, seq len 30
90%
55.6%
Block Missing
factor 0.1, block len 40, block width 4
10%
18.8%
Block Missing
factor 0.5, block len 40, block_width 4
50%
56.4%
Block Missing
factor 0.9, block len 40, block width 4
90%
71.9%
As seen in Table 8, the actual missing rates for block missingness tend to deviate from the target values. This is due to the nature block-based masking, where overlapping blocks and feature constraints introduce variability in the effective missing proportion. Instead of directly specifying the missing rate, the "factor" parameter is used to approximate the final missing rate, though it does not always match the intended level exactly. The structured missingness patterns allow for evaluating the model's ability reconstruct both small-scale gaps and large missing regions effectively.
thw.. hattas////ghabt...ooo////eee///d inder
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
E. Implementation and Reproducibility details
E.1. Lomb-Scargle implementation
The following listing provides a PyTorch implementation the Lomb-Scargle periodogram, designed to efficiently compute spectral estimates for irregularly sampled time series. This implementation is fully differentiable, allowing seamless integration into learning-based models for gradient-based optimization.
Import Lorch
class Lombsearglesett
:
def _init_ (self, f, omegas):
super (Lrrbssccrrrlccatcciiaaa self) -init_
self.omegas omegas Tensor of angular frequencies (omegas)
def (sele, mask-Nene):
10
Computes the Lomb-Scargle pariodogram for batch of time saries with maaking.
11
12
Arga:
13
(Tersori: Time va values, shape [8, N).
14
y (Tensozt: Observed data values, shape [B, N]. mask (Tensor, optional): Booleen mask, shape (B, N] valid, missing).
15
16
Returns:
18
Tensor: Lomb- Scargle periodogran values, shape [B, M].
19
20
N t.ahape
a wwl...omeaa..... shape [0]
22
23
mask is None: torch. ones__.
Default to valid points
25
Expand teasors
27
L.unsqueeze (1) $ [B,
28
y y.unsqueeze (1) [B, 1, 2]
mask mask. unsqusezeuuuee
[8,
30
omega = sel f. www.ma.................... M, 1) $ (1, M, 1]
32
# Compute tau for each frequency and batch. omega
[B, sin_2wt (torch.sin [two_omea__) maak) (dmidim22))
two_cmega_t
34
35
cos_2w= (torch. (two_omege_t) A mask) (dm(mim)2)
36
37
tan Len__oomeaa_tttn sin_2w: 2n: / torch of amp (cos_2wt, min-1e-10) Lorch.atan (Lan__2megaa_aeee
Lorch.clamp (oclaaaoooogeeeee (quega. (2), min-le 101
39
Compute Lemb-Scargle perisdogram
41
omega_i_tau omega tau. uns.ueezeeeeeee (2)) [B,
42
torch.cos (mmega_t_aa) mask
cin_omoga_t__aa torch.sin homego_t_taee maak
44
y_eos cos_cmega_t_tau y_sin sin_cmega_t_tau
46
48
PLoos myy_cos (dim=2) A+2 /torra. clamp ccos__cmea___taeee sum (dimm2) minmle-10) P_sin (y_sin. (mm 21 2/////// ftorch.clame sasii__oooa______ee t_t.a. (2) .oum (dim=2), mmin=1e-100
49
51
P 0.5 v cos P_ain)
[B, M]
53
return
Listing I: Batch Lomb-Scargle Periodogram with Masking
E.2. Model Architecture Details
Diffusion Model Architecture
Our approach adopts a diffusion-bur basedd generative model for time series imputation, implemented in a style consistent with standard score- based diffusion frameworks (Tashiro et al., 2021). Below, we outline the high-level architecture and key components:
Denoising Network: A U-shaped transformer that incorporates temporal convolution blocks and multi-heat head attention layers. The network accepts the following inputs:
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
Noisy target values x diffusion step
2. Observed context xi" (unmasked entries),
Spectral embeddings 2ş from the Lomb Scanlle encoder,
4. A positional encoding of the diffusion step L.
Irregular Sampling Handling: Timestamps into the feature representations at each layer, enabling the model to account for non-uniform time intervals.
mapped through 128-dimensional positional embeddings and fused
Network Block Details: For each denoising step, the architecture employs a series of transformer blocks to capture temporal structure:
Multi-Hal Head Self-Attentinn, which learns dependencies among time steps, Positional Encodings added along the time axis to preserve sequence order,
Feed-Ford Freed-oowwar layers with dropout and normalization encourage stable training and robust generalization.
Spectrum Encoder Transformer Blocks
As described in Section 4.2, we employ two transformer modules, one along the frequency axis, and one along the features axis, in order extract latent embedding zy from the Lomb-Scargle periodogram LS RKXL, Key hyperparameters include:
dmmoee (internal hidden dimension used by each attention module): 64.
nHeads (number of attention heads); 8.
depth (number of transformer layers): 4 blocks for each dimension (frequency/feattree).
Input Representation: Log-transformed power spectral density values log(1 + LS(x)") across J frequency bins, normalized to zero mean/unit variance per channel.
Training and Hyperparameters
We train for 400 epochs and select the best checkpoint via a validation set. The final 2s from our spectral encoder is concatenated with the other conditioning signals (e.g. partial observations) at every denoising step to guide the model.
Diffusion Steps: Tmax 50.
Batch Size: 16.
Learning Rate: 1e-3.
F. Evaluation Metrics
To ensure a fair evaluation, we compute reconstruction errors only on the target values, defined as the set of originally observed values that were artificially removed for evaluation. contrast, frequency-domain metrics computed using all observed values (condition plus target), ensuring that spectral estimates not influenced by imputed values outside of the ground truth.
In line with the conditional diffusion framework described in Section 3.3, let us recall that we split the data into conditional ((r) C X. When we refer to "originally observed values that
portion 2 m 09 O X and larget portion æ (M m²") were artificially removed," we mean win. At test time, the model imputes these missing target entries, producing a complete reconstruction X
F.1. Some--Dmain Metrics
For domeidomin metrics, we evaluate performance only on the target (imputed) entries, ie., the set of indices where (M- m. is
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
Mean Absolute Error (MAE)
Ex. [Ma. mxx)) [xx,1 Xx.1| Σx, (M - mx
MAE
(18)
MAE measures the average absolute deviation between the imputed values Xx. and the ground truth X2. where (Mx,1
mg)) 1.
Root Mean Squared Error (RMSE)
En. (Mai--ia - (Xx,1 - Xx.1) 2
RMSE
(19)
En,((MR, - m())))
RMSE penalizes large deviations more severely than MAE, again computed only over the target entries.
F22 Frequency-Domain Metrics
For -omainn metrics, we compute the Lomb Lomb-Scargle power spectral density (PSD) using only the originally observed points, i.e., where Ma.1 = 1. This ensures the spectral estimate is not influenced by imputed values outside the known ground truth.
Spectral Mean Absolute Error (S-MAE) To assess how well the imputed signals preserve the original frequency characteristics, we compare the normalized Lomb Scargle PSD estimates of the ground-truth series Par(w) against those of the reconstruction (w). Let be the set of evaluated frequencies:
Par(w) En Por(w")
(erea(w) En Prrcalw")
S-MAE
(20)
Here, (w) and (w) are both computed strictly from points where Mr.1 1.
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LSCD: Lomb-Seargle Conditioned Difflusion for Time series Imputation
G. Visualizations of spectral distributions
G.1. PM225 dataset
122
122
GT Prod
125
12:
Densty
Te.
Prwd
Densty
Densty 3F
Density
Prwd
3106
110
2319
211
Doum
Count
Count
1000
0:00
600
3.04
6:35
0.60
0.16
1602
0.00
0003
3.04
6:55
0.00
0.06
0002
0.60
003
3.04
0.65
3.04
3.00
000
0.00
002
6:04
0.66
0.08
330
trequency
Broquency
Prequency
Prequency
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
GT
GT
GT Pood
GT
Pred
Prod
Peod
Derett
Denuitt
Densaty
Dessiity
N 88
N
M
Count
Count
Count
6:36
0.08
3.0
8.06 0:72 Irequency
Lead. troquency
Lead threuueccy
Lead trrquenccy
(c) BRITS
(f) GP-VAE
(g)
ModernTCN
(h) US-GAN
Figure 6: Distribution of leading frequency on PM25 dataset.
PSDICTI PEDCPra)
PSDICTI BSDKPro)
PSDICT) PSEXPreD)
PRDIOT) PSDUPre))
3.5
3.3
P50000 2500baa)
3.0
21
30
81
0.5
3.0
0.4
Frequaicy
Frequancy
Prequarcy
Prequarty
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
PSDISTI PEDCProN
1.3
PRDISTI PEDOPMS)
1.5
FRDIET) PSDXPre))
1.5
PSDOST) PSDDIre))
124
PS0101 PSD00red)
1.4
6.3
6.5
0.5
5
0.0
0.1
1.3
0.4
0.5
3.0
0.1
0.3
0.4
0.5
3.0
0.1
02
6.3
0.4
0.5
1.0
0.2
122
0.3
0.4
35
(c) BRITS
(f) GP-VAE
(g) MCdernTCN
(h) US-GAN
Figure 7: Difference between the power spectral densities (PSD) of the ground truth and predictions, with the black line representing the mean difference and the shaded area indicating one standard deviation
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
G.2. Physionet dataset
I60
150 12.3
150
114
QT
GT
GT
GT
1oo
Denerg
160
Decater
Dessity
Count
Count
Load Preguenry
Load Prequency
Laad Fraguency
Land Frequency
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
150
153
185
G7
G7
G7
GT Prwd
Denaty
100
Prod
Donsty
100
Prwd
Donatr
10.3
Prwd
Donstty
13.8
Count
Count
02
24 03.
30
31.
06
00.
Prequency
Frequetcc
Fraquency
Frequency
(c) BRITS
GP-VAE
gg)
ModernTCN
(h) US-GAN
Figure 8: Distribution of leading frequency on Physio dataset with 10% missing data.
2.3
2.2
PSDIGTI PSDdPrad)
PSDIGT PS3/Pre)
PSDIGT) PSDXPred)
PSDIOT) PSIMPred)
1.5
3.5
3.5
pabeen (eDohaal
10
eSpOT PSbcced
1.3
PSDICT: (eDdrea
1.3
3.2
03
033
0.3
80
0.3
0.3
-13
-1.
Proquetiy
05
Prequery
Pooqumac
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
3.5
retron (504he0)
rebion (0500he)))
0.
211
0%
26
01
2.0
Eroquorry
Proquorry
Proquoory
(c) BRITS
GP-VAE
(g) ModernTCN
(h) US-GAN
Figure 9: Difference between the power spectral densities (PSD) of the ground truth and predictions, with the black line representing the mean difference and the shaded area indicating one standard deviation, on Physionet with 10% missing data.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
160
GF
GI Pred
G1 Prwd
300
Prwd
160
Pred
Denats
10.
Donstty
32.
50
Count
Count
05
8.4
0b
6.
9.11
32
13
23 03
3.
03 Frequency
64
83 on
Lead Prequenryy
Load Prequeacy
Lead Proquency
Ours
CSDI
SAITS
(d) TimesNet
135
12.5
12.3
12.6
Densty
Fred
Prod
Denety
Prod
Densty
Pred
422
Count
SL2
Dount
XLE
XLE
6.4
36
0%
45
Prequency
Lead
Lead Frequency
(c) BRITS
(f) GP-VAE
MCdernTCN
(h) US-GAN
Figure 10: Distribution of leading frequency on Physio dataset with 50% missing data.
(9D(ha))
PSDCT)
PSDOOT PSD/Pco))
PSDKDD PSDItwel
1.
Prseucsyy
Proquncc
Proquesay
Prequency
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
PSDIGT ISDrPrad)
PSDICTI (SDUPre))
PSDIGT) PSD/Pres)
PSDIGT) 753/Pred)
PS0DCT: 15000hed
0.0
0.4
Proquctic
Frequesay
Frequascy
Froquaicc
(c) BRITS
(f) GP-VAE
(g) ModernTCN
(h) US-GAN
Figure 11: Difference between the power spectral densities (PSD) of the ground truth and predictions, with the black line representing the mean difference and the shaded area indicating one standard deviation, on Physionet with 50% missing data.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
QT
07 Pred
GT Prwd
3oo
Prwd
160
Prod
Denatt
10.
Donstty
Count
Count
05
844
0b
6.
32
13
22
03
03 Frequency
64
65 on
Lead Prequencyy
Load Prequeacy
Lead Proquency
Ours
CSDI
SAITS
(d) TimesNet
135
12.5
12.5
12.6
Densty
Prwd
300
Prwd
Denety
Prwd
Densty
Prwd
Count
Countt
36
4.5
00.
Preguenry
Freguetry
Frequency
Frequency
(c) BRITS
(f) GP-VAE
(g)
ModernTCN
(h) US-GAN
Figure 12: Distribution of leading frequency on Physio dataset with 90% missing data.
ESPICT- (SD(Pra))
ISDIGT (SDIPaa)
PSDICT) PSDXhvd)
PSDICT PSD(hva)
Psp/on (500Pto0
PSDOCT) PStribvel
1/
Prseucsoy
Proqunccy
Prequesay
Prequency
Ours
(b) CSDI
(c) SAITS
(d) TimesNet
PSPICTI- (5D0hal)
PSDICTI (6DXPao)
PSDICTI PSDXive)
PSDIDD: PSDchve)
PSDDooll
PSOYCD- PSD0beed
4
4
4.
6
0.2
0.0
0.4
Proquetic
Prequesay
Frequascy
Proquatcc
(c) BRITS
(f) GP-VAE
(g) ModernTCN
(h) US-GAN
Figure 13: Difference between the power spectral densities (PSD) of the ground truth and predictions, with the black line representing the mean difference and the shaded area indicating one standard deviation, on Physionet with 90% missing data.
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LSCD: Lomb-Scargee Conditioned Diffusion for Time series Imputation
H. Visualization of time distributions
H.1. PM225 dataset
GT Pred
2.7
Pred
2-5
Pred
2.0
Pred
2.34
Cocntt
1.8
Countt
1.35
1.0
6.7
0.
Data valuue
Data valuu
Data Vallue
Data value
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
x127
X12
Xie
X1e
GT
GT
GT
GT
1.3
Count
1.3
Count
1.
Count
1.5
0.5
65
05
0.5
205
300
2.5
300
100
2:00
vallue
(e) BRITS
(1) GP-VAE
(g) ModernTCN
(h) US-GAN
Figure 14: Distribution of imputed values across different models on PM25 dataset.
H.2. Physionet dataset
2153
x1e
X107
14-
1.2
Pred
CT
1.0
GT
CT
1.75
3,50
Pred
Couls
Coust
1.35
2,00
04-
0.8
Data valuo
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
113
1.5
GT
2.00
CT
GT
1.4
Pred
L05
Pred
Prod
Prod
1.50
1.0
1.2
1.25
Count
Count
038
c.00
05-
2.75
02
635
02
-2
3.00
02
0.0
Data value
Data value
Data valuue
Data value
(e) BRITS
(1 GP-VEE
(g) ModernTCN
(h) US-GAN
Figure 15: Distribution of imputed values across different models Physionet with 10% missing data.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
Prod
Prod
Pred
Pred
Count
Count
+
-2
Data value
Data volue
Data value
Data value
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
GT
GT
GT
Pred
1.2
Pred
Prod
Pred
1.2
Count
Count
Count
0.4
62
Data value
Data value
Data value
Data value
(c) BRITS
(f) GP-VAE
ModernTCN
(h) US-GAN
Figure 16: Distribution of imputed values across different models on on Physionet with 50% missing data.
K10
X10P
GT Pred
GT Pred
GT Prod
GE Prod
122
1.2
1.5
1.0
Count
Count
04
54
0.-
022
02
Data value
Data value
value
Date value
(a) Ours
(b) CSDI
(c) SAITS
(d) TimesNet
2.5 -124
xint
2164
CT
GT
CT Prod
2.
Prod
Pred
Pred
Count
Count
0.7
0.
value
(c) BRITS
GP-VAE
ModernTCN
(h) US-GAN
Figure 17: Distribution of imputed values across different models Physionet with 90% missing data.
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LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation
I. Visualization of imputed examples
0.5
0.0
0.5
20
40
20
40
20
40
20
40
6mm
(a) Ours
(b) CSDI
0.5
0.2
1.0
0.5
-0.2
1000
1..0
0.0
-0.5
1.5
-0.5
1.0
-0.6
0.2
-1.01
20
40
20
40
20
40
20
40
Samee
SAITS
(d) TimesNet
Figure 18: Reconstruction results PhysioNet with 10% missing data.
2.5
1.0
valuo
0.5
20
40
20
40
20
40
20
40
20
40
Time
Bire
time
time
time
(a) Ours
(b) CSDI
0.2
0.1
2000
1.0
1.0
0.11
0.5
0.0
(c) SAITS
(d) TimesNet
Figure 19: Reconstruction results on PhysioNet with 50% missing data.
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Figure 20: Reconstruction results on PhysioNet with 90% missing data.
26


Reviews:
[{"review_id": "xJ9UFZp8Yr", "reviewer": "Reviewer_gtsg", "summary": "This paper introduces Lomb–Scargle Conditioned Diffusion (LSCD), an approach for irregularly sampled time series imputation. Unlike traditional frequency-domain methods that rely on the Fast Fourier Transform (FFT), which assumes uniform sampling and requires interpolation, LSCD leverages the Lomb–Scargle periodogram to handle missing data directly in the frequency domain. The method integrates a score-based diffusion model conditioned on the Lomb–Scargle spectrum, ensuring that the imputed time series aligns with its true spectral content. To enhance performance, LSCD employs a spectral encoder and a spectral consistency loss, reinforcing the coherence between imputed series and their frequency representation. Experiments on synthetic sine wave data and two real-world datasets, PhysioNet ICU patient records and PM2.5 air quality data, demonstrate that LSCD outperforms existing baselines in both time-domain accuracy (MAE, RMSE) and spectral preservation (S-MAE).", "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": "We thank the reviewer for their thorough and positive assessment of the manuscript. We appreciate the recognition of the relevance of the work, as well as the modeling and evaluation choices. Below are our detailed responses, we hope they address any remaining concerns.\n\n&nbsp;\n### Weakness 1 (Related Work)\nWe appreciate these valuable references in continuous-time modeling and specialized architectures for irregularly sampled data. In the revision of our paper, we will include an additional discussion on the comparison of score-based diffusion-based approaches and some of the aforementioned continuous-based approaches specifically w.r.t. the task of probabilistic time-series imputation. With respect to latent ODE/SDE models, a variational framework is typically used to train the model, and conditioning on a set of observed time-points to obtain the approximate posterior enables imputation by a forward pass of the learned model (filtering). More advanced variations, such as [6], perform an additional backward pass to obtain more accurate imputations (smoothing) based on all observed information. On the other hand, methods like CSDI and our method pre-define a fixed-window size for the generated time series and explicitly learn the conditional distribution $p(\\mathbf{X}^{\\text{mis}} | \\mathbf{X}^{\\text{obs}}, \\mathbf{C}^{\\text{obs}})$, where $\\mathbf{X}^{\\text{mis}}$ denotes the missing values we are trying to impute, $\\mathbf{X}^{\\text{obs}}$ denotes the observed samples at the time of imputation, and $\\mathbf{C}^{\\text{obs}}$ denotes a conditioning variable that provides additional information (based on $\\mathbf{X}^{\\text{obs}}$) to enhance the imputation of $\\mathbf{X}^{\\text{mis}}$. Examples in training are created using a mask. At inference time, filtering and smoothing are not used to impute, but a simple forward pass through the diffusion model with appropriate conditioning based on the observed values.\n\nIn addition to a deeper discussion of these models, we plan to include new experimental comparisons with a subset of these continuous-time approaches (e.g. Latent ODEs [1] or DSPD-GP [7]) in the final version of the paper as additional results. This will help illustrate how the LS-based diffusion framework fares relative to advanced continuous-time baselines, highlighting both advantages (e.g. explicit spectral preservation) and trade-offs (e.g. requiring a pre-defined grid).\n\n&nbsp;\n### Weakness 2 (Efficiency Analysis)\nPlease refer to the response to Reviewer 3 (**ZpA3**) for a detailed analysis of computational efficiency of our method.\n\n&nbsp;\n### Question 1A (Arbitrary timestamp imputation)\nLSCD, like most score-based diffusion models such as CSDI, operates on a fixed time grid and requires explicit placeholders defined by a mask to know where to perform imputation. It does not generate a continuous function that can be queried at any arbitrary timestamp. In contrast, models like Neural ODEs learn continuous latent trajectories that can be evaluated at any point in time, even those not seen during training or inference.\n\n&nbsp;\n### Question 1B (Effect of number of mask placeholders)\nOur model, like CSDI, performs joint conditional imputation over masked timestamps, and thus the prediction at a given timestamp may depend on the number and location of other masked points. However, we additionally condition on the spectrum of the observed time series, which provides global frequency information that can help stabilize the imputation process.\n\n&nbsp;\n### Question 2 (Limitations)\nOur approach presents the following limitations:\n\n- **Irregularly Sampled Data:** Our method supports missing time-steps but relies on a regular time grid. It also supports irregular sampling, but this requires upsampling to a fine-grained grid, which may be computationally expensive.\n\n- **No Continuous-Time Output:** Unlike neural ODEs, LSCD does not output a continuous function over time. This may limit its applicability in tasks requiring continuous-time interpolation or forecasting.\n\n- **Frequency Grid Dependence:** The model conditions on a fixed set of frequencies. An improper grid choice could lead to suboptimal conditioning, although this is mitigated by the spectral encoder and FAP-based filtering.\n\nWe will incorporate the discussion of the limitations of our approach in the manuscript.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No proofs in the paper", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The study highlights the potential of Lomb–Scargle-based spectral conditioning in machine learning applications involving incomplete time series.", "essential_references_not_discussed": "Key related works on irregularly sampled time series, particularly those for regression tasks, are missing.", "other_strengths_and_weaknesses": "Strengths:\n1. LSCD elegantly integrates the Lomb–Scargle periodogram into a diffusion model, allowing it to directly condition time-domain imputation on spectral information without requiring interpolation or zero-filling. This fusion preserves the frequency structure of irregularly sampled data while leveraging the powerful generative capabilities of diffusion models.\n2. By providing a differentiable implementation of the Lomb–Scargle periodogram, LSCD enables end-to-end learning, making it seamlessly compatible with modern deep learning frameworks and adaptable to various time series tasks with missing or irregularly sampled data.\n3. LSCD consistently outperforms both FFT-based and diffusion-based imputation methods by achieving lower MAE, RMSE, and better spectral fidelity (S-MAE).\n\n\nWeaknesses:\n1. This paper lacks discussion on related works addressing irregularly sampled time series, particularly for regression tasks such as imputation and forecasting [1-9] (a representative but non-exhaustive list). The authors should elaborate on the differences and advantages of their approach over these prior works and compare it with some state-of-the-art algorithms among them. \n2. This work lacks an efficiency evaluation of the proposed model, which is crucial for practical applicability and broader usage.\n\n[1] Latent ODEs for Irregularly-Sampled Time Series. NeurIPS, 2019.  \n[2] GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series. NeurIPS, 2019.  \n[3] Neural Flows: Efficient Alternative to Neural ODEs. NeurIPS, 2021.  \n[4] Multi-time attention networks for irregularly sampled time series. ICLR, 2021.  \n[5] Modeling Irregular Time Series with Continuous Recurrent Units. ICML, 2022.  \n[6] Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series. ICML, 2023.  \n[7] Modeling Temporal Data as Continuous Functions with Stochastic Process Diffusion. ICML, 2023.  \n[8] GraFITi: Graphs for Forecasting Irregularly Sampled Time Series. AAAI, 2024.  \n[9] Irregular Multivariate Time Series Forecasting: A Transformable Patching Graph Neural Networks Approach. ICML, 2024.", "other_comments_or_suggestions": "1. There is a typo: ‘Ls’ in the equation in Section 4.2.", "questions_for_authors": "1. Can this model impute measurements at any timestamp even without mask placeholders? Given the same observed time series, does the number of mask placeholders influence the imputation results for a specific timestamp?\n2.  What are the limitations of the proposed model?", "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": "G9qVw1em4t", "reviewer": "Reviewer_jzSS", "summary": "This paper introduces a novel diffusion-based time series imputation method. Specifically designed for irregularly sampled data, the proposed method leverages the Lomb-Scargle periodogram to enhance imputation performance.", "questions": "", "limitations": "", "rating": 2, "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": "We thank the reviewer for their thorough assessment of our work and for their useful comments. Below are our detailed responses, we hope they address any remaining concerns.\n\n&nbsp;\n### Relation To Broader Scientific Literature (Novelty)\nWe appreciate the presented references and recognize that the Lomb–Scargle periodogram has a longstanding history in signal processing and time series analysis. Our goal is not to claim it as a new method for time series, but rather to show how integrating a differentiable Lomb–Scargle operator into a diffusion-based architecture opens up new possibilities for modern machine learning pipelines. Unlike classical uses of Lomb–Scargle (e.g. for detecting periodicities in irregular or biological signals), our approach is the first to incorporate it into a trainable model. This enables gradient-based optimization of spectral features, allowing the network to learn periodic structure from irregular data in a way that has not been previously explored.  \n\nWe have also released our differentiable Lomb–Scargle implementation to encourage broader adoption of frequency-based conditioning in irregular time-series tasks. We will revise the manuscript to cite these prior works and to clarify the distinct contributions of our approach.\n\n&nbsp;\n### Weaknesses (Table 2 Results)\nThe results in Table 2 indicate that our method outperforms all other baselines, including CSDI, across all real datasets and evaluation metrics. However, CSDI consistently ranks second, with varying margins with respect to LSCD. A follow-up analysis using a one-sided Wilcoxon signed-rank test suggests that the improvement over CSDI is statistically significant with only moderate confidence ($p=0.065<0.1$).\n\nWhen combining both synthetic and real datasets from Tables 1 and 2, our method significantly outperforms CSDI according to a one-sided Wilcoxon signed-rank test ($p < 0.05$ for all metrics), confirming consistent improvements in imputation quality. In the following table we display the average MAE rankings of the methods for Table 1, Table 2 and the aggregate of both tables. Notably, multiple methods outperform CSDI in the synthetic datasets, while our approach remains consistently ranked first. We will incorporate this analysis as part of the supplementary material.\n\n**Table R3: Average MAE ranking per table**\n\n|        | MEAN | LERP | BRITS | GPVAE | US-GAN | TimesNet | CSDI | SAITS | ModernTCN | Ours |\n|--------|:------:|:------:|:--------:|:--------:|:---------:|:---------:|:-----:|:-------:|:-------:|:-------:|\n| Table1 | 5.9  | 9.6  | 2.6    | 6.9    | 6.8     | 8.7       | 6.1  | 2.2    | 4.6         | 1.8   |\n| Table2 | 10.0 | 4.9  | 4.0    | 7.9    | 7.5     | 7.1       | 2.0  | 3.6    | 7.0         | 1.0   |\n| Table1+Table2 | 7.2 | 8.1  | 3.0    | 7.2    | 7.0     | 8.2       | 4.9  | 2.6    | 5.3         | 1.6   |\n\n&nbsp;\n### Weaknesses (Theoretical Results)\nAs a starting point, we point out that without the spectral consistency loss, our setup is similar to [1]. The authors condition on the high and dominant frequency components of the FFT of the observed time series, whereas our technique conditions on the Lomb-Scargle periodogram. In [1], a theoretical result is presented that shows that the conditional entropy of the diffusion reverse process given the frequency representation of time series is strictly less than the conditional entropy of the reverse diffusion process without the frequency information:  \n\n$$\n\\mathbb{H}({\\bf z}|\\hat{\\bf X}_t, {\\bf X}^{\\bf C}, {\\bf C}^{\\bf H}, {\\bf C}^{\\bf D}) < \\mathbb{H}({\\bf z}|\\hat{\\bf X}_t, {\\bf X}^{\\bf C}),\n$$\n\nwhere ${\\bf z} = \\hat{\\bf X}_{t-1}$, ${\\bf X}^{\\bf C}$ denotes the time-series observation condition, and ${\\bf C}^{\\bf H}$ and ${\\bf C}^{\\bf D}$ denote the high-frequency and dominant-frequency condition, respectively. The essence behind this theoretical result is that incorporating additional conditional information reduces the entropy of the reverse process. This result extends to *any* conditional information, which means that if we condition on the Lomb-Scargle representation of the time series, it should also reduce the conditional entropy of the diffusion process. In the revision of our paper, we will include this analysis as a theoretical insight of the method.  \n\n[1] X. Yang et al. *Frequency-aware Generative Models for Multivariate Time Series Imputation*. NeurIPS, 2024.\n\n&nbsp;\n### Question (Computational Efficiency)\nWe have included a detailed analysis of computational efficiency in the response to Reviewer 3 (**ZpA3**).", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "No theoretical results are provided in the paper.", "experimental_designs_or_analyses": "Yes. The experimental design is reasonable.", "relation_to_broader_scientific_literature": "The idea of using Lomb-Scargle Periodogram for time series modelling is not novel. See\n\n[1] Glynn, Earl F., Jie Chen, and Arcady R. Mushegian. \"Detecting periodic patterns in unevenly spaced gene expression time series using Lomb–Scargle periodograms.\" Bioinformatics 22, no. 3 (2006): 310-316.\n\n[2] Ruf, T. \"The Lomb-Scargle periodogram in biological rhythm research: analysis of incomplete and unequally spaced time-series.\" Biological Rhythm Research 30, no. 2 (1999): 178-201.\n\n[3] Van Dongen, H. P. A., E. Olofsen, J. H. Van Hartevelt, and E. W. Kruyt. \"A procedure of multiple period searching in unequally spaced time-series with the Lomb–Scargle method.\" Biological Rhythm Research 30, no. 2 (1999): 149-177.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1) The idea of using the Lomb-Scargle periodogram to impute unevenly distributed time series data is reasonable.\n\n2) The proposed method is assessed on different time series datasets.\n\nWeakness:\n1) The novelty of the proposed method is limited. Although incorporating the Lomb–Scargle periodogram into the diffusion model is relatively new, the idea of using it to deal with time series data has been studied by many previous works and the proposed framework is still mainly based on CSDI with only small changes. \n\n2) From results presented in Table 2, the proposed method did not show significant empirical performance improvement compared to other diffusion-based method that did not use spectral method, i.e., CSDI.\n\n3)  The theoretical results regarding the effect of the Lomb-Scargle periodogram on the diffusion process are missing.", "other_comments_or_suggestions": "The authors could provide a theoretical analysis of how the Lomb-Scargle periodogram influences the diffusion process.", "questions_for_authors": "How is the computational efficiency of the Lomb-Scargle periodogram compared with FFT?", "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": "rLHHSmVF7A", "reviewer": "Reviewer_ZpA3", "summary": "The paper proposes a novel method designed for performing time series imputation when the input data either has missing data of is not measured at equal intervals.  The use of discrete Fourier transform in this case often leads to serious artifacts in the power density spectrum.  In contrast, the power density spectrum can be more accurately estimated using the Lomb-Scargle methods.  This paper shows how this can be incorporated into a diffusion-based method to ensure the frequency information is well captured in imputed time series.  The technique is thoroughly tested and compared against strong baseline models.", "questions": "", "limitations": "", "rating": 5, "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": "We sincerely thank the reviewer for their positive assessment and kind words about our work. We appreciate that you find the method’s contribution to be novel and substantial, and that our empirical evaluations appear fair and rigorous\n\n---\n\nBelow we include a general analysis of the computational efficiency of our method, requested by several reviewers.\n\n&nbsp;\n### Computation Time\nWe present an analysis of the computational speed of our method (LSCD). A common concern in the reviews was the increased computational cost of Lomb-Scargle with respect to FFT. Indeed, the computational complexity of FFT is $O(N\\log{N})$, while Lomb-Scargle's complexity is $O(N \\cdot J)$, where $N$ is the number of points in the grid, and $J$ is the number of chosen frequencies. In our method, we use $J = N$ thus the complexity of LS results $O(N^2)$. In effect, for the grid sizes used in PhysioNet and PM2.5, our differentiable implementation of **Lomb-Scargle has a computational cost that is $\\mathbf{50}$ times higher than FFT** ($1.25\\times10^{-2} s$ vs $2.46\\times10^{-4} s$). However, the cost of this operation does not translate directly to the computation time of our LSCD model. In Tables R1 and R2 we show a comparison of training and inference computation times respectively, for the LSCD and CSDI models, evaluated on PhysioNet and PM2.5 datasets. All computations in this analysis were performed using a g5.2xlarge AWS instance (AMD EPYC 7R32 CPU, with an Nvidia A10G 24 GB GPU). As shown in the tables, **the percentage increase in computation time is approx. $9\\%$ for training and $13\\%$ for inference**. However, the training of our method includes a final fine-tuning stage using the spectral consistency loss $\\mathcal{L}_{SCons}$ from Section 4.3, which takes $288.7\\,s/ep$ for PhysioNet and $430.3\\,s/ep$ for PM2.5, due to requiring running the inference pipeline as part of the computation. Considering this step together with the full training process, **our method resulted in an additional $43\\%$ training time for PhysioNet and $45\\%$ for PM2.5**. This analysis has been incorporated into the manuscript.\n\n**Table R1: Training time per epoch for CSDI and LSCD.**  \nThe last column shows the relative increase for LSCD. Measurements were averaged over $10$ epochs.\n\n|               | CSDI (s/epoch) | LSCD (s/epoch) | $\\Delta$Time (%) |\n|---------------|----------------|----------------|------------------|\n| PhysioNet     | 10.30          | 11.18          | 8.5%             |\n| PM2.5         | 14.82          | 16.23          | 9.5%             |\n\n**Table R2: Inference time per batch for CSDI and LSCD (batch size = 16).**  \nThe last column shows the relative increase for LSCD. Measurements were averaged over $5$ batches.\n\n|               | CSDI (s/batch) | LSCD (s/batch) | $\\Delta$Time (%) |\n|---------------|----------------|----------------|------------------|\n| PhysioNet     | 88.19          | 99.18          | 13.3%            |\n| PM2.5         | 69.36          | 78.58          | 12.5%            |", "claims_and_evidence": "The claim is that this novel approach provides better imputation of this type of time series data.  The empirical evidence appears compelling.", "methods_and_evaluation_criteria": "As far as I can see the evaluations are fair and rigorous.", "theoretical_claims": "N/A.", "experimental_designs_or_analyses": "No.", "relation_to_broader_scientific_literature": "The paper appears to be well versed in the literature around time series.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "The paper is very well written and accessible.  The contribution is novel and substantial.", "other_comments_or_suggestions": "None.", "questions_for_authors": "None.", "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": "PnV7nnCLWL", "reviewer": "Reviewer_y6X7", "summary": "The paper proposes LSCD for irregular time series imputation, which uses Lomb-Scargle periodograms to compute additional frequency domain loss in the diffusion model. LSCD uses Lomb-Scargle instead of FFT to transform irregular time series into the frequency domain and achieves more accurate imputation results.", "questions": "", "limitations": "", "rating": 2, "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": "We thank the reviewer for their careful assessment of our paper. Below, we provide detailed responses aimed at clarifying and addressing each concern.  \n\n&nbsp;\n### Claims and Evidence\nThank you for pointing out the discrepancy: in line 22, \"frequency domain\" should be \"time domain\". We have updated the text accordingly.\n\n&nbsp;\n### Methods And Evaluation Criteria (Figure 2)\n**Consistency loss:** We can confirm that the diagram correctly reflects the architecture, and that using $x_{co}$ instead of the \"observed time series\" is done by design: during training, we sample a conditional mask $m_{co}$ from the observed values and treat them as $x_{co}$. The consistency loss then ensures the reconstruction matches the frequency profile of these masked ‘observed’ values, which simulates the inference setting. This follows common practice in score-based diffusion training, including CSDI.\n\n**Noisy imputation target:** We have updated the figure so the effect of adding noise is more evident and does not resemble a shift.\n\n**Math symbols:** We have now increased the resolution of math symbols in the figure.\n\n&nbsp;\n### Experimental Design or Analyses\n**Use of Sine Datasets:** We acknowledge that the Lomb-Scargle method is well-suited to signals with sinusoidal components. The synthetic sine datasets were not intended to favor our method but rather to offer a controlled and interpretable setting where periodicity is known, in order to evaluate the recovery of ground truth frequency information. To mitigate potential bias, we also include evaluations on real-world irregular time series in Table 2. These results confirm that Lomb–Scargle conditioning remains helpful for imputation, even if the spectrum does not exhibit strong periodicity.\n\n**Figure 5:** We appreciate this observation and agree that the marginal distribution plots in Figure 5 do not, on their own, validate the model’s frequency reconstruction. The main purpose of these plots is to show that our imputed values align well with the observed data in the time-domain distributional sense (e.g. capturing the overall range and shape). For spectral validation, we rely on additional spectrum-based metrics (S-MAE) and Lomb–Scargle comparisons (Figure 4). We will clarify in the revision that Figure 5 primarily illustrates distributional consistency, while the Lomb–Scargle-based evaluations confirm how well our model preserves important frequency characteristics.\n\n**Computational Cost:** Please refer to the response to Reviewer 3 (**ZpA3**) for a detailed analysis of computational efficiency of our method.\n\n&nbsp;\n### Code Availability\nWe sincerely apologize for the delayed submission of the source code. The upload was unfortunately postponed due to an internal review process, but it is now publicly available in the anonymous repository linked in the abstract.\n\n&nbsp;\n### Weaknesses\nWe would like to highlight that Table 2 of our paper reports experiments on two real-world datasets (PhysioNet and PM2.5), demonstrating that our approach is not limited to synthetic scenarios. In these real-data settings, our method achieves consistent improvements over baseline imputation models, underscoring its practical applicability beyond the controlled environment of Table 1.\n\nFurthermore, like CSDI, our framework remains a probabilistic time series imputation method. Although we incorporate Lomb–Scargle conditioning, the underlying diffusion-based approach is unchanged. It still produces a distribution over missing values rather than a single deterministic estimate.\n\n&nbsp;\n### Question for Authors (Motivation)\nEven though missingness can reflect the inherent sampling schedule, imputation remains a standard practice in the medical time series literature for two main reasons:\n\n- Many predictive tasks, such as patient mortality prediction, rely on uniform or complete data inputs. Empirical results across multiple papers show that having an imputed time grid often improves classification or regression performance on these tasks, with respect to grids with missing data.\n\n- A large number of existing methods and neural architectures assume regularly spaced inputs. Imputation is thus commonly used to align varied medical measurements (e.g. vitals, labs) to a single grid, making it easier to integrate or compare multiple signals and to apply standard machine learning pipelines that are not natively designed for irregular sampling.\n\nWe will integrate this content into the manuscript as motivation for the task of irregular time series imputation.", "claims_and_evidence": "From line 21 to line 23, the abstract claims that the method imputes without requiring imputation \"in the frequency domain\", which seems to conflict with the statement \"prior to frequency estimation\" from line 17 to line 18. Also, imputing in the time domain instead of the frequency domain is more prevalent, so I suspect the claim is wrong.", "methods_and_evaluation_criteria": "In Figure 2, there are two questionable designs. First, in the calculation of consistency loss, the ground-truth spectrum is calculated using masked \"observed condition\" instead of the original \"observed time series\". Second, \"imputation target\" and \"noisy imputation target\" are plotted mostly the same, which is confusing. Since the code is unavailable in the provided anonymous github link, and LSCD is based on CSDI [1], I assume LSCD uses a similar approach in handling the diffusion and denoising process. According to the implementation of CSDI, \"noisy imputation target\" should be distorted with random noise instead of shifted compared to the \"imputation target\". \n[1] Y. Tashiro, J. Song, Y. Song, and S. Ermon, \"CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation,\" in Advances in Neural Information Processing Systems, Curran Associates, Inc., 2021, pp. 24804–24816.", "theoretical_claims": "I check the correctness of conditional diffusion in section 3.3.", "experimental_designs_or_analyses": "Experimental designs are not convincing. \nFirst, Lomb-Scargle is an approach relying heavily on the assumption of sinusoidal signal, which makes it unfair to compare the baselines on synthetic sines datasets. Also, unlike fully observed regular datasets, which usually exhibit clear seasonal patterns, real-world irregular time series usually lack such patterns, making them significantly different from synthetic sines datasets. Moreover, I think frequency domain patterns are not directly associated with data distributions shown in Figure 5, where data distributions are just statistical results on observed values. Therefore, visualizing data distributions cannot prove the effectiveness of  Lomb-Scargle on real-world irregular time series datasets. Lastly, since Lomb-Scargle is more computationally expensive than FFT, an analysis of training time and inference time is necessary to evaluate the model.", "relation_to_broader_scientific_literature": "The main contribution is the usage of frequency domain information in the diffusion model. However, compared to previous work CSDI [1], the introduction of the diffusion model in Figures and the experimental comparisons are not convincing enough. \n\n[1] Y. Tashiro, J. Song, Y. Song, and S. Ermon, \"CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation,\" in Advances in Neural Information Processing Systems, Curran Associates, Inc., 2021, pp. 24804–24816.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Other Weaknesses:\n\n• The code is not available in the provided URL at line 34.\n\n• Math symbols in Figure 2 are in low resolution.\n\n• I believe irregular time series are not equal to time series with missing values. The paper mainly discusses the missingness scenarios in Figure 3. However, unalignments and irregular time intervals in real-world irregular time series are caused by different sampling rates of different variables, which is not the same as regularly sampled time series with missing values. Therefore, Using incomplete synthetic data to simulate irregular time series is unreasonable. Although CSDI [1] also uses irregular time series datasets to prove its effectiveness, its task is \"probabilistic time series imputation\", which is a more general task that includes irregular time series imputation.\n\n[1] Y. Tashiro, J. Song, Y. Song, and S. Ermon, \"CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation,\" in Advances in Neural Information Processing Systems, Curran Associates, Inc., 2021, pp. 24804–24816.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "What's the research purpose of irregular time series imputation, especially on medical datasets? For example, in Physionet'12, samples are collected during patients' 48 hours of stay in ICU, where the values of different features are sampled at different frequencies as scheduled. The missingness is informative to indicate different sampling rates. Therefore, imputation for these medical irregular times series seems to lack proper motivations.", "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: The authors propose to use Lomb-Scargle periodograms as a new layer in a diffusion model for time series. The Lomb-Scargle (LS) periodogram is a technique that allows to estimate the spectrum of a signal even in the case of very irregular sampling. The model (called LSCD) proposed by the authors is essentially a variant of the popular CSDI model (Tashiro et al., 2021) that additionally leverages LS as a layer to handle missing values. On several data sets (several synthetic and two real), this addition seems give very good results (and in particular improve the results of CSDI).

Reviewers generally agreed that using LS within a deep learning framework (and, in particular, a diffusion model) was a sensible idea. However, there was significant disagreement on whether or not the paper was worthy of ICML. Two reviewers thought that the contribution was too simple, and that the novelty was not sufficient for ICML. One reviewer remained neutral (enjoying the elegance of the use of LS but acknowledging the limited novelty), and one championed the paper, arguing that simplicity was a quality and not a shortcoming.

It is indeed true that implementing LS as a differentiable layer in modern deep learning frameworks such as PyTorch is not very hard. Even though I think both sides of the argument had sensible arguments, I do believe that simplicity is a quality in that case. Moreover, I believe that there is a small chance that this paper might make LS more popular in the machine learning world. The LS periodogram (which is essentially equivalent to fitting a sinusoidal function to the signal using the mean-squared error loss) is widely used in astrophysics (e.g. VanderPlas, 2018) but not very well-known in machine learning. Its main strength is that it can deal seamlessly with irregular time series, and could be used in many contexts for dealing with time series. Because of this small chance of having a high impact on the community, I support accepting this paper.

While I recommend acceptance, I strongly encourage the authors to take into account all comments. In particular, adding more details on the theoretical/practical consequences of using sinusoidal functions (as was done in some of the discussions with reviewers) would clearly improve the paper.

# References mentioned in the submission

- VanderPlas. Understanding the lomb–scargle periodogram. The Astrophysical Journal Supplement Series, 2018.
- Tashiro et al., CSDI: Conditional score-based diffusion models for probabilistic time series imputation, NeurIPS 2021
Decision: Accept (poster)