We thank the reviewer for their careful assessment of our paper. Below, we provide detailed responses aimed at clarifying and addressing each concern.

 

Claims and Evidence

Thank you for pointing out the discrepancy: in line 22, "frequency domain" should be "time domain". We have updated the text accordingly.

 

Methods And Evaluation Criteria (Figure 2)

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.

Noisy imputation target: We have updated the figure so the effect of adding noise is more evident and does not resemble a shift.

Math symbols: We have now increased the resolution of math symbols in the figure.

 

Experimental Design or Analyses

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.

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.

Computational Cost: Please refer to the response to Reviewer 3 (ZpA3) for a detailed analysis of computational efficiency of our method.

 

Code Availability

We 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.

 

Weaknesses

We 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.

Furthermore, 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.

 

Question for Authors (Motivation)

Even though missingness can reflect the inherent sampling schedule, imputation remains a standard practice in the medical time series literature for two main reasons:

• 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.

• 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.

We will integrate this content into the manuscript as motivation for the task of irregular time series imputation.

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

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Below we include a general analysis of the computational efficiency of our method, requested by several reviewers.

 

Computation Time

We 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.

Table R1: Training time per epoch for CSDI and LSCD.
The last column shows the relative increase for LSCD. Measurements were averaged over $10$ epochs.

Table R2: Inference time per batch for CSDI and LSCD (batch size = 16).
The last column shows the relative increase for LSCD. Measurements were averaged over $5$ batches.

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.

 

Relation To Broader Scientific Literature (Novelty)

We 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.

We 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.

 

Weaknesses (Table 2 Results)

The 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 ($p=0.065<0.1$).

When 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.

Table R3: Average MAE ranking per table

 

Weaknesses (Theoretical Results)

As 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:

$$\mathbb{H}({\bf z}|\hat{\bf X}_t, {\bf X}^{\bf C}, {\bf C}^{\bf H}, {\bf C}^{\bf D}) < \mathbb{H}({\bf z}|\hat{\bf X}_t, {\bf X}^{\bf C}),$$

where ${\bf z} = \hat{\bf X}_{t-1}$, ${\bf X}^{\bf C}$ 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.

[1] X. Yang et al. Frequency-aware Generative Models for Multivariate Time Series Imputation. NeurIPS, 2024.

 

Question (Computational Efficiency)

We have included a detailed analysis of computational efficiency in the response to Reviewer 3 (ZpA3).

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.

 

Weakness 1 (Related Work)

We 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.

In 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).

 

Weakness 2 (Efficiency Analysis)

Please refer to the response to Reviewer 3 (ZpA3) for a detailed analysis of computational efficiency of our method.

 

Question 1A (Arbitrary timestamp imputation)

LSCD, 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.

 

Question 1B (Effect of number of mask placeholders)

Our 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.

 

Question 2 (Limitations)

Our approach presents the following limitations:

• 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.

• 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.

• 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.

We will incorporate the discussion of the limitations of our approach in the manuscript.