Single-Step Operator Learning for Conditioned Time-Series Diffusion Models

We thank Reviewer bj9n for the thorough feedback and thoughtful questions. We have carefully addressed each point below.

Q1: Why do the authors claim the high-frequency damping from DMs needs correction?

You are absolutely correct that frequency attenuation during forward diffusion is the intended mechanism. We apologize for the sloppy language. To clarify: we are not trying to prevent or avoid the attenuation during forward diffusion. Rather, our goal is to efficiently reverse this attenuation in a single step during generation. The main point is that while individual noisy samples gain high-frequency noise (as expected), the probability density function undergoes systematic frequency-dependent smoothing (Theorem 3.2). In multi-step reverse diffusion, in general, this smoothing is gradually undone across many steps. Since we compress the entire reverse process into a single operator, our operator must learn to restore the full frequency spectrum in one transformation. Our FAB module helps the reverse operator perform this restoration by operating in frequency space where it can amplify the attenuated components based on the noise level. We will revise line 202 to clarify that we are addressing the reverse process, not claiming the forward attenuation is problematic. Thank you.

Q2: Fig.2 is not clear.

Thank you for the very detailed feedback! We will revise Figure 2 to address these issues:

a) Unnamed block: This represents the semigroup consistency training (Stage 2 in Algorithm 1), where we enforce that composing two operations $\phi(\gamma,\tau,\cdot) \circ \phi(\rho,\gamma,\cdot)$ must equal the direct mapping $\phi(\rho,\tau,\cdot)$. We will add a clear label and connecting arrow to the main flow to avoid confusion.

b) Symbol notation: The repeated $\hat{x}$ symbols means outputs at different stages, which we realize is also confusing. We will revise this: $x_\sigma$ for noisy inputs at noise level $\sigma$, and $\hat{x}_0$ for the final denoised output.

c) FT vs Power Spectral Density: You are right: the visualization shows power spectral density as a cartoon illustration to just convey the concept of frequency-dependent processing. In practice, our FAB module operates on the full representation (both magnitude and phase for Fourier, or approximation and detail coefficients for wavelets). We will update the caption to clarify that this is a conceptual illustration, not the actual operation. Again, thanks for reading the figure carefully.

Q3: Some elements are in the appendix only.

This was primarily for space-saving reasons. If accepted, we will be allowed one additional page and this will allow us to absorb most of the implementation details and limitations into the main paper. Thanks.

Q4: Clarify the connection between the Fourier spectral theory and wavelet implementation.

Our technical analysis uses Fourier transforms because they provide a clearer/simpler description of frequency damping during diffusion (Theorem 3.2). However, as we show in Corollary 3.3, the same exponential damping and instant smoothing properties hold in any wavelet basis. For implementation, we chose wavelets because they are often preferred for time-series data: Fourier analysis assumes stationarity while wavelets can nicely capture time-localized patterns. But the important point is that our FAB module works with any orthogonal basis ${\chi_j}$. Both choices (Fourier/Wavelets) leverage the same underlying principle: diffusion attenuates high-frequency/fine-scale components that must be selectively restored during generation.

Q5: Claim of reducing computation overhead.

Thanks for encouraging us to emphasize this point. Our computational efficiency claim primarily comes from multi-step to single-step sampling. While some recent time-series diffusion models require 20 function evaluations during inference, SSOL achieves comparable results with a single denoising step. We performed additional runtime experiments and the key findings are as follows:

(a) SSOL achieves 3.5x faster inference than D3U and 25x faster than NsDiff.

(b) 35% less GPU memory than D3U, 85% less than NsDiff.

(c) Despite far fewer denoising steps, SSOL achieves comparable/better MSE/MAE.

The efficiency gains come from our single-step design. The additional training time for our second stage (semigroup consistency) is negligible compared to the inference savings. Additionally, as detailed in Table 2 in the manuscript, we demonstrate that our training approach can be applied to existing models (D3U, NsDiff), improving their inference efficiency.

Table: Efficiency and performance comparisons on ETTh1.

Q6: Error bars for the imputation experiment in Fig 5.

Yes, you are correct that our 90% credible intervals are quite wide, especially at high missing ratios. This actually suggest that our probabilistic approach is working well for the following reasons:

(a) These empirical quantile-based error bars (5th-95th percentiles from 100 samples) directly reflect the range of plausible values rather than standard error-based intervals. For a signal, when most of its values are missing (90% missingness), the conditional distribution given the limited observed points naturally becomes broad, and our probabilistic model shows this uncertainty.

(b) Even when we have wide intervals, our median predictions (red dotted) tracks the ground truth well. This is very good and shows our model correctly identifies the most likely trajectory while also correctly representing uncertainty.

(c) We find that the intervals appear to vary by signal characteristics: smoother channels (1 and 3) show narrower bands while other signals exhibit broader uncertainty, showing our model adapts its confidence.

We fully agree that the blue dots for missing points are non-standard. We will revise and use solid dots for observed values, with imputed regions clearly marked.

Q7: Fig 3 (right): Why does performance improve with more missing data (10% to 25%)?

This behavior likely stems from how our model (and the FAB module) leverages global temporal patterns during training. At 10% missing ratio with geometric masking, the model sees many training examples with sparse, isolated gaps that must be filled using primarily local context (finer scale wavelet coefficients). This trains the model to rely heavily on immediate neighbors. At 25% missing ratio, the training data contains longer contiguous gaps that force the model to learn and exploit global patterns to bridge these gaps. Basically, the FAB module should adapt to prioritize low-frequency components. This results in a model that better captures the overall signal structure. When evaluated on the test set, this more globally-aware model (trained at 25%) can outperform the more locally-focused model (trained at 10%) because many time series, particularly Weather and Energy, have strong global patterns that, once learned, make imputation more accurate.

Q8: Error bars.

Yes, Table 1 lacks error bars. The point is well taken and we will add error bars to the table our SSOL. The gap with the baselines is large, so the benefits will be very clear in the table.

For sampling stability, Table 6 reports standard deviations across inference runs, showing consistently small values. This confirms that our median-of-100-trajectories estimator produces statistically stable results. Table 5 shows, our method achieves better CRPS_sum performance on 5/6 datasets, confirming better joint distribution modeling. We agree that similar variability measures in Table 1 will reassure the reader. The additional results table shows performance across 3 random training initializations on extended horizons for ETTh1, with small standard deviations.

Q9: Place own contributions in context in related work.

We appreciate this constructive feedback. We will modify the text to clearly emphasize our contributions relative to existing approaches.

Q9: Suitability of the analysis in the spectral domain.

You are right that power spectral density (PSD) analysis must assume stationarity and discards phase information. We need to clarify two key aspects:

(a) We do not operate on PSD: Figure 2 PSD visualization was simply a cartoon illustration of frequency-dependent smoothing. Our actual FAB module operates on complex-valued spectral coefficients that preserve both magnitude and phase. The inner product $\langle x_\tau, \chi_j \rangle$ retains full complex information.

(b) Wavelet avoids stationarity problem: While our theory uses Fourier analysis for simplicity, our implementation uses wavelet transforms to avoid stationarity. Wavelets provide time-localized frequency analysis and our technical results carry through.

The phase reconstruction issue you raised is relevant for Fourier-based approaches but does not apply to our wavelet implementation, where both approximation and detail coefficients are preserved throughout the forward and inverse transforms. We will revise Figure 2 and make sure that this confusion is avoided.

I thank the authors for their responses to the points raised in my review. I am more satisfied with the clarifications provided now, however, I still find the explanations about computational complexity and error bars not fully convincing:

• It's not only for 90% of missing data that the error bars are large (see fig 5 in the Appendix). As this is shown only for the proposed method, it is difficult to assess if these wide error bars are the appropriate ones (as mentioned by the authors' response)
• In terms of computational complexity, from the table provided in the response, I see that the training time is slightly smaller for SSOL (vs D3U), in contrast to the paper's claim "significantly reducing computational overhead".

Despite these flaws, I recognise the value of this work and will increase my score to recommend (marginal) acceptance.

We thank Reviewer XQ6p for the constructive feedback and thoughtful questions. We have carefully addressed all points below.

W1: Is SSOL competitive with the latest approaches in the broader diffusion literature?

While a number of methods have shown impressive single-step generation for images, adapting them to conditional time-series tasks requires substantial re-engineering that goes well beyond simple adjustments to obtain a baseline.

In early stages of our work, the key adaptation challenges we encountered included sensible conditioning mechanisms, redesign of architectures and training objectives. For instance, image uses cross-attention with text/class embeddings. Time-series forecasting/imputation requires conditioning on irregular masks for missing data, or causal historical context, and partial observations. This is quite different from standard text prompts. Many image-focused models use U-Net with 2D convolutions operating on spatial features. Time series requires causal architectures that respect temporal ordering and handling of variable-length sequences with missing data. Image methods optimize perceptual quality, whereas time-series need probabilistic calibration. Again, these are not insurmountable challenges, but getting the architecture to work well and offer a strong baseline requires effort and experimentation. For example, with masked time-series data, defining valid trajectories between partially observed states and complete sequences needs domain-specific modifications. In our preliminary experiments, a direct application of Consistency Models to time-series yielded poor results due to these mismatches, which in some sense were the starting point of the formulation described here. For this reason, we focused our comparisons on recent time-series-specific methods (D3U, NsDiff) that already handle these domain challenges, providing more meaningful baselines for our community.

W2: Sensitivity to Fourier versus wavelet choices, and to irregular or non-uniform sampling.

To fully resolve this question, we performed a set of experiments comparing these bases on the ETTh1 dataset (prediction length 96), where wavelet transforms with db1 basis achieved better performance (MSE: 0.375, MAE: 0.396) compared to Fourier transforms (MSE: 0.391, MAE: 0.412). This performance difference aligns with what we would expect: Fourier analysis assumes global stationarity, and wavelets provide localized time-frequency analysis that better handles the non-stationary patterns and work a bit better.

Regarding irregular and non-uniform sampling, yes we agree that standard wavelet transforms require uniformly spaced samples and rely on dyadic scale relationships. However, our method can handle irregular missing patterns well, as shown through imputation experiments with 10%-90% missing ratios using geometric masks that simulate real-world sensor failures. This addresses one class of irregularity we encounter in practice. But yes, for truly irregularly sampled time series where sampling times themselves are non-uniform, extending our framework would require incorporating more specialized transforms like non-uniform FFT.

W3: How well does the approach scale to substantially longer horizons or to online, continually updated streams?

Thanks for the question! To address this concern, we conducted additional experiments on longer prediction horizons (336 and 720 steps) using the ETTh1 dataset. The results demonstrate that our method maintains competitive performance at extended horizons: SSOL achieves comparable or better deterministic predictions with consistently better joint distribution modeling (CRPS-SUM improvements of 31% at 336 steps and 54% at 720 steps compared with D3U). For streaming applications, our single-step design offers computational advantages. Each prediction requires only one forward pass rather than 20+ iterative denoising steps, reducing inference time. This efficiency enables real-time forecasting as new observations arrive. The conditioning mechanism enables the direct incorporation of new data points into updated lookback windows, allowing for continuous prediction updates in streaming scenarios.

Table: Forecasting performance for ETTh1.

Q1: Can you provide more insight or theoretical justification into when and why a single-step operator is sufficient for accurate approximation?

Our approach is based on the observation that the exponential smoothing of the data distribution's high-frequency modes during diffusion (Thm 3.2), can be effectively reversed by a model that understands the data's own frequency components. The FAB module is explicitly designed for this, learning to restore the fine-scale details in the data that correspond to the information lost in the distribution. The theoretical justification for why this single step is stable comes from the semigroup composition property. By training our operator $\phi$ to satisfy the consistency constraint $\phi(\rho,\gamma,\phi(\gamma,\tau,x)) = \phi(\rho,\tau,x)$, we are teaching it a direct mapping. This forces the model to learn the geometry of the globally consistent denoising trajectory, making the final one-shot reconstruction robust and well-behaved.

But this is not universally applicable. We expect its effectiveness is likely to diminish under certain conditions. As discussed in Appendix D, we work with linear drift and non-degenerate Gaussian noise. The performance on far more complex, state-dependent, or variance-expanding diffusion schedules remains open, a single step may not be sufficient to capture such intricate dynamics. Our FAB is specialized for the patterns found in regularly sampled time-series data. For entirely irregularly sampled sequences or data like high-frequency audio, the construction would likely be less effective without non-trivial modifications.

Q2: Have you explored hybrid settings where part of the operator is learned via single-step and part via recurrence or iterative refinement?

Thank you for this excellent question! Such hybrid settings are feasible within our framework through straightforward modifications to the semigroup composition property. A coarse-to-fine hybrid can be implemented by training the operator $\phi(\gamma, \tau, \cdot)$ where $\gamma > 0$ (e.g., $\gamma = 10$, $\tau = \sigma_{\max} = 80$) rather than $\phi(0, \tau, \cdot)$. This modification to the boundary condition in our two-stage training objective (Section 4.3) allows the single-step operator to map from maximum noise level $\sigma_{\max}$ to an intermediate noise level $\sigma(\gamma)$, while preserving the semigroup composition law $S^*_{\gamma \to \tau}$ for the high-noise regime. The remaining denoising from $\sigma(\gamma)$ to clean data can utilize standard iterative diffusion samplers.

Q3: How does the architecture scale with grid resolution and dimensionality? Any observed bottlenecks in training or inference?

Thank you for the question about scalability. As we mentioned in our previous response regarding temporal resolution, we conducted additional experiments on longer prediction horizons (336 and 720 steps), demonstrating that our method maintains competitive performance with CRPS-SUM improvements of 31% at 336 steps and 54% at 720 steps compared with D3U. Regarding dimensionality, our experiments covered a range from 7 channels (ETT datasets) to 862 channels (Traffic dataset) with various temporal frequencies from 15-minute to hourly intervals. However, we observe that computational costs do increase with both longer sequences and more channels. The FAB module uses wavelet transforms with $O(d \log d)$ complexity for sequence length $d$. For channel scaling, we process each of $C$ channels independently in frequency space ($O(C \cdot d \log d)$ total).

Q4: Would your method generalize to non-grid or graph-based operator learning tasks?

Yes, the core principles are generalizable to non-grid domains like graphs, though it would require adapting some components. The fundamental idea of learning a single-step inverse operator by enforcing the semigroup composition property is not inherently tied to a grid since the diffusion process itself can be defined on other data types. The primary challenge will be adapting the Frequency-Aware Block (FAB), which is currently designed for time-series using Fourier or wavelet transforms. For a graph-based task, this component would need to be replaced with perhaps Graph Fourier Transform (work from 2000s and early 2010, first by Maggioni and layer by Hammond; both these papers are excellent starting points). While the principle of modulating spectral coefficients to reverse the diffusion process remains the same, this adaptation would be a non-trivial architectural modification.

Q5: Could you release the code and data to support reproducibility?

Yes, we will release the full codebase and documentation to reproduce all experiments. We take reproducibility seriously and know that the easiest way to encourage adoption is to release code. Our code repository will be publicly available concurrently with the paper. Thanks.

Q6: limitations or societal impact.

We apologize for the inconvenience. We moved them to Appendix D for space saving and will relocate them to the main body.

We thank Reviewer uKCQ for the valuable feedback and insightful questions. We have carefully addressed each point below.

W1: What is the difference between the proposed single-step operator and the one-step diffusion that works for time series?

Thanks for pushing us to make this point explicit. The main distinction is in how single-step capability is achieved. Many existing approaches typically fall into three categories: (1) distillation methods like consistency models in the image domain need a pre-trained teacher model and iterative distillation to compress multi-step trajectories. (2) Post-hoc approximations like TSDiff will continue to maintain multi-step sampling but will use single-step guidance approximations within each iteration. But overall, 20+ NFEs are still needed. (3) Sequential sampling ideas like D3U and NsDiff reduce to roughly 20 steps through backbone architecture improvements. But still, they are iterative.

Our approach differs in that we build single-step capability directly into the model architecture and training objective. We enforce the semigroup composition property $\phi(\rho,\gamma,\phi(\gamma,\tau,x)) = \phi(\rho,\tau,x)$ during training. So, the operator learns globally consistent transformations across all noise levels rather than just endpoint mappings. We can appreciate that this is not a post-training compression because the model does learn from scratch to perform accurate single-step denoising. This has several benefits. First, there is no dependency on pre-trained models or distillation. Second, we can achieve true single-step generation (NFE=1) versus 20+ steps. Finally, the training optimizes what we want directly, i.e., single-step accuracy rather than hoping that multi-step trajectories can be compressed well. The frequency-aware design helps us exploit time-series structure, unlike generic acceleration (Table 2 shows some of these benefits). We hope this clarifies the doubt.

W2: Compare with one-step diffusion models from the image domain, such as [1].

Thank you for suggesting this recent result. Shortcut Models achieves one-step generation by learning direct mappings from noise to data, bypassing intermediate diffusion states. While conceptually this is similar to pursuing single-step generation, our approach and their formulation differ a fair bit. Some shortcut models train a separate network to directly predict clean data from pure noise, using pre-trained diffusion model only for generating training pairs. This needs a fully-trained diffusion model as a starting point as well as additional distillation-like training of the shortcut network. Finally, one would need to balance between shortcut and diffusion paths during inference. In contrast, we integrate single-step capability by baking it into the formulation via semigroup composition properties. This is from the ground up: no pre-trained model needed.

We also want to note that adapting Shortcut Models to conditional time-series tasks would require substantial modifications: handling partial observations through masking, incorporating temporal causality, and replacing image-centric architectures with time-series-appropriate designs. Our frequency-aware approach specifically leverages how diffusion affects temporal patterns (Theorem 3.2), achieving true single-step generation while maintaining time-series-specific inductive biases. A detailed empirical comparison would be valuable, but we hope that the reviewer will agree that it would essentially require developing a new time-series-specific variant of Shortcut Models. For example, adapting shortcut models[1] to conditioned time series generation requires domain-specific modifications beyond the core self-consistency framework. While the fundamental algebra of learning step-size conditioned shortcuts remains applicable, several adaptations are needed: (1) Instead of categorical labels, the model would condition on historical observations, missing value patterns, and forecast horizons through appropriate embedding schemes; (2) The spatial DiT transformer would be replaced with causal attention mechanisms or temporal backbones (like our FAB) designed for sequential dependencies; (3) Introducing mask-aware loss functions optimized only on future/missing indices. These modifications show that the direct application of shortcut models to time series is not straightforward and requires the adaptation approach we propose.

[1] Frans, K., Hafner, D., Levine, S., & Abbeel, P. (2025). One Step Diffusion via Shortcut Models.

W3: Report probabilistic metrics.

Thanks for the question. We evaluated our method using CRPS and CRPS-sum metrics, with results in Table 5 (Appendix C). SSOL achieves the best CRPS-sum performance on 5 out of 6 datasets and competitive individual channel CRPS performance. This shows our method's strength in modeling joint distributions across channels. We apologize for not including these metrics in the main results table.

We initially presented MSE/MAE to allow comparison with both probabilistic and deterministic baselines, as many recent time-series papers report only point estimates. However, we agree that probabilistic metrics are important. The CRPS results show an interesting pattern: while D3U excels at per-channel uncertainty quantification (best CRPS), SSOL better captures inter-channel dependencies (best CRPS-sum), likely due to our frequency-aware design that preserves correlation structures. If accepted, we are allowed one additional page of content. We will move these probabilistic metrics to the main results table in the revised version.

We thank Reviewer pJbC for the detailed feedback and thoughtful comments. We have carefully addressed each point below.

W1: Citation format.

Thank you for the feedback. We made a mistake because of a conflict due to using cite/citet with natbib and other packages. We have made this change, and it is now numeric, thanks!

W2: The paper lacks comparisons with existing step-reduction methods from image generation.

Thank you for this excellent comment. We agree. Indeed, our construction builds upon single-step generation principles from image-domain methods. For example, the semigroup setup is conceptually quite similar to consistency models since both enforce compositional properties across noise levels.

The reviewer will agree that adapting any shortcut method to a time series requires numerous domain-specific design choices:

1. Conditioning: Image methods use cross-attention with text/class embeddings. For time-series forecasting/imputation, we need to condition on (a) partial observations with irregular masks, (b) historical context with temporal ordering, and (c) missing data patterns. Our mask-aware conditioning is the result of many iterations of adjusting various ideas in image generation shortcuts, and turns out to be different from what works well in image domains.
2. Backbone: Many methods for image generation use U-Net, for time-series we need architectures that respect temporal causality. Again, conceptually there are many possible ways to do this, e.g., causal convolutions or masked attention, but many strategies do not directly work well. Separately, the FAB module exploits time-series structure by operating in frequency space. While we can start from various architectures, from image-generation shortcuts, finding the right combination that handles both causality and leverages time-series structure well, as well as gives good experimental results, is a bit involved.
3. Training Objectives: Image methods often use perceptual losses. Time series requires point-wise accuracy and probabilistic calibration (CRPS). Our two-stage training (Algorithm 1) balances these objectives without requiring a pre-trained teacher model.

Overall, we completely agree with the idea that adapting any image shortcut method to a time series is conceptually straightforward. But the implementation requires rethinking nearly every component. Each design choice cascades where conditioning mechanisms affect architecture choices, which impacts training objectives, which interacts with frequency processing. We describe one coherent path through this design space. For example, adapting MeanFlow[1] to time series would need several domain-specific modifications while keeping the core operations (flow-matching identity and one-step update) unchanged: (1) redefining the average-velocity field for 1-D temporal space; (2) replacing categorical conditioning with structured history plus mask embeddings; (3) implementing a causal 1-D Transformer or our FAB backbone with continuous-time position encodings; (4) introducing mask-aware loss functions optimized only on future/missing indices.

A direct comparison with consistency/flow-matching methods would require deriving entirely new time-series-specific architectures based on those principles and would need to create new methods rather than simple adaptations.

[1] Geng, Z., Deng, M., Bai, X., Kolter, J. Z., & He, K. (2025). Mean Flows for One-step Generative Modeling.

W3: Performance drops when inference noise ranges differ from training distributions.

Thank you for bringing this up. This behavior is expected. Our single-step operator $\phi(\gamma,\tau,x)$ is trained to implement the inverse semigroup mapping from specific noise levels. During training with $\sigma \in [0.002, 80]$, the operator learns the full transformation $\phi(0, 80, x)$, from maximum noise to clean data, by enforcing the semigroup composition $\phi(\rho,\gamma,\cdot) \circ \phi(\gamma,\tau,\cdot) = \phi(\rho,\tau,\cdot)$. But when $\sigma_{max}$ is reduced at inference time (e.g., to 10), we are essentially asking the operator to perform $\phi(0, 10, x)$, a transformation it can compose through intermediate steps but has not directly optimized for as a single jump. The forward diffusion at $\sigma=10$ hasn't sufficiently smoothed the data distribution (Theorem 3.2), so the prior is too close to the data manifold rather than being near-Gaussian. This violates the assumption that we start from a noise distribution with sufficient entropy for the reverse operator to shape successfully into the target distribution. Figure 4 (right panel) confirms this: performance remains stable for $\sigma_{max} \in [50,80]$ where the prior is sufficiently noisy, but degrades below this range. This shows our model is performing diffusion-based generation rather than learning a shortcut that ignores noise.

Q1: Actual training time.

We provide a runtime analysis in the table below, showing actual training and inference times measured on the same hardware setup. Our two-stage training shows comparable efficiency to existing methods, with Stage 1 (boundary denoising) and Stage 2 (semigroup constraints) requiring similar per-iteration times to baseline single-stage training. The additional semigroup constraint stage adds minimal training overhead compared to standard denoising training, while our training GPU memory usage remains lower than baselines. Although we do not achieve a pure 20× speedup in wall-clock time due to backbone architecture differences across methods, our approach shows strong efficiency gains during inference.

The key findings are as follows:

(a) Inference speedup: SSOL achieves 3.5x faster inference than D3U and 25x faster than NsDiff.

(b) Memory efficiency: 35% less GPU memory than D3U, 85% less than NsDiff.

(c) Performance parity: Despite far fewer denoising steps, SSOL achieves comparable/better MSE/MAE.

The efficiency gains come from our single-step design. The additional training time for our second stage (semigroup consistency) is negligible compared to the inference savings.

Table: Runtime analysis for ETTh1.

Q2: How does SSOL maintain temporal coherence with sparse historical data? Does it over-rely on priors in such cases?

Good question! Our SSOL method preserves temporal coherence through at least three key components/modules. First, the FAB module decomposes signals into multiple scales. So, even with sparse observations, low-frequency components can often be reliably estimated. This gives a sensible skeleton for reconstruction. High-frequency details can then be conditionally sampled consistent with this skeleton. Also, unlike unconditional generation that relies purely on priors, our operator $\phi$ is explicitly conditioned on both the mask $M$ and the observed values. During training, the model must figure out how to propagate information from observed regions through the semigroup structure. This allows learning how different masking patterns affect the conditional distribution. Finally, as observations become sparser, the model avoids blindly defaulting to generic priors. Instead, it adaptively widens its predictive distribution (Figure 5). The median estimator extracts the most likely trajectory while the widening credible intervals give some signal about when reconstruction is speculative versus data-driven. Rather than excessively relying on priors, we believe that SSOL learns observation-specific conditional distributions. With 90% missing data, the model must leverage learned temporal structures more heavily, but this is appropriate given the information constraints.

Q3: Explain why the proposed method can ensure performance with single-step sampling.

Our single-step performance guarantee stems from enforcing the mathematical semigroup composition property $S_{\gamma \to \tau}^* \circ S_{\rho \to \gamma}^* = S_{\rho \to \tau}^*$ during training. We learn an inverse operator $\phi(\gamma,\tau,\cdot)$ that must satisfy the semigroup composition constraint $\phi(\rho,\gamma,\phi(\gamma,\tau,x)) = \phi(\rho,\tau,x)$ for all $0 \leq \rho \leq \gamma \leq \tau$, meaning any noise transition should yield identical results whether taken as a direct jump $(\tau \to \rho)$ or decomposed through intermediate steps $(\tau \to \gamma \to \rho)$. Our two-stage training first uses boundary conditions via standard denoising loss (Stage 1), then explicitly encourages this composition property using an adaptive grid scheduler $N(\cdot)$ that progressively samples intermediate noise levels $\sigma_n = (\sigma_{\max}^{1/\upsilon} + \frac{n}{N(k)-1}(\sigma_{\min}^{1/\upsilon} - \sigma_{\max}^{1/\upsilon}))^\upsilon$ and minimizes $\mathcal{L}\_{cmp}=|\phi(\rho,\tau,x_\tau)-\phi(\rho,\gamma,\phi(\gamma,\tau,x_\tau))|\_2^2$ (Stage 2). This objective forces the operator to learn transformations that are consistent across all noise scales, while our frequency-aware block $B_\theta(x_\tau,\sigma(\tau)) = \sum_j \Psi_\theta(\alpha_\theta(\sigma(\tau)) \cdot \langle x_\tau,\chi_j \rangle)\chi_j$ specifically targets the exponential damping of high-frequency modes identified in Theorem 3.2. We are training the operator to approximate the semigroup composition, so $\phi(0,\tau,\cdot)$ learns to approximate inverse transformation $(S_{0 \to \tau}^*)^{-1}$ through semigroup property constraints rather iterative approximations.