Title: Single-Step Operator Learning for Conditioned Time-Series Diffusion Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time series; Diffusion model
Abstract: Diffusion models have achieved significant success, yet their application to time series data, particularly with regard to efficient sampling, remains an active area of research. We describe an operator-learning approach for conditioned time-series diffusion models that gives efficient single-step generation by leveraging insights from the frequency-domain characteristics of both the time-series data and the diffusion process itself. The forward diffusion process induces a structured, frequency-dependent smoothing of the data's probability density function. However, this frequency smoothing is related (e.g., via likelihood function) to easily accessible frequency components of time-series data. This suggests that a module operating in the frequency space of the time-series can, potentially, more effectively learn to reverse the frequency-dependent smoothing of the data distribution induced by the diffusion process. We set up an operator learning task, based on frequency-aware building blocks, which satisfies semi-group properties, while exploiting the structure of time-series data. Evaluations on multiple datasets show that our single-step generation proposal achieves forecasting/imputation results comparable (or superior) to many multi-step diffusion schemes while significantly reducing inference costs.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a model for time series. This is achieved by building on the observation that the noise-injection procedure in diffusion models removes/attenuates frequency components; therefore, the proposed model designs an operator over the frequency representation of the time series so as to ensure a correct treatment of the frequency components. Such operator, parametrised as a neural network, is implemented in a single step. The method is implemented over standard time series datasets and compared against a number of recent TS models.\n\nSTRENGTHS AND WEAKNESSES:\nThough the underlying observation of the paper is a well-known feature of denoising DMs (even with blogs written on the topic, see the link below and the references therein), the proposal is interesting, but it can be explained and motivated more thoroughly. Also, the experimental validation of the method is very thorough and promising. \n\nAs far as I understand, a hypothesis of this work is that implementing the denoising operator in the frequency domain should preserve the structure of the time series in a way that is more accurate than if done in the time domain. Why is this the case? Shouldn't the frequency and temporal representation be equivalent? Since both representations are one-to-one, any operation defined over one space should have an equivalence in the other space. \n\nI will refer to the weaknesses and strengths of the paper in more depth through questions in the next section. \n\nblog: https://sander.ai/2024/09/02/spectral-autoregression.html\n\nQUESTIONS:\nMajor: \n- It is not clear to this reviewer why the authors claim that (line 202) we need to **correct** the high-frequency damping introduced by DMs. That is the way DMs work, and it is not an \"artefact\", since intermediate (noisy) samples between source and target are not supposed to exactly follow the harmonic content of the data (but rather of the noisy data). I acknowledge that keeping track of the frequency components might be useful, but the call for avoiding their attenuation might be misleading\n- Fig2: The figure is not clear: \n    - What is the unnamed block in the bottom half of the figure? That block is disconnected from the rest of the diagram\n    - there are symbols repeated in the figure (\\hat{x}) - are those nodes the same?\n    - The diagram shows a Fourier Transform (FT) operator, but the plot shows a power density (which is the square absolute value of the FT, not the FT)\n- Some key elements in the paper are in the appendix only, e.g., the limitations and details about implementation\n- The presentation of the method is done using the Fourier spectral representation, but in practice, the method is implemented using wavelets. Could the authors clarify and comment on this?\n- One of the claimed contributions of the paper states: *significantly reducing computational overhead*, however, there is no experiment validating this claim. Note that the authors say that they refer to computational complexity in Appendix D, but this is largely uninformative.\n- Fig 5 reveals that the error bars are rather large for the imputation experiment - some of them as large as the entire range of the signal. Also, the authors seem to use a fairly nonstandard presentation of their results, where blue dots are missing points (rather than observed points)\n- Fig 3 (right): Why does the proposed method improve with more missing data (from 10% to 25%)\n- The main experiments do not provide error bars, which makes it difficult to (statsitically) confirm the claimed superiority of the proposal\n- The related work Section provides the right previous work; however, the authors do not place their own contribution in context\n- Critically, the authors do not refer to the suitability of the analysis in the spectral domain. For instance, modelling the signal using the power spectrum (as suggested by Fig2) is only applicable for stationary signals. Furthermore, \"coming back\" from PSD space to time space requires signing the signal's phase, which is also unaddressed. \n\n\n\n\nMinor: \n- There seems to be a formatting issue with the references, the authors have probably typed \\cite, but they use references as \\citep\n- line 344: Tabel -> Table\n- Table II: only the year of the previous work is provided, and not the full reference.\n\nLIMITATIONS:\nSee my questions above - also note that the limitations are only mentioned in the appendix", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 2, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Single-Step Operator Learning (SSOL), a diffusion-based framework that reconstructs or forecasts multivariate time-series in a single denoising step. SSOL first models how the forward stochastic differential equation exponentially dampens high-frequency modes, then counteracts this effect with a learnable Frequency-Aware Block that projects each noisy sample onto a spectral basis and re-weights its coefficients as a function of the noise scale. Building on semigroup theory, the authors define an inverse operator $\\phi$ that blends the spectral denoiser with the identity map and is explicitly trained to satisfy a Markov composition law, enabling a direct jump from the maximum-noise level to clean data without intermediate steps. A two-stage training objective aligns the model with the denoising boundary and progressively enforces the composition constraint over a refined noise grid. Experiments on six standard forecasting datasets and two imputation benchmarks show that SSOL matches or exceeds multi-step diffusion baselines while reducing function evaluations from 20 to 1, yielding roughly a twenty-fold speed-up.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper is technically solid and grounded in clear theoretical insights. The derivation of frequency damping in the forward SDE is rigorous, and the single-step inverse operator is carefully tied to semigroup composition, making the core idea both novel and well justified. \n\n2. The proposed Frequency-Aware Block is explained in sufficient mathematical detail, and the accompanying algorithms are easy to follow. \n\n3. Empirical validation is comprehensive: six forecasting datasets and two imputation tasks show that SSOL can match or surpass strong multi-step diffusion baselines while cutting the number of function evaluations from 20 to 1, yielding impressive practical speed-ups. \n\n4. The manuscript itself is well structured, with clear sectioning, helpful figures, and a thorough reproducibility checklist, so readers can trace claims without undue effort. \n\nWeaknesses:\n\n1. Despite its merits, the study omits direct comparisons with the most recent single-step or consistency-distillation samplers (e.g., Consistency Models 2023), leaving readers unsure whether SSOL is competitive with the latest approaches in the broader diffusion literature. \n\n2. The method’s reliance on a fixed spectral basis also raises questions about robustness: sensitivity to Fourier versus wavelet choices, and to irregular or non-uniform sampling, is not explored. \n\n3. While the paper demonstrates good results up to a horizon of 192 steps, it remains unclear how well the approach scales to substantially longer horizons or to online, continually updated streams.  \n\n4. Finally, although the authors promise to release code, the current submission does not yet include a public repository or seeding scripts, which limits immediate reproducibility.\n\nQUESTIONS:\n1.\tCan you provide more insight or theoretical justification into when and why a single-step operator is sufficient for accurate approximation?\n2.\tHave you explored hybrid settings where part of the operator is learned via single-step and part via recurrence or iterative refinement?\n3.\tHow does the architecture scale with grid resolution and dimensionality? Any observed bottlenecks in training or inference?\n4.\tWould your method generalize to non-grid or graph-based operator learning tasks?\n5.\tCould you release the code and data to support reproducibility?\n\nLIMITATIONS:\nThe paper does not explicitly discuss limitations or societal impact. Please consider adding a short section on cases where single-step learning may underperform (e.g., highly nonlinear or chaotic systems).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Single-Step Operator Learning (SSOL), an approach to time-series diffusion models that enables efficient, one-step generation by leveraging frequency-domain insights. Unlike traditional diffusion models requiring multiple iterative denoising steps, SSOL learns an inverse operator that restores clean signals from noisy ones in a single pass, using a frequency-aware block and enforcing semigroup consistency. Extensive experiments show that SSOL matches or outperforms state-of-the-art probabilistic and deterministic models in forecasting and imputation tasks, while achieving up to 20× faster inference.\n\nSTRENGTHS AND WEAKNESSES:\nstrengths:\n\n1. The studies problem is important as well as challenging.\n\n2. The idea of proposing semigroup consistency is noval, which avoids one-shot denoising from collapsing during inference or degenerating into a simple regression model. And the overall framework is techically sound.\n\n3. The experiments shows the effectiveness of the proposed method, interms of the sampling efficiency and prediction&imputation performance.\n\nweakness:\n\n1.  Lack of detailed comparison between the proposed method and other one/few -step diffusion works\nWhat is the difference between the proposed single-step operator and the one step diffusion works for time series?\nIn line 377-379,  the authors mentioned that \n“Some of these approaches often rely heavily on teacher-student training. In contrast, our work leverages  classical principles Pazy [1983], Henry [1981] to define a single-step inverse operator.” However, that does not address the benefits of the proposed method.\n\n2. Also there are other one-step diffusion methods like [1], it would be better to compare with them if applicable. \n[1] One Step Diffusion via Shortcut Models\n\n3. For the overall result table 1, to compare the probabilistic methods, why not reporting their performance in metrics like CRPS? Since for those methods, it is important to compare their probabilistic forcasting ability.\n\nQUESTIONS:\nplease see the weakness above\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces an operator-learning approach for conditioned time series diffusion models, enabling efficient single-step generation by leveraging the frequency-domain characteristics of both the data and the diffusion process itself. The core insight is that the forward diffusion process induces a structured, frequency-dependent smoothing of the data's probability density function, which can be effectively reversed by a module operating in the frequency space. By setting up an operator learning task with frequency-aware building blocks that satisfy semi-group properties and exploit time series structure, the proposed method achieves forecasting and imputation results comparable or superior to many multi-step diffusion schemes, while significantly reducing inference costs.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n1. SSOL reduces sampling steps from multistep to 1, achieving faster inference without sacrificing accuracy and showing efficient single-step generation ability. \n\n2. The Frequency-Aware Block (FAB) explicitly restores high-frequency components attenuated by forward diffusion, using wavelet transforms to capture local temporal patterns. This design aligns with the inherent frequency structure of time-series data.\n\n3. SSOL demonstrates consistent performance across diverse time-series domains, highlighting its adaptability to varying data characteristics.\n\nWeakness:\n1.The author's citation format seems to have a major issue; the reference entries are mixed directly into the main text, which makes reading very difficult.\n\n2. The idea of reducing denoising steps has already been extensively researched in image generation. \nMost of these methods can be directly utilized on time-series forecasting tasks. However, the paper lacks comparisons with these approaches, such as consistency sampling or flow-matching-based methods, among others.\n\n3. Performance degrades when inference-time noise distributions (e.g., narrower σ ranges) differ from training, indicating limited robustness to distribution mismatches.\n\nQUESTIONS:\n1. What is the actual training time for the two-stage training (boundary denoising and semigroup constraints) that requires iterative optimization?\n\n2. If the noise range during the inference stage is significantly smaller than the training setting and the performance seems to have dropped noticeably, how can this be explained?\n\n3. In imputation tasks with sparse historical data, how does SSOL leverage limited information to preserve temporal coherence? Does it rely excessively on prior distributions in such cases?\n\n4.Can you explain why your method can ensure performance with single-step sampling? \"avoids the iterative error propagation and hyperparameter tuning\" seems insufficient.\n\nLIMITATIONS:\nNot found", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}