Single-Step Operator Learning for Conditioned Time-Series Diffusion Models
Authors: Hui Chen
Vikas Singh
Venue: NeurIPS 2025 poster ()
Keywords: Time series; Diffusion model
Primary area: applications

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.

Single-Step Operator Learning for Conditioned Time-Series Diffusion Models
Hui Chen Department of Statistics University Wisconsin Madison hchen795@wisc. edu
Vikas Singh Department of Biostatistics and Medical Informatics University of Wisconsin Madison vsingh@biostat. wisc edu
Abstract
Diffusion models have achieved significant success, yet their application to time series data, particularly with regard 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 leverag- ing 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-dependen smoothing of the data's 's probability density func- tion. However, this frequency smoothing is related (e.g., via likelihood function) easily accessible frequency components of time-series data. This suggests that a module operating in the frequency space of the time-series can, poten- tially, more effectively learn reverse the beequency-dependee the data distribution induced by the diffusion process. We set up an operator learning task, based frequency-aware building blocks, which satisfies semi- group properties, while exploiting the structure of time-series data. Evaluations multiple datasets show that single-step generation proposal achieves fore- casting/imputation results comparable (or superior) to many multi-step diffusion schemes while significantly reducing inference costs. Our code is available htttp::/// //github www.wwiinnieeeccc SereiingggrouuSSS-ttiieeee Srtoup//S0U-ttimeerriss
smoothing of
Introduction
Ho al. 2020 has benefited immensely from the capabilities of modern diffusion models, which can capture complex data distributions across many different domains: from image synthesis to high-fidelity audio generation (Kong et al.) 2021 Chen et al. 2021]. However, the application of these models to time-series data is still in its early stages
Generative modeling [Sogg and Ermon 2019
Tashiro et al. 2021 Rasul al.
Li et al. 2022 Alearaz Strodthofi] 2023 Yuan and Qiao Ye et al. 2025 and poses unique challenges. One widely acknowledged obstacle, which applies more generally to diffusion models, the computational
2021
2024| Fan et al. 2024 Li et al.
2025
burden of the iterative denoising process [Song et al. 202 la Salimans and Ho 2022 Lu et al.) 2022 We need sequential evaluations during both training and inference, which makes the process slow and resource-intensive. A few of these issues are further compounded for time-series data with long time horizons. While excellent progress being made to mitigate these difficulties broadly [Salimans] and Ho| 2022 Lu al. 2022]. is natural to ask whether the specific properties and the structure generative modeling that retain the
of time-serics data may allow simplifications to diffusion- expressiveness/capccity of the models but can reduce the computational burden, especially for the sampling phase.
We seek approach the problem above by viewing the forward diffusion process as operators [Li et al.) 2021 Kovachki et al. 2023), where a suitably parameterized operator governs the evolution data dis- tributions across noise scales. This operator acts as semigroup [Pazy] 1983 Henry 1981] governing
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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how the data degrades, as noise is progressively introduced. We hypothesize that the character- istics of time-series data make them well-suited to benefit from this view. One observation is that the diffusion operator performs smoothing on the data's probability density function, sys- tematically attenuating different frequency com-
Higher Frequency Components Added
Tiaes Seiee
wyww
WMM
ponents at different rates, as shown Fig. I This frequency smoothing of the probability den-
sity is distinct from but related to modulation of the frequency-spacc representations of the
Clean Duta
Higher Frequency Noise Added
Pure Noise
Figure 1: Dual view: forward diffusion smooths out the frequency distribution (top). Simultane- ously, individual time-series trajectories gain high- frequency noise (bottom).
time-series data, and follows a predictable pat- smoothing via diffusion occurs broadly, its structure is simpler to interpret and verify in time-series data, where frequency components directly correspond to temporal patterns, such as daily or weekly cycles. This suggests that scheme to characterize and leverage this behavior could potentially invert (or learn to invert) the diffusion process more efficiently, This is the specific direction we explore here, checking if rapid generation possible while preserving the expressiveness that diffusion models are favored for.
tern. While rrqqeecc--depende
Our overarching goal is to learn an approximation of the inverse of the diffusion operator, in way that we can use it to effectively and efficiently reverse the noise addition process. Unlike conventional diffusion models that rely iterative denoising, we seek to achieve this in a single step, reducing the compute cost. This capability is due to a specialized (but obtained with a relatively small set of adjustments) architecture that includes an operator trained reconstruct and prioritize the distinet frequency components of the time-series. By conditioning this operator on both the starting and target noise levels, we can guide the model with a clear objective: to transform noisy signal at given scale to a cleaner representation at another scale, effectively traversing the trajectory defined by the diffusion semigroup. The training scheme enforces the composition property inherent to semigroups. ensuring that the learned transformations are consistent across different noise scales.
The main contributions of this paper are: (a) We introduce new construction for single-step sampling in conditioned time-series diffusion models, by inverting the diffusion process. The archi- tecture is based on operator learning principles and seeks to leverage the distinct roles of frequency components during denoising in the case of time-series data. (b) We validate the effectiveness of the idea on an extensive set of experiments where we match exceed the performance of baselines while significantly reducing computational overhead.
Preliminaries: diffusion models
A continuous-time diffusion model [Song et al. 2021b) defines forward process that gradually injects noise into data through a stochastic differential equation (SDE):
dx f(r)=d7 + g(T) dw.,
where {x(T))T=o C Rd, x(r)= (2)( xT-1(T)) ( denotes the univariate time-series trajectory of length d indexed by continuous diffusion time (or noise scale) E [0, 2T. w, is a standard d-dimensional Wiener process, f(r)æ is a time-dependent linear drift in æ, and g(T) is dime-dependent term control the diffusion amplitude. This process transforms clean data æp into increasingly noisy samples (TT). For notational simplicity, we write x,- = xTT).
Forward process. The forward process progressively obscures information in the data distribution. The marginal distribution pr (z) of the noisy data æ, at noise scale 7 can be expressed as a convolution of the original data, say Paata (ar) with a kernel. Specifically, the marginal distribution of the noised data x, evolves according to:
(a)+ = 5(7)d Paata (N (T))
1
(2)
where Pdaaa denotes the empirical data distribution of the "clean" data No. s(T) exp(Ir Io f(u) du),
and (²()) = fo
g(w)² a(u)z du are time-dependent scaling and variance functions respectively. The

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convolution with increasingly fatter Gaussian kernels controlled by (T) progressively suppresses high-frequency details by smoothing them out. The marginal distribution a, depends on 7 through noise scale (T), so we equivalently write Pa(r) (ar) Pr (x:). We denote a (T) by a or whenever its r-depennencce is clear from context.
Reverse process and loss. To reverse this information loss and generate samples from the original data distribution, one often estimates the gradient V log Pr (2z), which provides the local direction of the steepest ascent toward regions of higher likelihood under py. This is often modeled via a network that is trained to approximate the score function or equivalently to denoise the noisy data. The specific parameterization of the network can vary, with different choices leading to different reverse-tine time diffusion processes. One successful instance of this strategy, which we use in this work, is the EDM model in Karras et al. [2022] In this approach, one trains a neural network Hg to predict clean data from noisy samples To (I, I2) using a skip-residual architecture:
(z), Cakip a + Co) [Ba (Cin (o)(ao) Cnoise (n)is)
where By is the network block (c.g., U-net in images and wavenet in time-series) and Cakepp Cout- Cin. and Cnoise are scaling coefficients that depend on the noise level at a. For example, Cin and Cout can perform scaling to ensure that the input and output have unit variance. The model is trained by minimizing the weighted denoising objective:
(()) = E., E.,e, (w((]
(H(o))Ha((z)) Ho + oe, - x
(4)
with noise levels sampled from log-normal distribution, where w(-) is a signal-to-noise ratio weight function [Karras et al. 2022 and is sample from standard normal distribution.
3 Forward diffusion for time-series data
Reversing diffusion process is difficult, especially in single step. So, need to leverage all available structural information about the data and the diffusion process itself. In particular, we will exploit the semigroup property of forward diffusion, which gives compositional structure in how noise is added across time. Further, we want to use the distinct frequency- domain characteristics of time-series data and its link the smoothing induced by the forward diffusion. We begin by examining how forward diffusion transforms the data distribution. This will yield some structural constraints leverage. Next, delve into the trequency-domat demonstrating how the operator's 's action leads to structured, rrequency-depender attenuation of information in the data's probability distribution which emphasizes the difficulty of the problem.
implications of this forward process,
3.1 Time-inhomogeneous Markov evolution
We now analyze how the diffusion SDE (1 evolves the distribution of data over time (since our data time-series, "time" is overloaded). By classical SDE theory [Øksendal 2003). for each initial condition 2p, there is a unique strong solution Φr. Denoting the law (distribution) of æ; by pr, we know the evolution of pr is governed by the forward Kolmogorov (Fokker-Planck) equation (Henry
1981
Evans 1998.
Grafakos 2014]:
Öp-
Òr (T) Pir=0 ==
Pdatas
(5)
where the operator (T) captures both the drift and diffusion effects: p²(r)p=-V) [[(])a]]
+
((T))Ap,, with A denoting the Laplacian in Rd. We denote Pdata simply by Po-
To capture how the density evolves from any time (noise scale) 7 up to a later time т, we can define a time-inhomogeneous Markov semigroup. Under mild regularity assumptions of globally Lipschitz drift (f(T)æ is linear in x) and uniformly nondegenerate diffusion coefficients (g(7)² bounded away from 0), classical SDE theory [Oksendal 20033 guarantees pathwise existence and uniqueness of solutions to the forward diffusion process
(I). Equivalently, the corresponding
equation (5) is well-posed under these same conditions, which ensures the induced semigroup uniquely determines the transition distribution.
Definition 3.1 (Two-parameter Markov family T.A). For each S 7 S r, define the operator S LL L (IRd) to be the unique linear operator that sends an initial density p., at time
3

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to the solution p, at time T of the Fokker-Plaane equation (5). i.e. Pr = (p,r) We call
(o the two- ww-parameter Markov family generated by The family satisfies the semigroup composition law for any f p S
S pp-7 S ST = Sptri p-t S = Identity.
Moreover, each - is positivity- and mass- mass- preserving.
Relevance. The semigroup property captures the Markov nature of the process - the evolution from time P to T can be decomposed into the evolution from p to y followed by evolution from y to This Markov perspective will be useful later (Sec. 4 when we attempt to invert the forward diffusion with reverse operator, where this property can offer constraints estimating those primitives that will compose to give us longer-range transitions. However, recovering the original distribution is not simply applying the same semigroup in reverse, because diffusion is dissipative and thus not inherently invertible. We can see this by evaluating how extensive the dissipation in practice.
3.2 Forward diffusion: exponential damping
We now check that the dissipative process leads to exponential suppression of high-frequency detuils. In essence, when the forward SDE spreads out and smooths the data distribution, it "dampens" or attenuates fine-scale modes. This behavior becomes clear by examining the distribution's 's Fourier transform. We formalize this in Theorem 3.2 below.
Theorem 3.2 (Exponential decay and instant smoothing). Let po be initial probability density on RR, and assume that pr solves the forward Kolmogorov equation (5). Define the spatial Fourier transform (+) J2= e- -g- (x)) dæ,, & Rdd Then, for each fixed & + O and >> 0, the quantity Pr(E) is suppressed by Gaussian factor in ||E]|2:
(E) (o(((T) exp(
(i)=)(T)3) 22)))
(6)
f(u) du) and (²( fo stuf du. Hence, each nonzero Fourier mode in where s(T)= exp(So =xp)=
pr (E) decays exponentially in flr fixed time r 2 0 and in T for fixed ||E|| * 0, provided that ((T) grows in T.
Example 3.1. Consider the simple SDE dx = bdw,, b > O. with solution 2, + b W-. Its distribution is normal with mean zg and variance 12r, so the PDF spreads out as t grows, flattening the peak and widening the tails.
This reflects the damping of high-frequency components (in terms of the data distribution). In other words, while the overall density becomes progressively smoother and more diffuse (losing high-frequency details), individual realizations exhibit increasingly large random fluctuations (for time-series data, this means introduction of high frequency noise). It is worth emphasizing that the smoothing of the PDF alongside increased noise in individual samples arises because we are looking at the distribution of possible trajectories, not the trajectory of any single sample path. Recognizing this dual behavior motivates a frequency-awan approach that exploits the information the transform space of the time-series data, where can analyze and manipulate the frequency components of the data itself. By understanding how the diffusion process affects the data distribution in the frequency domain, we can design model to selectively restore and reweight dominant frequencies, more effectively reversing the smoothing effect on the PDF This selective reweighting, if we can learn it, can help restore essential features of the initial distribution.
As one would expect, we notice that the same exponential damping of high-frequency components also manifests in other transform spaces, including wavelet spaces. In particular, we find that the solution pr (-) to equation (5) exhibits an instant smoothing property: as soon as t > 0, pr belongs to every Sobolev space H (..).. This implies rapid decay of its coefficients in any standard multi-scale representation (c.g.. wavelets), and so for other types of data, our frequency informed/awan /awre module can be adjusted based on other transforms best suited for the data at hand.
Corollary 3.3 (Instant smoothing). With the assumptions in Thm 3.2 assume additionally that
Pr E L (IR)) ) each t 2 O. Then for every T > and for every real K 2 O, we have pr E H (Rdd)
Equivalently, the wavelet coefficients b (p+) ) at scale position k satisfy j.b 203 16 (px())) < 0o
for each k 2 0. Hence p. is infinitely differentiable at any 0.

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Frequency-Aware Block (o(:.))
Scaling
vo6))
H
Basis Oxl
Vin
F(Rau))
Fout
Flow Function
Eo(r)
p(p, T
Semigroup Consistency Training (Stage 2)
Co(r
o (T)
Bo(r)
6(7 T,
((p: 7
To(p)
(p(p)
Time
Noise
((pTT,))
Denolsed
Series
Level
Figure 2: Overview of our single-step inverse operator for conditioned time-series diffusion models. A noisy input Xo(r) is passed through frequency-aware block to selectively restore high-frequency components. A single-step operator & then enforces the semigroup property to denoise across noise levels in single step. The bottom block illustrates the semigroup consistency training that enables this single-step capability by enforcing p(7, T, p 7. = T,
Relevance. The above results show clear picture of forward diffusion: (i) it forms Markov semigroup S that is not trivially invertible, and (ii) exponentially suppresses high-frequency details in both Fourier and wavelet (and other) domains. To recover such lost details, standard diffusion models resort to many small reverse steps. We will seek restore these details one step by exploiting information regarding the frequency domain of the time-series described above and the composition property, conjunction.
4 Single-step inverse operator
The discussion suggests that we need two ingredients: (i) frequency-awarr block that uses the frequency-domanl characterization of the time-series to restore the damped high-frequency compo- nents (of the data distribution), and (ii) semigroup- different time intervals. We begin by describing how the block selectively boosts suppressed modes in 4.2
reverse operator that is well-defined across
then show how embed it within flow operator that approximates (5+ from 3.1
4.1
Use case: time series completion
We will use time-series completion task an umbrella example describe the components of our construction. Let a RCxd be multivariate time series with C' channels over discrete timesteps, and let M E {0, 10 Oxd be binary mask indicating observed versus unobserved entries. We then write Tobs = M ( æ and Dtarget = (1 - M) O x, where C denotes clementwise multiplication. This formulation unifies various data completion tasks: forecasting arises when M masks future timesteps, while imputation arises when M zeroes out scattered entries within æ.
1xxd
Our noisy sample I, is generated by injecting noise at level e.g., x + with e (O I). Our goal learn an inverse mapping á that approximately recovers (or its unobserved portion) from H, in a single step when conditioned on the observations Lobs- To achieve this, $4.2 describes a Frequency-Aware Block that adaptively smoothes out unplausible frequency components and reweights dominant components operating in the frequency domain of the data (time-series). Then, $4.3 we construct a single-step operator á that enforces the semigroup properties of the diffusion process, ensuring coherent reconstructions under varying noise levels.
5

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4.2 Design of frequency-awarr block (FAB)
Let (-(-) be a neural embedding function (a small stack of residual layers) for spectral coefficients de- rived from a noisy input time series Z,. We first project the input onto a chosen spectral basis {xy} to obtain coefficients (zr, X3) This spectral basis is chosen to align with the inherent frequency-domane structure of time-series data. We use localized freeuenyyy transforms such as wavelet transform in practice, as it works well at capturing lime-localized patterns. A Fourier basis decomposes signals into their constituent frequencies globally, while a wavelet basis provides localized time-freee nee-ffruy representation. The specific choice will depend on the time series at hand, considering its dynamic or piecewise-stational frequency content (e.g., seasonal patterns, transient bursts). Thus, a localized time-frequue fine-frequency transform allows our model to adaptively emphasize different frequency bands at specific time points (here, time refers to timesteps the time-series). A learnable modulation function (T) (a MLP with positional embeddings) scales these coefficients based on the noise level (r(). We define the block simply as:
Bo (e(x,, = Σ Ve (o(((()) (Br) Xs.
(7)
Recall that the forward diffusion process progressively attenuates/smooths the high frequency com- ponents of the data's PDF. This block provides signal to counteract this effect by operating in the frequency domain of the time-series data itself. The projection onto the spectral basis decomposes the time-series into its constituent frequency components. The learnable modulation function adjusts the amplitude of each data frequency component based on the noise level since the degree of smoothing in the PDF depends 7 (and so, (r)). At high noise levels, must learn use the higher data frequencies, recognizing that this is related to the smoothing of the PDF. This block helps associate how the attenuation pattern the PDF space (due to diffusion) corresponds to the observed frequency content in the noisy data. Next, in 44. we integrate FAB into our single-step operator to enable the direct recovery of clean observed time series from noisy samples.
4.3 Design of inverse semigroup operators
We will now leverage the semigroup property. While conventional diffusion-based samplers approxi- mate (Sp=)))1 via many small reverse steps, we describe learnable inverse semigroup operator 6 that reconstructs these lost components in a single step. This approach directly maps a noisy observation (r(r) at noise level (T) to cleaner sample ka(n) at lower noise level (7)), where y < r, guided by the frequency-domain information, while preserving the algebraic properties of the forward semigroup.
Flow operator construction. To ensure consistency with the semigroup structure, our learnable flow operator 6(7, -) is a time-dependent convex combination:
$(7, T, x) = 7 Ho(x)
7 T
+
(8)
T
where Ho(-) is a neural network that incorporates the FAB block Bg in internally via (3) and is conditioned on the noise level (7)) The linear interpolation ensures that the identity Φ(T, T. holds (no denoising), maintaining consistency with the forward semigroup at zero step size (i.e., no noise added). Intuitively, H-)-) attempts to denoise based on the amount of denoising needed, and leverages the FAB block to enable frequency-specins re-weighting and reconstruction of the fine-scale features lost by the smoothing effect of S
Two-stage training objective. We train & through a two-stage process that enforces both endpoint accuracy and global consistency with the semigroup property: (i) Boundary/Endpoint. We ensure that mapping from noise level r to does recover the clean sample: @(0, T, (T()) = zg- This translates to the denoising loss in (4), where we sample (T) from a log-normal distribution and &.(r)) =
2o + (.) This objective aligns the single-step operator with standard diffusion model training at the boundary = O, preventing trivial solutions or model collapse. (ii) Semigroup composition property. Next, we ensure á approximates (S -1 globally by enforcing the composition property:
(9) This enforces semigroup coherence: two single-step moves from 7 + 7 and 7 > p should match single move from p. To facilitate training, we use linear adaptive schedule N-)) that
o(r7, 7, ((),T,))) = x)), 4OS p S0SpS>S S

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progressively refines the temporal discretization grid as training proceeds, allowing & to learn multi-step inversions. We begin with larger intervals between noise levels to encourage coarser approximation initially, then refine toward smaller intervals; directly starting with small intervals could trap the model in local minima by overtly focusing on fine-scale noise differences. Alg. Tand Alg. 2 in Appendix B show the complete training and sampling procedures.
5 Experiments
In this section, we present our experimental findings. Our evaluation protocol consists of two stages: (i) assessing the quality of multivariate time series (MTS) completion on both forecasting and imputation tasks with conditions on a variety of datasets, and (ii) conducting an ablation study to analyze the effectiveness of our model components.
Table 1: Performance comparison of probabilistic diffusion models for conditioned time series generation (forecasting). Deterministic point forecasting methods are included as reference baselines to contextualize current forecasting capabilities. Bold values indicate best performance per dataset, while underlined values represent second-best results. ()).
Model
Method
ETTml
ETTm2
Weather
Solar-Energy
Electricity
Traffic
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSB
MAE
MSE
MAE
NSformer
0.440
0.430
0.343
0.270
0.191
0.295
0.653
0.360
Times/et
0:374
0.387
0.248
0.309
0.219
0.261
0.259
0.318
0.184
0.289
0.617
0.336
Point
DI
3024
0.380
0.380
0.284
0.362
0.237
0.299
0.320
0.318
0.106
0.285
0.508
0.377
forccasting
PatcTTTT
0.370
0.312
0.223
0.259
0.463
SpuneVQ 40.2
0.363
0.380
0.242
0.302
0.225
0.258
0.256
0.286
0.182
0.267
0,480
0,300
lTransformer
[2124]
0.377
0.301
0.250
0.300
0..21
0.264
0.233
0.261
0.164
0.255
0.418
(1.844
TinzGrrad 2021
0.732
0.865
1.001
CSDI [202]
0.867
0.690
1.291
0.576
0.842
0.523
0.848
0.818
0.553
0.795
0,921
0.678
Probubilistic
TimeDai 2:25 25
0.796
0.577
0.284
0,542
0.277
0.331
1.169
0.936
0.730
0.690
1.465
0.851
forecasting
Diffusion 22
1,030
0.744
2.372
1.232
0.563
0.574
0.749
0.740
1.072
0.856
1.473
0.815
TMDM M124
0.607
0.358
0.524
0.493
0:244
0.286
0.2955
0.317
0.222
0.529
0.721
0.411
D3U (0225)
0. 0.968
0.387
0.241
0.302
0.222
0.264
0.237
0.270
0.179
0.267
0.468
(1.299
N&Dil 2025
0.488
0.455
0 0.281
0.352
0,248
0.293
0.242
0.307
0.209
0.306
0.637
0,373
SSOL (OUS)
0.369
0.314
0.209
0.255
0.223
0,550
0.259
5.1 Setup: datasets and baselines
Baselines. We benchmark method against two groups of baselines (a) diffusion-bused probabilis-
tic models: NsDiff (Ye et al. 2025). D3U [Li et al.
2025), TMDM [Li al. (024), Diffusion-TS
Yuan and Qiao 2024). TimeDiff [Shen and Kwok 2023]. CSDI [Tashiro et al., 2021), TimeGrad
Rasul et al. 2021]. DifrWave [Kogg et al.
2021
(b) deterministic models: TTrnnforrmr [Liu
al. 2024]. PST (Nie et al.] 2023 DLinear Zeng al., 2023), TimesNet [Wu al. 2023].
NSformer Liu et al. 2022] Sparse VQ Zhho et al.
2024]. Forecasting and imputation experiments
use task-ipec subsets for appropriate comparisons. More information is provided in Appendix C
Datasets.
We evaluate our model on six widely used benchmarks: ETTh1, ETTm1 [hou et al. 2021].
Weather, Electricity, Traffic [Wu et al. 2202]. J. and Solar-Energe (Lai al. 2017). For forecasting, mainly follow the experimental configurations in [Wu et al. 2023]. including the same data processing and splitting protocol. All experiments for baselines that report at this horizon, or fixing lookback at 96 otherwise for fair comparison. For
a 192-step prediction horizon, adopting default lookbacks
imputation, we fix the window length 48, following Yuaan and Qiao| 2024
5.2 Evaluations on MTS completion tasks
Forecasting, Forecasting results from our experiments are summarized in Table with detailed prob- abilistic evaluation using the Continuous Ranked Probability Score (CRPS) provided in Table 2 Our method demonstrates robust performance across these diverse benchmarks, matching or surpassing diffusion-basee probabilistic baselines. To give context for our method's probabilistic performance, we include comparisons with deterministic Transformer-based forecasting models (e.g., PStchTST, iTransformer). Although deterministic forecasting is not our primary focus, our approach remains competitive with these strong deterministic forecasters, frequently attaining top or near-top results. Summary. Overall, these findings suggest that our single-step design can handle complex temporal dependencies effectively in a 192-horizon prediction setting.
7

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Table 2: Performance evaluation of probabilistic forecasting models using CRPS and CRPS Bold
values indicate best performance per dataset, while underlined values represent second best results.
ETTml
ETTm2
Weather
Solar-Ean Solar-Energy
Electricity
Traffic
Model
Method
CRPS
CRPS
CRPS
CRPS
CRPS
CRPS
CRPS
CRPS.s.
CRPS
CRPS
CRPS
CRPS...
TimeGrad (2021)
0.996
0.785
1,051
0.482
0.503
0.783
1.167
0.503
0.657
Probabilistic
CSDI (2021)
0,773
0.852
0.625
0.782
0.508
0.465
0.649
0.681
0.465
0.823
0.612
1.275
TimeDiff (2023)
0.454
0.846
0.316
0.180
0.293
0.400
0.900
1.164
0.475
0.594
0.671
0.823
Forecasting
TMDM (2024)
0.429
0.633
0.380
0.226
0.226
0.292
0.375
0.267
0.446
0.137
0.552
0,179
D3U (2025)
0.285
0.749
0.243
0.141
0.207
0.283
0.186
0.266
0.202
0.160
0.232
0.186
NsDiff (2025)
0,350
1.614
0.256
1,315
0.244
1,873
0.300
27.64
0.290
29.65
0.378
11944
SSOL. (Ours)
0.376
0.553
0.184
0.131
0.348
0.384
0.211
0.195
0.224
0.116
0.268
0.176
Imputation. We follow the same geometric-mast strategy setup as in Zerveas et al. [021 Yuan and
Qiao [2024). where each missing segment's is length is sampled from geometric distribution, ensuring that the data is masked in consecutive segments rather than at random individual points. This setup aligns with real-world sensing scenarios, such as sensor failures or intermittent transmission losses, where data typically drops out bursts. We evaluate our approach on the ETThl and Energy datasets under missing ratios ranging from 10% to 90%. In Figure 3 we observe that our proposed method consistently achieves lower MSE than all baselines throughout the entire range of missing ratios. In Figure 5 in Appendix
C
we show single-channel imputation examples.
How can single-step outperform diffusion- baselines? Our algorithm is efficient yet powerful and only needs a single denoising step (NFE
1) while consistently matching or outperform- ing multi-step (e.g., NFF- 20) diffusion based approaches on all tested datasets. Conventional methods often rely on multi-step backward diffu-
Squared Enror
0.10
0.008
0.00
0.0G
0.006
0.04
0.004
sion, where even moderate time increments can be ill-conditioned due to the exponential decay of high-frequency modes in the forward process (discussed in Thm 3.2). This necessitates many small reverse steps and fine-tuning of step sizes noise schedules to mitigate error accumula- tion. In contrast, we learn a global operator. Of course, the composition property in ( over all noise intervals is essential: a direct jump from one noise level to another must match the outcome of traversing any intermediate levels. This ensures the operator remains well-defined for partial intervals and helps stable one-shot reconstruction of fine-scale details, helped by the FAB block. Our model avoids the iterative error propagation and hy- perparameter tuning (e.g., step sizes or scheduling) common in multi-step diffusion. Our end-to-end training over the full noise range yields faster inference and lower forecasting errors.
0.002
0.00
10%
25%
50%
75%
90%
25%
50%
75%
30%
Missing ratio
Missing ratio ETTh1
Energy
550L
Figure 3: Imputation results. MSE of different imputation models under various missing ratios for the ETTh1 (left) and Energy (right) datasets.
5.3 Digging deeper into the operating profile of the model
Ablation study. We evaluate the impact of key components in our single-step conditioned generation model through systematic ablation experiments on forecasting benchmarks. All experiments maintain identical training configurations while selectively removing specific components: (i) Complete model: Full framework with composition property and FAB; (ii) No composition property: Omits composition training stage; (iii) No frequency-awar block: Uses standard residual block instead of FAB.
Observations. (a) Composition. The ablation results in Table show that removing the composition property consistently increases both MSE and MAE across all benchmarks. This is because the composition property enforces the semigroup structure, ensuring stable transitions from high low noise levels, even with large step sizes. Otherwise, the model degenerates into a direct mapping from high noise to clean data, and so cannot handle intermediate noise transitions effectively. (b) FAB block: The frequency-ane aware block is equally useful for model performance. While forward diffusion inherently suppresses high- figg-freyuency modes the probability density function, our block's sample- level frequency adjustments enable the cumulatíve restoration of these modes at the distribution level. The increased error rates observed when removing this block, as shown Table show that precise per-sample frequency modulation is meaningful signal.
8

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Table 3: (a) Ablation study: impact of composition property and frequency-aware block removal. (b) Runtime gains due to single-step generation. (t).
(a) Ablation study (Pred. Len 95)
(b) Runtime gains (ETTh.)
Modids/Metric
ETTbi
ETTml
Weather
Model Approach
Steps
Prod. Len 96
Pred. Lon 192
MSE MAE MSE MAE MSE MAE
MSE MAB CRPS CRPS
MSE
MAB CRPS CRPS...
w/o Composition 0.102 0.133 0.362 0.394 0.190 0.268
Naluff Original
0.543 0.496 0.371
1.539
0.603
0.525
0.394
1.638
2025
SSOL
(200)
0.527 0,484 0.364
14777
0.580
0.522
0.389
1,682
FAB
0.406 0.336
0.178 0.221
DAU
Original
SSOL.
(20x)
0,415 0,426 0.317
0,400
0:412 0.369
0.699
0.813
0.471
0.440
0.463
0.499
0.340
0.381
0.823
0.799
Complete Model 0.375 0.396 0.325 0,352 0.153 0.309
SSOL
Reference
0.375 0.496
0.377
0.564
0.421
0,422
0.405
0.600
Efficiency, As shown in Table 3 we achieve strong efficiency gains by reducing the sampling steps to only one (NFE = 1) rouuhhy 20x improvement. When applied to recent diffusion models NsDiff and D3U, ours consistently improves prediction accuracy, reducing MSE and MAE in both 96 and 192-step forecasts while maintaining comparable CRPS metries.
Runtime analysis. The additional runtime analysis table is shown in Appendix C.I We provide actual training and inference times measured on identical hardware to address efficiency claims in Table 8 We used a computationally efficient variant of our method with reduced model size and shallower residual layers, while preserving the core frequency- aware block and semigroup composition training methodology. Our two-stage training exhibits comparable per-iteration times to baseline methods, with Stage (boundary denoising) and Stage 2 (semigroup constraints) adding minimal overhead compared to standard single-stage training. 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: (a) inference speedup of 3.5 x faster than D3U and 25x faster than NsDiff, (b) memory efficiency with 35% less GPU usage than D3U and 85% less than NsDiff, and (c) performance parity, where, despite far fewer denoising steps, SSOL achieves comparable/better MSE/MAE. The efficiency gains come from single-step design, while the additional training time for semigroup consistency is negligible compared to the inference savings.
Are single-step outputs based on actual denoising? In the single-step framework, one concern is whether the model can collapse to a trivial solution simply by ignoring the injected noise and behaving like a conventional regressor in the conditioning window. In that case, drawing multiple noisy samples would not meaningfully change the model's predictions, since each forward pass would converge to the same deterministic output. To check this, we conducted an experiment in which we varied the number of generated samples n (see panel 3 of Fig. 4.. We observe that as increases, the median of these n samples consistently lowers MSE and MAE, indicating the model does react to each injected noise draw, i.e., it is genuinely denoising from different noisy realizations. If the model were ignoring noise, its outputs (and thus errors) would not improve with additional samples, suggesting that it learns the conditional distribution over possible future trajectories.
Impact of shifted noise distribution. We also tested conditional forecast generation with a much smaller o max at inference time, which produces a narrower prior distribution, creating a mismatch in the types of perturbations the model must denoise. The right panel of Figure 4 shows that the distribution shift degrades performance, indicating that the single-step operator depends access to a broader range of noise levels during inference to capture the distribution's is variability. This behavior is expected: our operator is trained to map from high noise levels, where forward diffusion has sufficiently smoothed the data distribution (in Thm 3.2), but reducing Fmax at inference violates this assumption by starting from insufficiently noisy priors. The degradation confirms our model performs diffusion- biffusiin-bbsedd generation and is not learning shortcuts that ignore the noise structure.
Does single-step denoising recover the high-frequency components in distribution? We can ask if our model can recover the fine-grane fine-grained, high-freuuency details via the frequency-aware block. In Fig. 4 the t-SNE plot (left) shows close proximity in the embedded space, suggesting that we can cover the overall data manifold in the forecasting setting. Further, the power spectral density curves in the second-left panel show that the generated distribution captures high-frequency components that align closely with the real data, indicating that FAB can denoise the noise in individual observations and re- injineect the fine-scale details in the conditional distribution.
9

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t-SNE Component: 2
Sepels
Density Estimate
0.5
srgets
0.33
04
50
0.300
03
0.253
0.7
0.2
50
0220
0.1
-50
0
50
0.0
-5
Witt
t:SNE Component
Power Spectral Density
Sample Numbers
Maximum Noise Level
Figure 4: Left: 1-SNE manifold comparison and power spectral density recovery on ETTml. Right:
Error analysis of conditioned generation on Electricity and ETTh1.
6 Related Work
Single-step denoising. Most diffusion models rely on multiple iterative denoising steps in the reverse process [Song and Ermon 2019 Ho et al. 2020). which can be expensive at inference time, Recent research has therefore explored strategies for reducing the number of sampling steps. For instance, Nichol and Dhariwal [2021 propose a strided sampling schedule to skip certain reverse-time intervals, while (Salimans and Ho 2022 proposes progressive distillation that halves the diffusion steps each iteration through a teacher-student paradigm. More recently, [Song et al. (023) learn a direct mapping from any intermediate noisy point to the clean signal in single step via consistency distillation.
Similar single-step approaches have been developed for image generation [Frans al. 2025 Geng ctt al. 2025], though these require domain- specifcc adaptations for time-series applications. Our work leverages classical principles in Pazy 1983]
Henry 1981 to define a single-step inverse operator.
Diffusion models for conditioned time series generation. Time-series forecasting has evolved from classical statistical models such as ARIMA [Box and Jenkins [1994] to advanced Transformer- based approaches (Liu al.
2024 Nie et 2023 and more recently, diffusion based approaches.
Proposals like Diffusion-TSS TS Yuan and Qiao, 2024 MG-TSD [Fan et al. 2024), TSDiff Kollovieh
al. 2023 SSSD [Alcaraz and Strodthoff] 2023]. J, D3VAE (Li et 2022 TimeGrad [Rasul et
2021 CSDI Tashiro et al. [021], J. and Imagen'iime Naiman et al. 2024 leverage diffusion- based
generative modeling in interesting ways for probabilistic forccasting/mmpuattion Some recent results have run diffusion in the frequency domain: Crabbé et al. (2024) performs score matching on mirrored Brownian motion (in Fourier basis) and improves sample fidelity, and FIDE [Galib et al., 22024] inflates high-freyuency coefficients and conditions on block maxima to preserve extreme events. However, these approaches still rely on multi-step sampling, efficient single-step generation for time-series diffusion models is largely unaddressed in these works.
7 Conclusions
We introduced novel single-step operator learning approach for time-series diffusion models that enables efficient conditioned generation while maintaining high-quality results. By using insights from the frequency-domaii characteristies of both time-series data and the diffusion process itself, our model can effectively reverse the smoothing effects of forward diffusion in one step. Extensive experiments across multiple datasets shows that our approach achieves comparable or better performance than existing multi-step diffusion and deterministic methods, while reducing computational costs quite drastically, which can make diffusion models more practical and efficient for numerous time-series applications.
Acknowledgments and Disclosure of Funding
The authors are grateful to Sourav Pal, WonHwa Kim, Melanie Boly, and Giulio Tononi for discussions and input. Partial support for this work came from a contract UW-Madison under the DARPA Strengthen program.
10

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A Attenuation of high-frequency modes
key observation of forward diffusion processes is the attenuation of high-frequency modes. In the non-degenerate linear drift diffusion setting ( the associated Fokker-Planck operator (5) enforces exponential damping of the Fourier transform of the density at large wavenumbers, In the following. we provide detailed proofs of these properties (which are also stated in Sect. 3.2p, then connect them smoothing Sobolev and wavelet spaces.
Assumptions. Throughout this section, we fix measurable coefficient functions f.g: o0) + R with g(T) 2 8 and work under the following hypotheses:
(A1) Initial data. The starting density satisfies po L (R.) n (RR.).
(A2) Coefficient regularity. f.g (10, oo)), ensuring that all integrals So f(u) du and So g(u)² du are finite for every т > O.
(A3) Existence of an L L2 density. For each T > 0, the Fokker-Plann in equation 5
0-p+ = [p)) V [f(r)))),) Pl = Po:
admits a solution E (TR)) n T.2((RR)) with sufficient decay at spatial infinity to justify term-wise Fourier transforms and integration by parts.
A.1 Exponential damping in the Fourier domain
To analyze the tiequecyyyency behavior, let us take the spatial Fourier transform of the probability density (x). Define
(E) = [ e 1 e- Pr (x) dæ, E E Rd
The following theorem shows that for each fixed & * these modes decay an exponential rate determined by the diffusion amplitude.
Theorem A.1 (Exponential decay of Fourier modes). Assume (A1)-(A3) hold. Put
S(T) exp(< du))
a()) = : 1/ g(u)² s(u) -2 du,
2
which is also stated in (2.. For every 0 and every frequency Rdd the Fourier transform of the solution to
O-pr(x) gT) (pr()))))))))))))))))))))()))( (p))) pl.-0= po:
satisfies the identity
(E) (=()) exp [ r(7)2 [[6)]]
Consequently.
< po (=))) exp 1 ' g(>)2 ||= 12 12
pr (
Proof. Step 1: Fourier transform of the Fokker-Plancl operator. Because p- E L n L and enjoys Gaussian upper bounds (see assumption (A3)), we apply the spatial Fourier transform and integrate by parts. Writing (E)
= F. F. (p,)(E) we use the standard identities:
F[Ap-1(E)
|| (E).
J(V F(v.(xp+) = -(S. VgPT (E))
Transforming the Fokker- Planck equation therefore gives the first-order PDE
0, (E)
( (T()) + [(7) (E, VeTT(E)),
to be solved in the (T. variables. Or, equivalently
0 (g) f(r) (E. VEPr(E)) - 9 (|E]? (E).
15

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Step 2: Solve the equation via characteristic curves in É-space. Fix 7 > 0 and & E R For y E [0, define the backward characteristic ((>) by
d (n) = - f(r) E(),,
-|,=, = &.
dy
Since f Li (10, x)) by Assumption (A2), the ODE well posed and has the explicit solution
( exI( ex [(w)
s(T)
[())du) du) E
0 S T
x(7)
where s(-) is the scaling factor defined in (2) Along this curve in g-space, using the chain rule and the PDE obtained in Step 1 we compute:
d
p. (E(r))
(9( [<>]] | (, (=(7))
d-
This linear ODE integrates to
(= = po (=(() exp L g(v)² [[w]]
0 SSST. T.
Setting 7 in the equation obtained above, we have
(F(E) = no (0)) exT - [ s(n)² |[]l]] du)
Recall that (uu)
=(x) a(r) & where s(T) exp( So f f() du). Since (0) 1, we have
(()) +()
E and pa{ (0)) = No (x( E)
For the integral in the exponent, we use | (z7)
|| obtain
L' s(u)) [())]l du = ||11 (T) [ g(u² du.
s(n)
Defining d(7)² == So 2(u)² <(x)7 du, this simplifies
I. gq(u)² |= (1)] du =
Substituting these results into our expression for p. (E) yields
(E) = ()(((T))) 2 (
which matches the identity in Theorem A.1
Takeaway. The final expression
Pr (E) ()()(T) exp(5 | ((T))2(((T)
shows that for each fixed r > 0, the Fourier transform of pr inherits a Gaussian-type decay factor in || Provided 9(-) does not vanish identically and s(-) remains finite and nonzero, the product s(T) (²) is strictly positive, thus giving an exponential decay in ||E|| for each t > 0. In other words, no matter how f()) and g(.) vary over time, the factor exp( Ell s(T)² (T))² suppresses freeuffryyency modes of p, in Gaussian manner.
Moreover, we can view this suppression in two complementary ways: (a) Decay in frequency for fixed T. For any fixed > 0, as || + co, we have the factor exp( || s(r)² (T))²) forcing rapid (Gaussian) damping of high-frequency modes. This is the source of instant smoothing: one can show pr H (Rd) for all O. (b) Decay in time for fixed *f 0. For any nonzero frequency E. the same exponential factor shrinks (E) to zero in т, typically at an exponential rate, provided ((T))o²(T) grows in This illustrates how each individual mode is damped time evolves.
These two types of decay reflect the main smoothing and mode-attenuating nature of forward diffusion processes. (a) First, p- becomes instantly smooth for any > 0: indeed, p- E H" (Rd) for all 2 0. implying Pr is C the spatial variables. (b) Second, from an inverse perspective (e.g. generative modeling), the loss of high-frequee frequeccyy content means that recovering Po from p- becomes ill-conditioned as grows. Thus this exponential decay factor captures both the smoothing and the mode mode-attenuating nature of forward diffusion processes.
16

---

A.2 Instant smoothing in Sobolev and Wavelet spaces
In the previous subsection, we established that p- (E) decays exponentially at large wavenumbers ||||| (see Theorem (.IIp Here, show how this exponential decay in the Fourier domain leads directly to the conclusion that p, belongs to all Sobolev spaces H" (R") ) for each 0. Then, by leveraging the known equivalence between Sobolev norms and wavelet-coefficient decay, we confirm that p, is likewise instantly smooth with respect to any transform that localizes high- fegg-freuuency (or fine-scale) components, such wavelets.
Corollary A.2 (Instant smoothing in Sobolev and Wavelet spaces), Let p+ solve the linear-drift Fokker-Planck equation (5) and assume that p+(+) remains in (TRRR) (A3)). Then for each fixed T > O and for every real we have
) for all T 2 0 (see Assumption
E (RR))
Equivalently, in wavelet basis {03,)). if (p=)) denotes the wavelet coefficients of Pr, then
ΣΣ 22 (b)(((=)) 0o for each K 2
JEZ REFA
Hence Pr is in all Sobolev space H* (MRR)) for > O, and is therefore infinitely differentiable.
Proof of Corollary A.2 Step 1: Exponential decay in the Fourier domain. By Theorem A.1 there
exist constants C, 0, O, and Rg > depending 7, such that
pr (E) C exp( at [E] for all ||||| >
Ro.
(If f()) = 0, this is immediately seen from the heat kernel's Gaussian decay. If f(-) * 0, one uses the scaled-argument form of the solution in Theorem A.
Step 2: Sobolev-norm boundedness. Recall that Pr E H (TR)) for all x. > 0 if and only
(1
||E| (E) de < co.
(See, e.g., Grafakos 2014]. Evans 1998 for this classical characterization.) We split the integral in (+) into two regions:
{|k & Ro} and {llE| > Ro}.
Since pr (TR() (by Assumption (A3), Pr E (.2(RR)) and Plancherel's theorem), the portion over
{llEll < Ro} is trivially finite. On [l|E|l > Ro ), we use the exponential bound:
| 7f (E) < (1 1 O exp( 22,,,, |E(17)
As || + co, the polynomial factor (1 + || 2) exp(-2 1pp(-2ap
xd is dominated by the superpolynomial decay of
at a2 I2. Hence the integrand is integrable for all 0. Thus the integral in (*) is finite,
yielding p. E HRR)). Since is arbitrary, p, E H1 (Rd) for all > 0.
Step 3: Equivalence with wavelet decay. Next, we recall (e.g. from Meyer [193]) that a function's inclusion in H²RRd) is equivalent having sufficiently small wavelet coeffcients large scales:
I|p 1 (1P
ΣΣ JEZ
223N (o),((=))
where by.4 (p,) denotes the wavelet coefficient of at scale j and position k. Since exponential decay of (E) at large ||E|| implies fast convergence of p, expansions in localized bases (such as wavelets), one sees directly that p+ wavelet coefficients decay sufficiently rapidly for it to lie in all H" for ς > 0. Equivalently, from the purely wavelet viewpoint, each increase in K imposes an extra factor of 2 on the sum, yet exponential decay in frequency/seale ensures convergence for every K.
In conclusion, Pr lies in all (RRd) and thus C (Rd) for each r >> 0.
17

---

Remark A.3 (Implications for generative models). From the perspective of generative modeling, Theorem A.I and Corollary A.2 formalize the idea that 'noise injection" destroys seghffreuencc information about the initial distribution Po- The exponential shrinkage of P+(E) at large ||E|| makes recovery of po from p- ill-conditioned as increases. Hence effective methods may require learned priors or neural architectures that reconstruct the lost high-frequency details in a stable way. From a practical standpoint, the generative reverse process must reconstruct precisely those exponentially damped (high) frequencies, which explains in part why carefully designed neural network architectures and training procedures are needed.
Remark A.4 (Conditions for smoothing properties). Consider the forward SDE defined in equation
(a) C -mmoothness When the diffusion coefficient is strictly non-degenerate, i.c., g(T) for
all T > 0, Theorem A.1 establishes that Pr (E) = O(e -. llell".. Consequently, Pr E H*(R") for every 2 0 and exhibits C -moothhesss with respect æ. The strict positivity condition 9(T) is necessary; in cases where g = O (purely deterministic flow) or where g is rank-deficient and Hörmander's bracket condition is not satisfied, pr may reduce Dirac distribution without any smoothing effect. (b) Real analyticity. If, furthermore, the coefficients satisfy uniform parabolicity on each compact time interval,
inf 9(7) S sup g(7) < 00, USySo"
sup (f()) < Do (Va 0) 05y50
05y50
then we can strengthen the regularity conclusion to real-analyticity of pr in the spatial variables for all > O. (c) Practical implication. Many score-based diffusion models (c.g., EDMKarras et al. (2022) specifically choose a full-rank Brownian motion with g both bounded and bounded away from zero. Consequently, both the C -moothhnsssss property and the analytic regularity are automatically satisfied in practical implementations.
B Algorithm
In the following, we collect the implementation details that turn the approach in Secs into step-by-step algorithm. The algorithm details the two-stage optimization that equips the inverse operator ó with the boundary condition and the semigroup constraint that connects all intermediate noise levels. The algorithm @illustrates how, once trained, the same operator can be used to generate (or complete) time series in single jump from the highest noise level max the data manifold at 0.
C Experiments
Datasets. We evaluate our model on two main tasks: forecasting and imputation. For forecasting, we utilize five benchmark time series datasets: Electricity Transformer Temperature (ETT) Zhou
et al. [202T), which contains electricity transformer measurements at hourly (ETTh) and 15-minute (ETTm) intervals; Weather Wu et al. [2021]. comprising 21 meteorological parameters collected at 10-minute intervals; Solar-Energe Lai et al. [2017]. which records power generation data from multiple solar plants at 10-minute intervals throughout 2006; Electricity Wu et al. [202T,, tracking consumption patterns across 321 clients; and Traffic Wu et al. [2021]. which monitors hourly road occupancy rates through 862 sensors in San Francisco from 2015 to 2016. For the imputation task, use subset of these forecasting datasets along with Energy [Candanedo] [2017), which captures energy consumption at 10-minute intervals over approximately 4.5 months, collected via a ZigBee wireless sensor network in Belgium.
Experimental setup. Our experimental configurations follow the protocols established in Wu al. 2023], including identical data processing and splitting methods. Details for each dataset are provided in Table 4 Aligned with fair comparison settings outlined in Liu et al. [2024). Li et al. 22055. we fix the lookback window length to 96 time steps for most datasets and baselines, with the prediction horizon set to 192 time steps. For baselines that are reported under similar settings and have published results for 192-step prediction, such as NsDifiYe al. 22025) we adopt their default configurations, regardless of whether they utilize longer lookback windows (which potentially provide more historical context for forecasting or conditioned generation). For imputation tasks, we fix the window length to 48 time steps, following the protocol in Yuan and Qiao (2024). For probabilistic
18

---

Algorithm Two-stage training of the single-step inverse operator Ø
Input: dataset D of clean sequences æ6; mini-batch size N; log-normal noise prior ptn (a);; weight-
ing function w(a) (Eq. learning rates 1/, 12: EMA rate Semm's adaptive linear grid scheduler
(-) identity-)): noise scheduler in second stage o, [N (k) (a N( (a 1/v max)) )
min
with max 80, min 0.002, and =
Initialize: random parameters (shared by Ho in Eq. (3) and @ in Eq. 8 + 0
Stage boundary denoising (Eq. (4)
while not converged do
Sample { [ J1=1
D
Sample for 1=1 Pon and { Ji=1 I)
for
to do
(1)
+ Holw no , (1)
end for
denoisee f-
w (
N
10
+ 0 1n V
end while
Stage semigroup consistency (Eq. (
12
while not converged do
13
Increment global step l I Tim1 ~
14
Sample { {
15
Choose Unif [1, ...., N())
set set)rsstt 0 on P 0
16
Sample {e 1=1 (0..
17
for to do
18
19
a direct (1)
- ( T, xt
20
Fcomp
2o p. 7. 0, T, 2,
21
end for
1
(i)
comp 12
22
Lemp
direct
N
23
+ 0 1/2 &eCemp
24
+- Bema 0 + (1 Boma)
25 end while
Return:
> final EMA parameters
Algorithm 2 Single-step sampling with inverse operator
Input: number of samples Ñ; trained EMA parameters 0 maximum noise level mmax' observations Robe: mask M
Return: completed sequences (a N J==1 for i = to N do
I- independent draws
draw (8) ~ I0.
a..()) <- Tmax (f) if observations are given then
initialise highest noise
b completion case
me + M) @ (0))
+ @ Pobs
end if
(())
dbg (0, Tmax, x!!))
single jump ++0
end for
return (5) N
completion tasks (including both forecasting and imputation), we sample 100 trajectories per instance and use the median trajectory our point estimator.
19

---

Table Dataset characteristics for forecasting and imputation tasks. Dim. denotes the number of variates (channels); Size specifies the sample counts in the training, validation, and test splits: Freq. indicates the data sampling interval.
Task
Dataset
Dim.
Pred. length/Miising ratio
Size
Freq.
Domain
ETThl
7
192
(8545, 2881, 2881)
1 hr
Temperature
ETTml
7
192
(34465, 11521, 11521)
15 min
Temperature
ETTm2
7
192
(34465, 11521, 11521)
15 min
Temperature
Forecasting
Weather
21
192
(36792, 5271,, 10540)
10
Weather
Electricity
321
192
(18317, 5261)
1 hr
Electricity
Solar-Encrgy
137
192
(36601, 10417)
10 min
Electricity
Traffic
862
192
(12185, 1757, 3509)
1 hr
Transportation
ETThl
7
(10,25,50,55,90
(8545, 2881, 2881)
1 hr
Temperature
mT
(10.25.50.55.90)
(34465, 11521, 11521)
15
Temperature
Imputation
Weather
21
110,25,50.75.90 90.5,999
(36792, 5271, 10540)
10
Weather
Energy
28
(10,55,50.55.9088
(13797, 1972, 3942)
10
Sensor network
Baselines. For the time series forecasting task (conditioned generation with masks identifying the lookback and prediction horizon), we conducted comprehensive comparisons with both point
forecasting models and probabilistic models. The point forecasting models include NSformer Liu et al.
(2022). TimesNet Wu et al. (2023), DLinear Zeng et al. (2023). PatchTST[Nie et al. [0233] Sparse
Zhao et al. [2024], and iTransformer Liu et al. [2024] The probabilistic models include TimeGrad
Rasul et al. 2021 J. CSDI et [002]]
J. TimeDiff Shen and Kwok [2023], DiffusionTS Yuan
and Qiao 2024]. TMDM Li et al. 2024]. D3U Li et al. [2025]. and NsDiff Ye et al. 2025 For
models that explicitly addressed this forecasting task under mostly identical settings, specifically
D3U et al. 2025 and NsDiff Ye et al. [2025], J, directly referenced their published results to
ensure accuracy and consistency in our comparisons. For baselines that did not originally report results under exact experimental conditions,
either reproduced their outcomes using official code repositories where available, or referenced reproduction results reported by D3U and NsDiff to maintain comparability and fairness across all evaluations.
Implementation details. All experiments were conducted using Py PyTorch|Pazkke Paszke et al. (019 on single NVIDIA A100 40GB GPU. We trained model using a two-stage approach with the Adam optimizer Kingma and Ba (015). In the first stage, we trained the model using the EDM-type
loss Karras et al. [202 defined in Eq. 4 The second stage focused optimizing the composition
property loss, as shown in Algorithm Summary of the experimental configurations for all datasets
listed in Table
Table 5: Hyperparameter settings used for training and evaluating the diffusion models.
Dataset
Epochs
Batch Size
Learning Rafe
Layers Wavelet Levels
Wavelet Type
Sampled Trajectories
Aggregation
ETTml
median
ETTm2
50
32
0.001
dhl
100
median
Solar-Fnergy
median
Electricity
50
0.001
dh1
100
median
Weather
0.001
db1
100
median
Trallic
0.001
dbl
100
median
Energy
50
32
0.001
3
dbl
100
modian
Probabilistic metrics. For evaluating probabilistic predictions, we employ the Continuous Ranked Probability Score (CRPS) [Matheson and Winkler 1976 Tashiro et al.) [202T], which measures compatibility between an estimated distribution and observations. CRPS is defined as:
. 2A.(P (u),
CRPS(P-1.22
(P) x)
(10)
where Au(g, .x) (u (x - q) is quantile loss function, P-1 the inverse CDF of predicted distribution, and æ is the observed value. In practice, we approximate this integral using dis- cretized quantile levels with 0.05 intervals and generate 100 samples to estimate the distribution. For multivariate time series, also evaluate CRPS-sum, which assesses the joint predictive performance
20

---

by calculating CRPS for the distribution of the sum across all channels:
CRPS E, CRPS(IP - E. Fc,t)
(11)
Ee,e [Ze,el
where e indexes channels and l indexes position timesteps. Both metrics are normalized by the sum of absolute observed values ensure fair comparison across datasets, with lower values indicating better model performance.
C.1 Additional experiments in probabilistic forecasting
Probabilistic metrics. While our main evaluation (Table D employed deterministic metrics (MSE/MAE), we further assess probabilistic forecasting performance. Table preseetsssssnss both CRPS and CRPS metrics, evaluating distributional accuracy at individual channel and aggregate levels, respectively.
Analysis of Taleaall that SSOL achieves state-of-the-art CRPS performance on four of six datasets, demonstrating superior accuracy aggregated channel forecasting. This advantage stems directly from our channel permutation invariance design. While D3U excels in per-channel CRPS, SSOL maintains competitive performance, ranking first on ETTm2 and second on several datasets. This performance pattern illustrates the trade-off between optimizing for individual channels versus aggregated behavior, confirming the effectiveness of our approach for multi-channel forecasting tasks.
Statistical robustness. We report the statistical significance of our model's performance by com- puting the standard deviation across 5 different consecutive random seeds during the sampling stage, consistently
while keeping the trained model fixed. As shown in Table 6 the standard deviations small (on the order of 10-4 or 10-5), demonstrating that our results are robust to the stochasticity in the sampling process. This stability is expected since use the median of 100 trajectories per instance as our point estimator, which provides statistical robustness.
Table 6: SD of each metric for prediction lengths 192 and 96 on ETTh1, computed over 5 random seeds during the sampling stage using the same trained model.
Pred. Len
RMSE
MSE
MAE
CRPS
CRPS
192
0.649 = 1.19 10-1
0.421 = 1.54 x 10-1
0.422 9.88 x 10-5
0.405 10-5 0.600 10-1
0.612 + 1.43 x 10-4 0.375 + 1.76 x 10-4
0.306 1.77 x 10-4 0.377 + 9.72 x 10-5
0.564 2.05 10-4
Table 7: Comparison of imputation performance with Diffusion- - Yuan and Qiao reported: MSE (x10-3) and MAE 3). Diffusion-TS and SSOL (ours) results are based on 100 samples using the median as point estimator, while SSOL-S uses a single trajectory sample as estimator. (4)
2024 Metrics
ETTh1
Energy
Weather
ETTml
50%
50%
10%
50%
50%
Diffusion-TS
MSE
2.62
3.13
19.6
22.6
23.9
23.9
6.22
3.58
3.24
0.74
1.13
1.43
MAB 3,11
3.35
3.71
4,06
4.07
10,0
10.2
10.4
10.8
10.8
1.64
1.37
1.25
1.22
1.23
2.71
2.99
3,36
3.76
3.79
SSOL
MSE 1.83
1,31
1.6
2.14
2.14
4.41
4.74
4.09
5.75
5.65
1.34
1.29
1223
1.28
1.34
0.44
0.50
0.62
0.77
0.81
MAE
2.56
2.40
2688
3.05
3,03
281
2.90
2.52
3.07
2.85
0.90
0.87
0.78
0.79
0.83
1.45
1.49
1.61
1.78
1.89
SSOL-S
MSE
2.92
2.08
2.70
3.51
3,51
7.83
8.84
7.69
10.9
10.8
2.20
2.07
2.30
2.12
2.15
0.65
0.86
1.07
1.39
1.56
MAF
3.31
3.08
3,51
3.97
3.43
3.74
3.15
3.72
1.21
1.14
1.14
1.79
2.17
2.44
2.64
Runtime analysis. We provide actual training and inference times measured on identical hardware to address efficiency claims in Table g We used a computationally efficient variant of our method with reduced model size and shallower residual layers, while preserving the core frequency-awar block and semigroup composition training methodology. Our two-stage training exhibits comparable per-iteration times to baseline methods, with Stage (boundary denoising) and Stage 2 (semigroup constraints) adding minimal overhead compared to standard single-stage training. Although we do not achieve a pure 20x speedup wall-clock time due backbone architecture differences across methods, our approach shows strong efficiency gains during inference.
21

---

Table 8: Runtime analysis and computational efficiency comparison of SSOL, D3U, and NsDiff models on ETThl dataset for prediction lengths 336 and 720.
Forecasting Performance
Computational Efficiency
Resources
Model
Pred. Len
MSE
MAE
CRPS
CRPS atmm
Train (s/iter)
Inference (min/batch)
NFEs
GPU Mem
+
(MiB)
336
0.491
0.477
0.458
0.639
0.025
0.022
364
SSOL
720
0,539
0,535
0.508
0.672
0.032
0,044
1
646
336
0.512
0.478
0.351
0.922
0.025
0.078
20
582
D3U
720
0.533
0.505
0.371
1,458
0.030
0,164
20
884
336
0.728
0.583
0.431
1.083
0.105
0.544
20
2560
NsDiff
720
0.704
0.613
0.440
2.086
0.179
0.608
20
5380
The key findings are: (a) inference speedup of 3.5x faster than D3U and 25x faster than NsD- iff, (b) memory efficiency with 35% less GPU usage than D3U and 85% less than NsDiff, and (c) performance parity, where, despite far fewer denoising steps, SSOI achieves comparable/better MSE/MAE. The efficiency gains come from our single-step design, while the additional training time for semigroup consistency is negligible compared to the inference savings.
Longer horizons. conducted additional experiments on longer prediction horizons (336 and 720 steps) using the ETTh] dataset, as shown in Table X The results demonstrate that our method maintains competitive performance at extended horizons: SSOL achieves comparable better deter- ministic predictions with consistently better joint distribution modeling (CRPS-SUM improvements of 31% at 336 steps and 54% at 720 steps compared with D3U).
Fourier versus wavelets. We compared Fourier and wavelet transforms as the spectral basis in our frequency-aware block using ETThl with prediction length of 96. 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 difference aligns with expectations: Fourier analysis assumes global stationarity, while wavelets provide localized time- frequeeyyy analysis better suited for non-stationary time-series patterns.
C.2 Additional experiments in imputation
Experiment results. We provide additional details on the time series imputation experiments shown in Figure 3 Our empirical evaluation demonstrates that the proposed SSOL method significantly outperforms existing time series imputation approaches, including the of-e-of-the-an Diffusion- TS model Yuan and Qiao [2024], across four evaluated datasets and missing ratios. As shown Table 7. SSOL consistently achieves lower error metries, with MSE reductions 16-50% on ETTh1, 76-82% on Energy, 59-80% on Weather, and 40-44% on ETTml compared to Diffusion-TS. TS. Both Diffusion-TS and our primary SSOL method use the median from the multiple trajectories as estimator, which is standard pipeline in time-series diffusion models. To further validate approach, we also evaluate SSOL-S, a variant of our method that uses single trajectory sample as the estimator rather than aggregating multiple samples. While SSOL-S primarily serves reference point, it remains competitive with or outperforms Diffusion-Te TS on several datasets, particularly on ETTml and Weather.
Showcase of the imputation task. The figure demonstrates that our imputation method maintains high accuracy across varying missing data ratios (10% to 90%), the median imputation (red lines) almost accurately reconstructs long, continuous missing segments (regions with blue dots) by retaining the shape of the ground-truth curve (solid blue line) and staying within the 90% confidence interval (shaded red area) for most channels. This pattern suggests that our model effectively learns the temporal dependencies, such as periodic or seasonal trends, in each dataset and uses them to bridge prolonged gaps. The confidence bands expand around regions of sharp ground-trutt fluctuations, indicating the model captures higher uncertainty in those transitions and reflects it in broader intervals.
22

---

The shaded areas represent credible intervals (5th-95th percentiles across 100 generated samples) that show the range of plausible imputed values given the observed data. Note that these differ from confidence intervals for the median estimator, which would be much narrower and indicate the precision of our point estimate rather than the inherent uncertainty the imputation task. This pattern suggests that our model effectively learns the temporal dependencies, such as periodic or seasonal trends, in each dataset and uses them to bridge prolonged gaps. The confidence bands expand around regions of sharp ground-truth fluctuations, indicating the model captures higher uncertainty in those transitions and reflects it in broader intervals.
D Limitations and broader impact
Limitations. Our theoretical results rely standard regularity assumptions of linear drift and non-degenerate Gaussian diffusion (see Remark. (A.4). which are satisfied by most modern diffusion frameworks Karras al. (2022]. For multiplicative (state-dependent) noise models, which arise in variance-expanding or data-adaptive diffusion schedules, relaxing these assumptions and investigating the smoothing behavior of forward operators remains an open question. While our frequency-ave -aaenccyywwee block effectively captures temporal patterns in regular time-series data, its extension to more complex frequency structures requires further investigation. Our empirical evaluation focuses on six publicly available multivariate time-series datasets with minute-to-hourt granularity. Adapting our single-step operator to irregularly sampled sequences, longer contexts (e.g.. high- gigh-freuencc financial data audio signals), or entirely different modalities would require more non-trivial architectural modifications to properly capture domain- speecifcc structures.
Broader impact. Our single-step operator learning approach for conditioned time-series diffusion models reduces the number of function evaluations from a typical 20 [i et al. [2025] Ye al. [225] to just 1, yielding computational speedup during inference. This efficiency can translate to reduced energy consumption while maintaining comparable or superior performance across multiple benchmarks. The proposed method also naturally produces confidence intervals alongside point estimates, as shown our showcase experiments (Fig. [). allowing more informed decision-making in domains where trend identification is needed (e.g.. energy forecasting, traffic monitoring). Since we focus on structured time-series data generation, potential negative impacts remain limited primarily to cases where historical data is biased erroncous.
23

---

3.00
0.0
0.0
1.90
3.00
0.23
3.50
6.2
0.10
0.22
3.0
1.39
12:00
28:00
10.00
00:0
0.00
3.70
0.70
0.73
1.70
3.00
0.00
2.50
0.5
0.23
0.3
2.00
0.16
0.43
2000
23:00 49.30
200CE
2006
2000 43.00
10.33 2010
10.00
2013
20.0
0.90
4.11
1..12
2.33
Value
0.50
0.53
3.50
0.43
3.3
0.30
0:35
3.30
0:33
1.00
1.10
0.1
0.72
2:30
0.9
0/60
Volue
3.50
024
8.59
2.40
0.43
1.40
0.00
2000
20000
13:00
49.20
0.0
2005
2006
2000
10.00
2.30
10.0
20.0
2005
1000
0.30
3.00
12.00
28.00
3040 nous
11:00
23:00
28.00 1642
0.55
3.50
0.52
Volue
0.42
0.30
2.40
0.42
0.4
0.15
0.25
0.30
0.30
0.12
0.79
1.00
0.55
0.5%
11
2.50
0.53
ZUe 20.31 12.01 43.31
6.0
Jone
43.00
3.3
19:22 2010 wire
mice
6.5
12.00 18:0
6:40
12:00
32.55
42.3
0.28
0.30
0.34
2.35
0.00
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Figure 5: Imputation comparison for the ETTh1 dataset with a 48-step window using a time series sample of 7 features. The red shaded area represents the 90% confidence interval, the blue line denotes the ground truth, the dotted line indicates the median imputation, and the blue points mark the imputed ground truth values. Columns from left to right correspond to missing ratios of 10%, 25%, 50%. 75%, and 90%. Rows from top to bottom correspond to different channels.
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Reviews:
[{"review_id": "gGbbkMfZmy", "reviewer": "Reviewer_pJbC", "summary": "This 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.", "questions": "1. 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.", "limitations": "Not found", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "I think the author's response addressed most of my questions, but for one point, I cannot further increase my score: based on many methods in the image domain, I believe transferring them to the time series prediction field is not difficult. The author's rebuttal mentioned three points: 1. Differences in data structure. Data distribution is implicitly learned, and structural differences can at most lead to performance loss, which can be fully compensated for with additional techniques. 2. Model architecture choice. As far as I know, DiT is currently widely applied and does not require any substantial modifications to the Transformer. 3. Conflict in evaluation systems. MSE can be used for training in both cases. In summary, I believe the author's avoidance of discussing existing methods in the image domain is a debatable behavior, but considering the overall quality of the paper is good, I maintain my score of 4.", "rebuttal": "We thank Reviewer pJbC for the detailed feedback and thoughtful comments. We have carefully addressed each point below. \n> W1: Citation format.\n> \nThank 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!\n> W2: The paper lacks comparisons with existing step-reduction methods from image generation.\n> \nThank 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.\n\nThe reviewer will agree that adapting any shortcut method to a time series requires numerous domain-specific design choices:\n\n1. 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.\n2. 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. \n3. 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.\n\nOverall, 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. \n\nA 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. \n\n[1] Geng, Z., Deng, M., Bai, X., Kolter, J. Z., & He, K. (2025). Mean Flows for One-step Generative Modeling.\n> W3: Performance drops when inference noise ranges differ from training distributions.\n> \nThank 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  $\\phi(\\rho,\\gamma,\\cdot) \\circ \\phi(\\gamma,\\tau,\\cdot) = \\phi(\\rho,\\tau,\\cdot)$.\nBut 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.\n\n> Q1: Actual training time.\n> \n\nWe 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.\n\nThe key findings are as follows:\n\n(a) Inference speedup: SSOL achieves 3.5x faster inference than D3U and 25x faster than NsDiff.\n\n(b) Memory efficiency: 35% less GPU memory than D3U, 85% less than NsDiff.\n\n(c) Performance parity: Despite far fewer denoising steps, SSOL achieves comparable/better MSE/MAE.\n\nThe 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. \n\nTable: Runtime analysis for ETTh1.\n\n| Model | Pred. Len | MSE | CRPS_SUM | Train T₁ (s/iter) | Train T₂ (s/iter) | Inference (min/batch) | NFEs | GPU Mem (MiB) |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| SSOL | 336 | 0.491 | 0.639 | 0.023 | 0.027 | 0.022 | 1 | 364 |\n| D3U | 336 | 0.512 | 0.922 | 0.025 | — | 0.078 | 20 | 582 |\n| NsDiff | 336 | 0.728 | 1.083 | 0.105 | — | 0.544 | 20 | 2560 |\n| SSOL | 720 | 0.539 | 0.672 | 0.028 | 0.036 | 0.044 | 1 | 646 |\n| D3U | 720 | 0.533 | 1.458 | 0.030 | — | 0.164 | 20 | 884 |\n| NsDiff | 720 | 0.704 | 2.086 | 0.179 | — | 0.608 | 20 | 5380 |\n\n> Q2: How does SSOL maintain temporal coherence with sparse historical data? Does it over-rely on priors in such cases?\n> \nGood 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.\n> Q3: Explain why the proposed method can ensure performance with single-step sampling.\n> \nOur 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."}, {"review_id": "HfiMjck2DZ", "reviewer": "Reviewer_uKCQ", "summary": "This 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.", "questions": "please see the weakness above", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "strengths:\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.", "was_revised": true, "final_justification": "I appreciate the author's response. My concerns have been addressed, I've raised the score to 5.", "rebuttal": "We thank Reviewer uKCQ for the valuable feedback and insightful questions. We have carefully addressed each point below.\n\n> W1: What is the difference between the proposed single-step operator and the one-step diffusion that works for time series?\n> \n\nThanks 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.\n\nOur 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.\n\n> W2: Compare with one-step diffusion models from the image domain, such as [1].\n> \n\nThank 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.\nSome 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.  \n\nWe 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.\n\n[1] Frans, K., Hafner, D., Levine, S., & Abbeel, P. (2025). *One Step Diffusion via Shortcut Models*.\n\n> W3: Report probabilistic metrics.\n> \n\nThanks 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.\n\nWe 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."}, {"review_id": "ExzwCg3yE6", "reviewer": "Reviewer_XQ6p", "summary": "The 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.", "questions": "1.\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?", "limitations": "The 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).", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The paper shall be acceptable based on all the clarification and new experiment results.  I rated the paper as 5.", "rebuttal": "We thank Reviewer XQ6p for the constructive feedback and thoughtful questions. We have carefully addressed all points below. \n\n> W1: Is SSOL competitive with the latest approaches in the broader diffusion literature?\n> \n\nWhile 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.\n\nIn 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.\n\n> W2: Sensitivity to Fourier versus wavelet choices, and to irregular or non-uniform sampling.\n> \n\nTo 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.\n\nRegarding 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.\n\n> W3: How well does the approach scale to substantially longer horizons or to online, continually updated streams?\n> \n\nThanks 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.\n\nTable: Forecasting performance for ETTh1.\n\n| Model | Pred. Len | MSE (std) | MAE (std) | CRPS (std) | CRPS_SUM (std) |\n| --- | --- | --- | --- | --- | --- |\n| SSOL | 336 | 0.491 (0.004) | 0.477 (0.003) | 0.458 (0.004) | 0.639 (0.006) |\n| D3U | 336 | 0.512 (0.006) | 0.478 (0.003) | 0.351 (0.002) | 0.922 (0.109) |\n| NsDiff | 336 | 0.728 (0.136) | 0.583 (0.046) | 0.431 (0.038) | 1.083 (0.171) |\n| SSOL | 720 | 0.539 (0.014) | 0.535 (0.010) | 0.508 (0.013) | 0.672 (0.045) |\n| D3U | 720 | 0.533 (0.044) | 0.505 (0.022) | 0.371 (0.017) | 1.458 (0.862) |\n| NsDiff | 720 | 0.704 (0.062) | 0.613 (0.027) | 0.440 (0.020) | 2.086 (0.159) |\n\n> Q1: Can you provide more insight or theoretical justification into when and why a single-step operator is sufficient for accurate approximation?\n> \n\nOur 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. \n\nBut 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.\n\n> Q2: Have you explored hybrid settings where part of the operator is learned via single-step and part via recurrence or iterative refinement?\n> \n\nThank 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. \n\n> Q3: How does the architecture scale with grid resolution and dimensionality? Any observed bottlenecks in training or inference?\n> \n\nThank 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). \n\n> Q4: Would your method generalize to non-grid or graph-based operator learning tasks?\n> \n\nYes, 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.\n\n> Q5: Could you release the code and data to support reproducibility?\n> \n\nYes, 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.\n\n> Q6: limitations or societal impact.\n> \n\nWe apologize for the inconvenience. We moved them to Appendix D for space saving and will relocate them to the main body."}, {"review_id": "BNDo2vo3AK", "reviewer": "Reviewer_bj9n", "summary": "The 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.", "questions": "Major: \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.", "limitations": "See my questions above - also note that the limitations are only mentioned in the appendix", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 4, "originality": 2, "strengths_and_weaknesses": "Though 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", "was_revised": true, "final_justification": "I thank the authors for their responses to the points raised in my review. I am satisfied with the clarifications provided and thus increased my score.", "rebuttal": "We thank Reviewer bj9n for the thorough feedback and thoughtful questions. We have carefully addressed each point below.\n\n> Q1: Why do the authors claim the high-frequency damping from DMs needs correction?\n> \n\nYou 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.\n\n> Q2: Fig.2 is not clear.\n> \nThank you for the very detailed feedback! We will revise Figure 2 to address these issues:\n\na) 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.\n\nb) 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.\n\nc) 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.\n\n> Q3: Some elements are in the appendix only.\n> \n\nThis 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. \n\n> Q4: Clarify the connection between the Fourier spectral theory and wavelet implementation.\n> \n\nOur 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.\n\n> Q5: Claim of reducing computation overhead.\n> \n\nThanks 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:\n\n(a) SSOL achieves 3.5x faster inference than D3U and 25x faster than NsDiff.\n\n(b) 35% less GPU memory than D3U, 85% less than NsDiff.\n\n(c) Despite far fewer denoising steps, SSOL achieves comparable/better MSE/MAE.\n\nThe 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.\n\nTable: Efficiency and performance comparisons on ETTh1.\n\n| Model | Pred. Len | MSE (std) | MAE (std) | CRPS (std) | CRPS_SUM (std) | Train T₁ (s/iter) | Train T₂ (s/iter) | Inference (min/batch) | NFEs | GPU Mem (MiB) |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| SSOL | 336 | 0.491 (0.004) | 0.477 (0.003) | 0.458 (0.004) | 0.639 (0.006) | 0.023 | 0.027 | 0.022 | 1 | 364 |\n| D3U | 336 | 0.512 (0.006) | 0.478 (0.003) | 0.351 (0.002) | 0.922 (0.109) | 0.025 | — | 0.078 | 20 | 582 |\n| NsDiff | 336 | 0.728 (0.136) | 0.583 (0.046) | 0.431 (0.038) | 1.083 (0.171) | 0.105 | — | 0.544 | 20 | 2560 |\n| SSOL | 720 | 0.539 (0.014) | 0.535 (0.010) | 0.508 (0.013) | 0.672 (0.045) | 0.028 | 0.036 | 0.044 | 1 | 646 |\n| D3U | 720 | 0.533 (0.044) | 0.505 (0.022) | 0.371 (0.017) | 1.458 (0.862) | 0.030 | — | 0.164 | 20 | 884 |\n| NsDiff | 720 | 0.704 (0.062) | 0.613 (0.027) | 0.440 (0.020) | 2.086 (0.159) | 0.179 | — | 0.608 | 20 | 5380 |\n\n> Q6: Error bars for the imputation experiment in Fig 5.\n> \nYes, 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:\n\n(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.\n\n(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. \n\n(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.\n\nWe 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.\n\n> Q7: Fig 3 (right): Why does performance improve with more missing data (10% to 25%)?\n> \nThis 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. \n\n> Q8: Error bars.\n> \n\nYes, 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.  \n\nFor 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. \n\n> Q9: Place own contributions in context in related work.\n> \n\nWe appreciate this constructive feedback. We will modify the text to clearly emphasize our contributions relative to existing approaches.\n\n> Q9: Suitability of the analysis in the spectral domain.\n> \n\nYou are right that power spectral density (PSD) analysis must assume stationarity and discards phase information. We need to clarify two key aspects: \n\n(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.\n\n(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.\n\nThe 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."}]

Author rebuttal:

Meta-review: This paper proposed a diffusion model for time series based on operator learning.  The approach reduces the generation problem from the usual multi-step diffusion methods to a single step. The authors developed their approach to mitigate the frequency-dependent smoothing induced by the forward operator of the diffusion by working in the frequency domain. The paper contains extensive experiments demonstrating the method's improved performance and computational complexity.

The reviewers highlighted a number of strengths
* The novel approach encouraging semi-group consistency to reduce the sampling steps to 1 makes inference faster without losing accuracy.
* The paper is technically solid with rigorous derivations and motivations.
* The comprehensive experimental evaluation was well done where the method was demonstrated to produce consistent predications across diverse domains.

The reviewers did bring up a few weaknesses related to comparing the method to existing work, some design choices, and evaluations on other relevant tasks (e.g. long-horizon prediction). All of these concerns were addressed.

One concern raised by a reviewer related to not comparing to existing working on single-step generation from the image generation literature. After substantial discussions between the reviewer and authors, these comparisons would require substantial changes to the existing methods at which point the authors would just be creating a (potentially sub-optimal) competitor to their proposed algorithm.

Despite this last concern, the proposed method and comprehensive evaluations make this a strong paper that should be accepted.
Decision: Accept (poster)