Latent-to-Data Cascaded Diffusion Models for Unconditional Time Series Generation
Authors: Lifeng Shen
Kai Syun Hou
Weiyu Chen
James Kwok
Venue: ICLR 2026 Poster (2026)
Keywords: time series
unconditional
synthetic
Primary area: learning on time series and dynamical systems

Abstract: Synthetic time series generation (TSG) is crucial for applications such as privacy preservation, data augmentation, and anomaly detection. A key challenge in TSG lies in modeling the multi-modal distributions of time series, which requires simultaneously capturing diverse high-level representation distributions and preserving local temporal fidelity. Most existing diffusion models, however, are constrained by their single-space focus: latent-space models capture representation distributions but often compromise local fidelity, while data-space models preserve local details in the data space but struggle to learn high-level representations essential for multi-modal time series. To address these limitations, we propose L2D-Diff, a dual-space diffusion framework for synthetic time series generation. Specifically, L2D-Diff first compresses input sequences into a latent space to efficiently model the distribution of time series representations. The distribution then guides a data-space diffusion model to refine local data details, enabling faithful generation of time series distribution without relying on external conditions. Experiments on both single-modal and multi-modal datasets demonstrate the effectiveness of L2D-Diff in tackling unconditional TSG tasks. Ablation studies further highlight the necessity and impact of its dual-space design, showcasing its capability to achieve representation coherence and local fidelity.

Published as a conference paper at ICLR 2026
TA-ANTATO-DATA CASCADED DIFFUSION MODELS FOR UNCONDITIONAL TIME SERIES GENERATION
Lifeng Shen Kai Syun Hou?, Weiyu Chenf, James T. Kwok?
1 Chongqing Key Laboratory of Computational Intelligence,
Chongqing University of Posts and Telecommunications
2 Department of Computer Science and Engineering, Hong Kong University of Science and Technology
shen f0 recuppt... edu. cn, kshou ust.uut.... .kk wchenbxêcse, ust hk, j jame s sk@ cse. ust hk
ABSTRACT
Synthetic time series generation (TSG) is crucial for applications such as privacy preservation, data augmentation, and anomaly detection. A key challenge in TSG lies in modeling the multi-modal distributions of time series, which requires simul- taneously capturing diverse high-level representation distributions and preserving local temporal fidelity. Most existing diffusion models, however, are constrained by their single-space focus: latent-space models capture representation distribu- tions but often compromise local fidelity, while data-space models preserve local details in the data space but struggle to learn high-level representations essential for multi-modal time series. To address these limitations, we propose L2D-Diff, a dual-space diffusion framework for synthetic time series generation. Specifi- cally, L2D-Diff first compresses input sequences into latent space to efficiently model the distribution of time series representations. The distribution then guides a data-space diffusion model refine local data details, enabling faithful generation of time series distribution without relying on external conditions. Experiments both single-moda and multi-modal datasets demonstrate the effectiveness of L2D-Diff in tackling unconditional TSG tasks. Ablation studies further highlight the necessity and impact of spacespace design, showcasing its capability achieve representation coherence and local fidelity.
INTRODUCTION
Time series data is critical in domains such as finance, healthcare, biotechnology, and climate science. However, restricted access temporal datasets, especially in privacy-sensitivy contexts, often limits the progress of machine learning models. Synthetic time series generation (TSG) has emerged as promising solution, leveraging deep learning techniques create realistic data that replicates complex temporal dependencies and multidimensional correlations (Zhou et al., 2023; Alaa et al., 2021; Ang al., 2023; Yuan Qiao, 2024). These synthetic datasets retain their utility for downstream tasks such as classification and forecasting (Esteban et al., 2017; Ang et al., 2023; Yuan & Qiao, 2024).
Generative adversarial networks (GANs) (Goodfellow al., 2014) were the preferred approach for TSG (Esteban et al., 2017; Li et al., 2022; Mogren, 2016; Pei et al., 2021; Yoon et al., 2019). Despite their success, GANs face challenges such as adversarial training instability and mode collapse, limiting their effectiveness in generating diverse and robust time series. Recently, diffusion models (Yang al., 2023), particularly the denoising diffusion probabilistic model (DDPM) (Ho al., 2020), have gained prominence due to their superior perceptual quality and stable training dynamics. These advancements have led to significant progress in generative tasks (Yang et al., 2023), with diffusion models excelling in areas such as image editing (Huang et al., 2024), image generation (Cao et al., 2024), and video generation (Xing et al., 2024).
While diffusion models have achieved remarkable success in images and videos, their application to time series presents unique challenges. Unlike visual data, time series generation requires the simultaneous modeling of multi-modal latent structures and the preservation of local temporal fidelity.

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This involves capturing intricate temporal relationships and managing complex interdependencies across variables, both of which are essential for accurately modeling real-world time series patterns. Addressing these challenges is crucial for extending the capabilities of diffusion models to TSG.
Recent works on time series diffusion primarily focus on conditional generation tasks such as fore- casting (Rasul et al., 2021; Shen & Kwok, 2023; Kollovieh et al., 2024) and imputation (Tashiro et al., 2021; Alcaraz & Strodthoff, 2022). For instance, TimeGrad (Rasul et al., 2021) employs recurrent neural networks to summarize history as conditions for denoising future values. Similarly, TimeDiff (Shen & Kwok, 2023) introduces autoregressive initialization and future mixup to enable efficient non-autoregressive prediction. CSDI (Tashiro et al., 2021) adopts self sepe-sueeeeeeed masking techniques, while Alcaraz& & Strodthoff (2022) enhance CSDI by replacing transformers with structural state space models (Gu et al., 2021), improving long-range temporal modeling. These studies primarily focus on leveraging conditional information, designing robust conditioning networks, and constructing effective denoising architectures to address specific supervised tasks. In contrast, synthetic time series generation focuses on unconditionally producing high-quality time series (modeling the data distributions) that replicate the statistical properties of the original dataset (Ang et al., 2023).
Recent approaches to unconditional generation can be broadly divided into two categories. (i) sata--pacee diffusion models (Park et al., 2024; Yuan & Qiao. 2024; Crabbé et al., 2024; Naiman et al., 2024a; Sikder al., 2025), which directly model the raw time series distribution; and (ii) latent-space diffusion models (Qian et al., 2024; Feng et al., 2024), which operate on compressed representations obtained through predefined transformations (e.g., Fourier transform) learned nonlinear encoders. While latent-space diffusion models excel at capturing high-level semantic structures through compressed representations, they often struggle to preserve subtle temporal dynamics. The process of dimensionality reduction can result in the loss of fine-grained details, thereby reducing the diversity and fidelity of the generated outputs. On the other hand, data-space diffusion models perform iterative denoising directly on the raw time series, effectively capturing localized temporal patterns with high precision. However, their focus on local details makes it difficult to comprehensively model representation distributions.
To address these challenges, we transition from un- conditional diffusion in the data space to latent- to-data conditional diffusion, which balances rep- resentation distributions with local temporal data distributions. Specifically, we propose L2D-Diff, a latent-to-data diffusion framework that integrates the strengths of latent-space modeling and data- space refinement overcome the limitations of unconditional generation. L2D-Diff operates in two complementary stages: (i) Latent-space coarse gen- eration: A latent diffusion model captures repre- sentation distributions by representation learning techniques. (ii) Data-spuce refinement: A subse- quent denoising process integrates the global latent codes into the data space, enabling fine-grained tem- poral precision and ensuring consistency with the original data distribution. This two-stage approach ensures both global consistency and local precision. enabling realistic, semantically rich, and high-fideliyy time series generation. To the best of our knowledge, we are the first to study the latent-to-data cascaded diffusion model for synthetic time series generation.
data
latent
cascaded
TimeGrad
x
X
CSDI
x
X
TimeDiff'
X
x
TSDE
X
TimeLDM
X
V
X
LDT
X
V
X
DiffusionTS
X
X
L2D-Diff (proposed)
Table 1:
Comparing related diffusion methods. "data" refers to directly modeling the time series distribution in the data space. "latent" indicates learning the distribution of representations in latent space. "cascaded" denotes using multiple diffusion models for generation.
Some initial attempts have been proposed in the contexts of image generation and graph modeling. For example, in the ppreeentationncondiioon
generation (RCG) framework (Li et al., 2024), pre-trained image encoder is used to first obtain image representation distributions, which then condition the image distributions. This further extended the generation of graphs in (Wang et al., 2024). Another model EDDPM (Liu et al., 2019) uses parameterized encoding- unified space to generalize the Gaussian noising-denoising denoising in standard data-space diffusion. However, the development of hybrid models for time series generation is still under-explored. In (Ge et al., 2025), text-to-series diffusion model (T2S) developed that leverages textual features to assist in time series generation. In contrast to T2S, we do not utilize additional textual descriptions. Our
in a
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focus is on establishing effective representation-to cascaded model for unconditional time series generation. The advantage our approach lies in its independence from the text processing mechanisms of large language models, making simpler and more efficient.
Note that some cascaded time series diffusion models exist, including (Fan et al., 2024; Shen et al., 2024). However, these models not originally designed for unconditional generation. In contrast, the proposed L.2D-Diff formulates a cascade between latent-space diffusion model and data-space diffusion model, and to the best of our knowledge, the first framework to explicitly bridge latent representation diffusion and data-space diffusion for unconditional time series generation. Table provides a comparison between the proposed method and related works.
RELATED WORKS
This section reviews diffusion models for unconditional time series generation. Classical generative models, such as GANs, VAEs, and flow-based approaches, are covered in Appendix A.
Data-space diffusion models directly model the raw time series distribution. Park et al. (2024) employ diffusion bridges to map prior distributions to time series, enabling flexible and accurate synthesis. Diffusion-TS -S (uaan & Qiao, 2024) integrates seasonal-trent decomposition with diffusion models and introduces a Fourier-bask based objective to better capture periodic patterns. Similarly, FourierDiffusion (Crabbé et al., 2024) operates in the frequency domain, replacing the traditional Brownian motion with mirrored Brownian motion to enhance its ability to model periodic behaviors. Naiman et al. (2024a) transform time series into images and apply vision-based diffusion models to synthesize data. Sikder et al. (2025) recently develop TransFusion for long sequence generation.
Latent-space diffusion models operate on compressed representations obtained through predefined transformations (e.g., Fourier transform) or learned nonlinear encoders. Representative methods such as TimeLDM (Qian et al., 2024) and latent diffusion transformer (LDT) (Feng et al., 2024) achieve computational efficiency by working in nower-dimensional latent space. This compression helps preserve structural representations within the data distributions. However, the reliance on encoder- decoder architectures introduces an information bottleneck, which risks discarding fine-grained temporal details and limits the fidelity the generated outputs.
PRELIMINARIES
Problem definition. Let T (x(1) x(N) be dataset with N multivariate time series samples.
Each (0) = (x1 (x.,),,,,> ( with x E RD,, can be represented as D-by-L matrix, where D is the number of variables and L is the time series length. The goal of synthetic time series generation (TSG) create a synthetic dataset Tgen (x x(^")) such that its distribution g(T9en) is similar to the true distribution p(T), exhibiting consistent statistical properties and temporal dynamics. This is an unconditional generation task. Importantly, we require each synthetic time series x to be also of length L and contain D variables, ensuring compatibility with the original dataset structure.
3.1 DENOISING DIFFUSION PROBABILISTIC MODELS
Denoising diffusion probabilistic model (DDPM) (Ho et al., 2020) is latent variable model with forward diffusion and backward denoising processes.
Forward diffusion. A time series input' x" is gradually corrupted to a Gaussian noise vector. At the ktth step, x is generated by corrupting the previous iterate x" (scaled by vI -3) with zero-mean Gaussian noise (with variance Be E [0, 1)):
qxx* | x-1 = Nx*:; VI 3,x+ 1- 1x == k= 1.....,,K K.
It can be shown that xx* x²) ; v Vāxx", (1 az I]. where āy II as and as 1 Bk. Thus, x can be simply obtained as
/ axxx /1 ā+e,
(I)
Here, superscript 0 means the original input without diffusion.
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where is noise from (O., This equation also allows x" to be easily recovered from x".
Reverse denoising. At the kth denoising step. x*~1 is generated from x² by sampling from the normal distribution:
1o(x~>1/>") = Ni NN No k), Eo(x", (x(xxxx)))
(2)
Here, the variance (x k) is usually fixed as LLL while the mean pa k) is defined by neural network (parameterized by 0). This is usually formulated as a noise estimation or data prediction problem (Benny & Wolf, 2022). For noise estimation, network co predicts the noise of the diffused input x*, and then obtains px(x,, .xK,, k)) vay x yoEvI irevi-ay (x k)) Parameter 0 is learned by minimizing the noise estimation loss f. X
9
[||e to (x* 1)
Alternatively, the data prediction strategy uses denoising network xa(-) to obtain estimate xx(x*, & k) of the clean data x" given x², and then set
px k)= Varll 0k-
BR BRVOR xo (x* R k).
x
Or
1 Crp
Then, is learned by minimizing the following loss
Lx Exa,,, R 11 xx A)]
(4)
When condition is accessible, the following distribution considered (Rasul al., 2021; Tashiro et al., 2021; Shen & Kwok, 2023)
K II (x F(c)),
po 0: (c)
(5)
where x ~ NN(,,) I), and F a conditioning network that takes the condition с as input. Corre- spondingly, the denoising process at step k
(x - c) c)xN(x) 3 (FFF))
(6)
where k- K, K
Data sampling. During inference, let the generated sample corresponding to x be & We first initialize * as noise vector from ((00 I). By repeatedly running the denoising step in Equation (6) till k = 1. the final generation is X²
3.2 LATENT-SPACE DIFFUSION MODELS
Latent-space diffusion models (LDMs) (Rombach et al., 2022) consist of two main components: (i) pretraining process and (ii) latent diffusion. The pretraining process is commonly performed based on optimizing representation learning task, such as masked modeling or contrastive learning. involves encoder, which maps time series I DXL to a nower-dimension space r E Rd, and decoder, which generates x from r. Subsequently, a diffusion model is applied on the latent code r. The forward diffusion process in latent space is:
(fied-length)) latent
Vagr
age.
The reverse process is performed by recursively sampling R from the following normal distribution:
1r*) = N mo (+ R),, (r* k)).
(8)
The training objective minimizes the following loss:
C. Ep |r (1 1)1
(9)
Here, rg is denoising network in the latent space.
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Data-space diffusion
Man bln x
Encoder
largel
time series
(for training)
Pa(x x
F(F)
Lateni-to-data condition
representation distribution
F
g0
Denoising
samplnng
Latent-space diffusion
(al interence)
inpul
pole "1e"
Figure 1: Framework of the proposed L2D-Diff for time series generation.
METHODOLOGY
Overview. The proposed cascaded diffusion model, L2D-Diff, is illustrated in Figure integrates two collaborative diffussion/eenoising branches: in the latent space and the other in the data space. The latent-space branch models the distribution of high-level representations in the time series, offering compressed yet structured understanding of temporal patterns. To construct the latent space, an encoder-decode, pair is pretrained using masked modeling-basee representation learning optimization, ensuring that the latent representations are meaningful and informative. The data-space branch then models the probability density function of the time series data guided by the representation distributions, capturing fine-grained temporal details. To bridge these two branches, latent-to connttto-data conditioning mechanism is introduced. This module enables latent representations to guide the denoising process the data space, ensuring seamless coordination between the representation distribution and data distribution.
This design enables L.2D-Diff to effectively capture high-level temporal patterns in the latent space while achieving data detail generation in the data space, guided by the latent variables. In Appendix B, we provide a thcoretical understanding of L2D-Diff through the lens of the Information Bottleneck (IB) principle (Tishby et al., 1999). Intuitively, divide-and-on andiie--ad-coqquer strategy. With a rich latent space, L2D-Diff utilizes latent diffusion to capture high-level semantics of the data while allowing the data diffusion process to more easily focus on modeling local details and residual uncertainties, resulting in higher-quality generation.
tatent-to-data cascaded structure presents
Although there have been attempts study representationncondiin such as image and audio, time series generation presents unique challenges not fully addressed by existing cascaded diffusion frameworks, particularly concerning temporal consistency and multi- channel correlations. Time series data has inherent temporal dependencies, meaning that the sequence of data points affects future values. Ensuring temporal consistency is crucial, as high-quality generated series must reflect realistic trends and relationships. Moreover, many time series comprise multiple channels that may interact complex ways, necessitating the capture of these correlations to produce coherent and high-quality outputs. By addressing these challenges, our work aims provide robust framework for unconditional time series generation, ultimately raising the quality standards for generated time series with multiple modes that leverage their full potential.
generation data modalities
4.1 DUAL-BRANCH DIFFUSION DESIGN
Latent Space Construction. Given an input time series E P where D is the number of channels and the sequence length, we perform a pretraining task based on masked modeling to derive compact, high-level representation Rd, with d & L x D.

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In this process, random subset of the input tokens masked according to binary mask {0, 1) 15xx,, where ma,j = 1 indicates that the token (i:j) is masked. The masked input Xmasked is obtained by replacing the masked positions with special mask tokens. The encoder E processes the corrupted input Xmasked- generating a latent representation m'asked = E mmmuee))) The decoder D reconstructs the original input from the latent representation m'msked- The optimization objective is designed to minimize the reconstruction error at the masked positions only: Lpretranning = ||m @ (x - D(E((maakkea)) 12, where @ denotes dlement-wiss multiplication, ensuring that only the masked positions contribute to the loss.
After pretraining, input x is encoded into the representation r r= E Intuitively, encapsulates the high-level temporal characteristics of x. We then introduce a latent- space diffusion model, denoted ro (where & denotes its parameters), to model the distribution of over 5 diffusion steps. The diffused representation r' is obtained from (= r) following (7):
Latent-Space Diffusion.
v + -I-
(10)
where e the Gaussian noise, ds governs the noise level at step s (1 S 8 S 5), and S is the total number of latent diffusion steps.
To train rg we minimize the denoising loss (9), which encourages rs to recover the original representation r from its noisy counterpart r":
Liatent ,e. |r00 Po(r",))]
(11)
Data-Space Diffusion. The data-space diffusion model regenerates the full-resolution series Dxx guided by the representation encoded in the latent space. This latent-to-data diffusion mechanism allows each position in the data-space series x, attend the latent code r, effectively injecting structural priors into local refinements.
Following (4), the diffusion model is optimized by minimizing the denoising loss
Laata Exo, xx - ,x, ,(,)))))
(12)
where xy is the denoising network, is the condition, and F is the conditioning network. In practice, F is implemented as a convolutional neural network (with 5 layers by default). At each denoising step ki X4 takes three inputs: noisy input x* DxL, timestep h and the conditioning network's output F(c), to produce a data estimate X1 (x K,, F(c)).
4.2 LATENT-TO-DAYE CONDITIONING
Conditioning Network. In L2D-Diff, unconditional time series generation is reformulated as conditional generation problem, where the latent representation sampled from the learned latent distribution, denoted i, serves as the condition for the data-space diffusion model. this way, the global structure encoded in the latent space is explicitly injected into the data-space denoising process, enabling the diffusion model to generate samples consistent with the learned representation distribution.
During training, the conditioning network F takes the latent code r E Rdd as input, i.e., c = r, and transforms it into conditioning signal compatible with the data-space diffusion model. Intuitively, F learns to project latent representations into a structured guidance signal that modulates the denoising trajectory at each diffusion step.
This latent-to-data conditioning mechanism ensures that the representation distribution captured in the latent space effectively guides the local refinements in the data space, leading high-quality time series generation that aligns with both representation distribution and data representation.
Denoising Network. The denoising networks rg in Equation (11) (reps. Xy in Equation (12)) is trained learn to denoise the representation r (resp. the diffused data x") into r=11 (resp. x 1).
The key distinction between the two denoising networks lies in their conditioning mechanisms: X9 incorporates latent-ppace-deriveed conditioning signal guide the refinement process.

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4.3 SYNTHETIC TIME SERIES GENERATION
On inference, we start from 1 ~ (N( in the latent space. Based on the data prediction strategy in (3),
,8) 8) + ost, p va, as-1) r + Vas- ass) ro (r(rr"
(13)
1 02s
as
where (N when > 1, and e 0 otherwise. Till 8 1, we obtain the sampled representation f Then, we have с 8 to guide the data denoising process. Specifically, we start from & ~ NN I) in the data space, and we have the reverse denoising step equation
O'k kk-1)
vāk- axk) xx(x", ks F(c)) dge.
*-1
(14)
ak
akk
Till 1, we obtain the sampled time series data x². The pseudocodes for the training and sampling procedures are in Algorithms and of Appendix C, respectively.
5 EXPERIMENTS
In this section, we perform experiments on number of datasets to demonstrate the performance of the proposed L2D-Diff.
5.1 SETUP
Datasets. Previous works, such as Diffusion- TS (Yuan Qiao, 2024), mainly use time se- ries datasets with relatively limited distribu- tional diversity (Stock, Energy, ETTh, Riverflow). These datasets are typically constructed by us- ing a sliding window over the whole time series. While temporal variations exist, these patterns are drawn from the same underlying data distri- bution, making them less suitable for evaluating performance across distinct labeled categories.
dataset
#training
#testing
D
I.
C
Stock
2,928
733
6
24
Energy
15,768
3,943
28
24
ETTh
13,801
3,451
7
168
Riverflow
18,858
4,715
168
Two Patterns
1,000
4,000
128
ECG500)
500
Medical Images
381
760)
1
99
10
Arabic Digits
Arrial Fibrillation
4,832
185
2
45
Japanese Vowels
270
Character Trajectories
300
2,558
3
205
20
To better evaluate generative performance, include seven more datasets with explicit class annotations from the time series classification archive². Hence, they have inter-class variabil- ity and the underlying data distributions exhibit multiple modes.
Table 2: Summary of dataset statisties (D: dimen-
sion; L: time series length; C: number of classes).
Table 2 shows statistics of the datasets. ensure fair evaluation of generalization in the uncondi- tional generation setting, the data are uniformly split into training and test sets. The test samples are not used during training, preventing data leakage and enabling evaluation on unseen data.
Baselines. We include baselines from various categories: (i) Diffusion models operating in the time domain: Diffusion-TS (Yuan Qiao, 2024) and TSDE (Senane et al., 2024); (ii) Latent diffusion models: TDMel.DM (Park et al., 2024) and EDDPM (Liu et al., 2019); (iii) Diffusion models operating in the Fourier domain: Fourier Diffusion (Crabbé et al., 2024) and Imagen'iime (Naiman al., 2024a). (iv) Flow- generative models: FourierFlow (Alaa et al., 2021) and its time-domain adaptation, referred to as TimeFlow, which applies the FourierFlow framework dircctly in the time domain for comparison. (v) Generative adversarial networks (GANs): We include two popular baselines in (Ang et al., 2023): TimeGAN (Yoon et al., 2019) and GTGAN (Jeon et al., 2022); (vi) Variational autoencoder (VAE) models, including VAE (Naiman et al., 2024b) Time (Lee et al., 2023) LS4 (Zhou et al., 2023) and the original VAE (Kingma & Welling, 2014).
"Though those datasets contain label information, such information is not used during the generation process. 'htpps: / www.. meseriesch assi fi cat on. com/
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Energy
ETTh
Riwerlow
Twe
ECG5000
Medicul Images
Arabic Digits
Atrial Fibvillation
Japanere Vowels
Churucter Trujectorics
L2D-Diff
0.31
0.53
0.45
0.32
0.21
0.11
0.08
1.29
1.15
0.46
0.25
Diffasion-TS
1.24
3.10
1.66
1.93
TSDE
3.90
4.13
0.26
2.83
1723.0
0.85
2.60
3,97
4,70
TimeL.DM
6.17
3.51
9.52
1.01
1.40
0.76
0.88
5.99
5.48
0.99
2,00
10.76
28.29
FoarierDiffusion
0.21
0.48
3.38
3.54
L.66
0.32
0.41
1.36
1.14
0.49
3.58
ImagenTime
4.23
2:22
7.72
0.50
4.82
6.38
2.99
2.98
1.66
1.08
12.02
FourierHlow
1.15
0.38
3.17
1.843
1.21
0.98
1.52
2,84
2.37
0.74
5.07
TindFlow
041
0.85
3.19
2.177
1.17
0.20
0.65
8.40
173.12
3.45
TimeGAN
0.88
0.87
20.32
2,00
2,26
3.88
0.70
4.73
6.63
1.30
3.97
GTGAN
0.70
2.55
26.60
3,23
25.53
3.39
2.82
16.23
3.23
2.24
10.01
KoVAE TQeEVVVEE
1.72
8.82
0.30
2.45
6.05
8.40
0.74
5.06
4.20
2.93
8.17
3.77
4.62
3.98
LS4
5.85
10.97
23,47
3:47
15.81
24,21
31.81
14.45
8:15
11.34
24.67
35.65
1.67
3.42
2.62
15.70
Table 3: Contextual-FID results on 11 time series datasets. The lower the better. Bold and underline indicate the best and second best performance, respectively. The symbol "_" indicates that the method suffers from unstable training and significantly inferior performance.
Evaluation Metrics. Following Ang et al. (2023), we evaluate generation quality using the following three metries on the set: (i) Contextual-FID (C-FID) (Zhou et al., 2022), which measures how well the synthetic time series align with the local context of the original data; (ii) Discriminative Score (DS), and (iii) Predictive Score (PS). The use of DS and PS follows (Yuan & Qiao, 2024). For DS, a 2-layer LSTM is trained to classify sequences as "real" (original) or "not real" (generated), with the classification error measuring dataset similarity. For PS, a 2-layer LSTM trained on the generated data to predict next-step temporal vectors, and its mean prediction error on the original dataset reflects how well predictive patterns are preserved. However, we consider DS and PS as secondary metries due to their sensitivities to model setup and dataset size. Besides, we also provide visualizations using t-SNE and distribution plots to compare the distributions of the original and generated time series.
Implementation Details. We train the model using Adam with a learning rate of 10 batch size of 128, and early stopping for up to 100 epochs. Both the latent and data diffusion models in L2D-Diff are trained with K 100 diffusion steps under a linear variance schedule (Rasul et al., 2021), with 3, = 10-4 and Ax 10-1 The CNN of TS2Vec (Yue al., 2022) is pre-trained our encoder, with a default latent dimension of for high-level representation learning. The decoder utilizes be 50% as in (Dong al., 2023).
three-layer convolutional network. The masked ratio Implementation details of the denoising network are in Appendix F. Experiments are run an Nvidia RTX A6000 GPU with 48GB of memory. All baselines are evaluated using their official implementations.
5.2
MAIN RESULTS
Results on Contextual-FIID are shown in Table 3. To validate the statistical significance of method rankings, we employ the Friedman test (Friedman, 1937) and Conover's post-hoc (Conover &
US44
L20-BU
Iman, 1979). Figure shows the average rankings and the corresponding critical differences (CD) for each method. The average ranking reflects the overall performance of each method (the lower the better). The CD value indicates the smallest dif- ference in rankings that statistically significant, as determined by the post-hoc test.
Wi
GTOWN Bo00
FoutorFow
Orhakn-T5
TiraLMM
EDePy
ThraGN
InagenTime EX0T
As can be seen, the proposed L2D-Diff achieves superior overall performance with average rank of 1. 1.45, significantly outperforming all the base- lines. D.ff- is simple yet effective. As cas-
Figure 2: Critical difference diagram of TSG methods. The lower the better.
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60
Orlginal Generatec
Origial
40
20
-70
-0
-40
-20
80
-40
20
-40
-20
-40
-20
20
(a) L2D-Diff.
(b) FourierDiffusioon
(c) FourierFlow.
60
Original
Original
40-
Generated
20-
-20
-20
-40
-50
Original
-60
-40
-0
60
-50
-40
-20
20
40
60
40
20
20
40
50
(d) Diffusion-T'S
(c) TimeGAN.
(f) KoVAR
Figure 3: 2D t-SNE embeddings of data (not representations) generated vs the real data of Character Trajectories with multiple modes.
caded diffusion model, it bridges the latent and data diffusion processes, transforming an uncondi- tional time series generation problem into conditional one. Specifically, the latent diffusion model first captures the high-level representation distribution, which then guides the data-space denoising process. This approach not only reduces complexity but also ensures efficient and robust generation of time series, particularly when dealing with complex or multimodal distributions.
Among the baselines, FourierDiffusion (Crabbé al., 2024) ranks second with an average rank of 3.55, followed by FourierFlow (Alaa et al., 2021) with an average rank of 5.36. Fourier Diffusion introduces the concept of mirrored Brownian motion and performs data generation in the frequency domain, while Fourier Flow leverages discrete Fourier transform to convert time series into fixed- length spectral representations and applies a data-dependent spectral filter to these transformed series. Moreover, TSDE underperforms due to its lack of high-level latent guidance. Results for Discriminative Score (DS) and Predictive Score (PS) are presented Table 6 of Appendix D.
5.3 VISUALIZATION RESULTS
Figure shows the 2D t-SNE embeddings produced by the proposed L2D-Diff and several popular baselines the Character Trajectories dataset. As can be seen, this dataset is challenging due its complex multi-modal distribution across 20 classes, limited training samples, and the need to model numerous modes effectively. The proposed L.2D-Diff overcomes these difficulties by leveraging its latent-to-dats dual-space framework, capturing global structures while preserving local fidelity, and generating time series that replicate the original data's features and patterns with high accuracy. In contrast, competing baselines tend to capture the overall distribution center while failing to adequately represent its diversity. More visualization results are in Appendix E.
5.4 ABLATION STUDIES
In this section, we perform an ablation study on the effectiveness of latent-space and data-space diffusion using the Stock and Character Trajectories datasets. We compare the proposed L2D-Diff with two variants: (i) Latent-space only, which uses only latent-space diffusion by removing the data-space branch and decoding with pretrained decoder, and (ii) Data-space only, which uses only the data-space diffusion by replacing the latent condition with zeros. More ablation results on the effects of the latent dimension and conditioning network are in Appendices G and H, respectively.

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Stock
Character Trajectories
C-FID
DS
C-FID
DS
PS
L.2-DDfff (full)
0.310
0.048
0.041
0.284
0.179
0.333
Latent-spas space only
3.682
0.204
0.089
1.829
0.355
0.353
Data-space only
0.385
0.049
0.052
2.368
0.380
0.369
Table Effectiveness of the latent-space-only and data-space-only variants.
Table shows the C-FID, DS and PS results, As can be seen, on Stock, the data-space variant outperforms the latent-space one. We speculate that it is because this time series short (L 24) and the distribution is simple (as shown in Figure 5(a) of Appendix E). On the other hand, for the more difficult Character Trajectories dataset, the latent-space variant performs better, indicating the effectiveness of global semantics. In both cases, L.2D-Diff consistently outperforms the two variants. This demonstrates that combining latent-space and data-space diffusion crucial for achieving both global coherence and local fidelity in time series generation.
5.5 EFFICIENCY
In this section, we evaluate the efficiency of L2D-Diff against four representative diffusion models: (i) Diffusion-TS. recent popular data-space model; (ii) TimeLDM, latent-space diffusion model; (iii) Diffiir the most competitive baseline.
training (ms/sample)
inference (ms/sumple)
# of trainable parameters
type
L2D-DifT
data latent
0.52
3.47
2.2M
Diffusion-TS
data
14.28
5.10
25M
TSDE
data
2.10
5.05
1.3M
TimeLDM
latent
0.51
4.85
1.9M
EDDPM
latent
0.51
3.80
1.9M
FourierDiffusion
frequency
0.36
9.66
1.6M
ImagenTime
frequency
1.22
2.82
1.IM
Table 5: Training & inference time, and number of trainable parameters on the Character Trajectories.
Table compares their training time, inference time, and number of trainable parameters on Character Trujectories dataset. Results on additional datasets are in Appendix I. As can be seen, compared to existing time series diffusion models, the proposed L2D-Diff efficient because its latent-space diffusion process is learned in low-dimensional latent space (d < D x L, d 8 by default). By leveraging convolution layers and the acceleration technique DPM-Solver (Iuu et al., 2022), L22 achieves significant reduction in computational costs without compromising generation quality. This demonstrates its efficiency in handling complex multi-modal time series data while maintaining a lightweight model design.
CONCLUSION
We proposed L2D-Diff. simple yet efficient dual-space diffusion framework for high-fidelity time series generation. By integrating dual-space diffusion processes, L2D-Diff learns representation distribution compressed latent space and generation time series in the data space under latent guidance. This streamlined design effectively balances simplicity, fidelity, and efficiency, achieving state-of-the-art performance across a wide range of datasets. Extensive experiments validate L2D- Diff's ability to generate realistic and coherent time series while preserving multi-modal distribution structures. In summary, L2D-Diff exemplifies how a simple yet efficient approach can address the challenges of unconditional time series generation. We hope this work inspires the development of more lightweight and scalable diffusion models. The official implementation of L2D-Diff is publicly available at link to support reproducibility and further exploration.
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ACKNOWLEDGMENTS
This research was supported in part by the National Natural Science Foundation of China (Grants 62221005, 62576056) and the Research Grants Council of the Hong Kong Special Administrative Region (Grants 16200021, 16202523 and HKU C7004-22G).
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A RELATED WORKS ON TIME SERIES GENERATION
Generative models aim to learn intricate patterns and temporal dependencies in time series datasets, enabling the generation of new data that reflects the statistical properties of the original dataset (Ang et al., 2023). In addition to the recent time series diffusion models discussed in Section 2, this section reviews three major categories of classic generative models for time series generation: generative adversarial networks (GANs), variational autoencoders (VAEs), and flow-based generative models.
Generative adversarial networks (GANs) (Goodfellow et al., 2014) consist of a generator and a discriminator, trained through two-player minimax game. The generator takes random noise as input and learns to produce synthetic data that is indistinguishable from the real data, while the discriminator is tasked with classifying real and generated samples. In the context of time series generation, GANs have been enhanced by incorporating specialized generator architectures, such as LSTMs or Transformers, to improve the modeling of temporal dependencies (Esteban al., 2017: Li et al., 2022; Mogren, 2016; Pei et al., 2021; Yoon et al., 2019). Additionally, various strategies have been proposed to improve the training process, including novel loss functions, extra discriminators, classification layers, and data augmentation techniques, which aim to achieve better temporal alignment and enhance performance (Ni et al., 2021; Jeha al., 2022: Scyfi al., 2022; Wang et al., 2023). Despite their effectiveness, GAN-based models are often challenging to train due to instability in the adversarial process and are computationally expensive, requiring significant resources and time (Jeon et al., 2022; Ang et al., 2023).
Variational autoencoders (VAEs) (Kingma& Welling, 2014) offer an alternative approach by minimiz- ing a combination of reconstruction loss and the divergence between the learned latent distribution and prior standard Gaussian distribution. VAFs effectively leverage variational inference to capture complex temporal relationships in time series data (Desai et al., 2021; Lee et al., 2023; Li et al., 2023). A notable example is Time VQVAE (Lee et al., 2023), which integrates vector quantization (Van Den Oord et al., 2017) to preserve both the general shape and fine-grainets details of time series. Another recent work, LS4 (Zhou et al., 2023), models latent space evolution using a state space ordinary differential equation (ODE) and is trained with standard sequence VAE objectives.
In addition to GANS and VAES, flow-based generative models have also been extended to time series generation (Dinh al., 2024; Alaa et al., 2021). Unlike GANs and VAEs, flow-based models directly model the probability density function of time series, avoiding the computational challenges of sampling from latent representation distributions. For instance, Fourier Flow (Alaa et al., 2021) employs a novel class of normalizing flows combined with discrete Fourier transforms (DFT) to convert variable- length time series with arbitrary sampling periods into fixed-length spectral representations. A data data-dependent spectral filter is then applied to refine the frequency-transfommed time series, enabling explicit likelihood estimation.
Most recently, Sikder et al. (2025) developed a Transformer-based diffusion model called TransFusion for long-sequence generation. However, our research focus differs; we emphasize how to better facilitate time series generation with multiple modes using novel latent-to-data cascaded structure.
B THEORETICAL UNDERSTANDING
In this section, we provide theoretical justifications of the proposed latent-to-date method using the Information Bottleneck (IB) principle
B. ENTROPY DECOMPOSITION AND CASCADED GENERATION
First, we decompose the time series generation process using the chain rule of entropy. Let X be the observed time series and be the latent representation. The total entropy of the data (XX can be decomposed into two components:
H(X)
1(X;2)
H(X 22)
(15)
Global Information
Local Detail / Residual Uncertainty
This decomposition provides a formal basis for our latent-to-data diffusion architecture, assigning distinct and complementary roles each model.
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Stage (Latent Diffusion P(Z)): Modeling [[;; Z). The first stage learns the distribution of the latent code Z. Since Z is a compressed representation, [[x; Z) corresponds to the global semantic structure. By modeling P.Z), the first diffusion model captures the high-level semantics of the data.
Stage 2 (Data Diffusion Z)) Modeling H (X (2). The second stage models the conditional distribution of X given Z. This term represents the local details residual uncertainty not captured by Z. Crucially, we show that this conditional distribution often has high entropy and is multi-modal Our framework employs diffusion model here precisely because diffusion models excel at sampling from such complex distributions, effectively "synthesizing" the missing high-frequency details that deterministic decoders would average out. The marginal distribution is thus recovered via the integral:
Po(X) = Po(X(Z)P(Z) dZ =s (X).
This proves that the cascaded approach is theoretically capable of recovering the true data distribution, provided both conditional and latent models are well-learned.
B.2 INFORMATION BOTTLENECK TRADE-OFF AND IMPACT OF LATENT DIMENSION
Next we provide the theoretical understanding of the trade-off imposed by the latent dimension through the lens of the Information Bottleneck principle.
The Bottleneck Constraint: A compact latent dimension d imposes a capacity constraint C(d) on the mutual information, i.e., T(X; Z) C(d). As d decreases, C(d) decreases, foreing Z to discard more information. Using the entropy decomposition in (15), a reduction in T(X; Z) (due to compression) necessarily increases the conditional entropy [XX] 2):
H(X - ();
-
-(d)-
This inequality quantifies the "burden shift":
Small d (High Compression): Z with a much smaller d retains only very coarse global semantics. The burden of modeling data complexity shifts to the conditional model P(X which must now model a highly uncertain, multi-modal distribution. Our use of conditional diffusion model is critical here, it prevents the "blurriness" or information loss typical of deterministic decoders by effectively sampling from this high-entropy distribution.
Large d (Low Compression): In this case. Z retains more high- information, reducing H (2) However, modeling P(Z) with larger d becomes more challenging as the latent space becomes nigh-dimensional and complex, potentially leading overfitting or difficulty in learning the global manifold.
The theoretical insight emphasizes crucial mechanism: as the small latent representation Z com- presses (leading to decrease in [(X; Z)), the onus of capturing data complexity shifts to the conditional term Po X Z). Our framework excels this regard, as the data-space diffusion model adeptly captures the heightened conditional entropy, thereby mitigating the information loss typically associated with deterministic decoders.
Moreover, even with rich latent space, cascaded structure provides distinct advantages over single-stage models, particularly for multi-modal time series generation. Directly modeling the marginal P(X) necessitates model that navigates diverse semantics and complex local variations, often resulting in mode mixing. By decoupling the generation process, our framework delegates global semantic modeling represented by [[X; to to the latent model, facilitating clearer and more learnable abstract structures. This duivide-and-conquee strategy ensures generated time series exhibit coherent high-level semantic structures and realistic local details.
C TRAINING AND SAMPLING ALGORITHMS
Algorithms and show the pseudocodes for the training and sampling processes, respectively.
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Algorithm Training of L2D-Diff.
Require: Training dataset T, noise schedules {
Ensure: Trained latent-space denoising network rg latent-space denoising network Xg. and condi- tioning network F.
while not converged do
Uniform( (.. k Uniform Uniform((122,.... (1,,2..... K});
Sample T:
Generate latent embedding Exx);
Generate noised latent r v vā, + vI- āge, where ~ I) and e R
Compute latent denoising loss Listen in Equation (11).
Obtain latent-to-data condition c =
Generate noised data x V vI cite where (O and RDxL; Compute data denoising loss Ldaaa in Equation (12);
Update á, 0 via Vo,e(Liateut X d.daaa (A is by default);
end while
Algorithm 2 Sampling of L2D-Diff.
Require: Trained denoising models ræ, and conditioning network F.
Ensure: Generated sample x Latent-space generation:
Sample 1 ~ (0, I).
for downto 1 do
Denoise latent: p end for
= =r 8) by Equation (13).
Data-space refinement: Sample x (N, I).
Compute the condition c F(ro). for T downto do
Denoise data: & k-11_ (x c) by Equation (14).
end for return 0
D DISCRIMINATIVE SCORE AND PREDICTIVE SCORE
From each category of the baselines, use the top one from Figure 2. Table 6 reports the evaluation results on discriminative and predictive scores. As can be seen, L2D-Diff consistently achieves the best overall performance, outperforming all the baselines. This demonstrates the effectiveness of the L2D sual-space framework in tackling the key challenge of TSG: capturing global structures in the latent space while preserving local fidelity in the data space.
Stock
Energy
ETTh
Riverfow:
The
ECG3000
Medical
Anahic
Atrial Eibrillution
Jananese
Churanter Tiajectories
Win/
Diftusion
TmmeDDM
0.893
0.495
0.589
0.000
0.000
0,000
0.000
0:475
0,000
0.326
0.323
18
FouserDiffasion
0.174
0.316
0.153
0.000
(.800)
0,000
0.000
0.451
0.000
0.205
01521
FourierFlw
0.221
0.304
0.361
0.000
0.000
0.800
0.000
0.481
0.000
0.16
0.553
TimeGAN
0.218
0.496
0.994
0.000
0.000
0,000
0.000
0,492
0,000
0.142
0.164
RoVAE
0.054
0.314
0.082
0.000
11.000
0.000
0.000
11.220
11,000
0.392
D1207
1.20-IMIT
0.041
0.251
0.654
0.129
0.754
0.556
0.623
0.333
0LE3)
0.351
IL333
Diffision-TS
0.48
0.369
0.905
0.049
0.755
5.589
0.801
0.338
0.530
0.365
0 8.568
IimeLDM
0.078
0.278
0.889
0.064
0.755
0.671
0.631
0.365
1,081
0.355
0.347
PS
FourierDiftasion
0.051
0.252
0.780
01500
1.755
0.551
0.626
0.338
[1.40
0.340
01355
FourterFTow
0.108
0.268
0.623
0.064
0.755
0.554
0.654
0.343
0.540
0.338
0.355
TimeGAN
0.025
Table Results on the time series datasets. Top: Discriminative Score (DS); Bottom: Predictive Score (PS). The lower the better. The last column counts the number of wins or ties for each method.
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For several datasets (Riverflow, Two Patterns, ECG5000, Medical Images. and Atrial Fibrillatio), all baselines achieve a DS value O. DS value of 0 indicates that the discriminator cannot reliably distinguish real from generated sequences, meaning the classification enror is close random guessing. This suggests that under the specific LSTM-bassd discriminator setup (Section 5.1) the generated samples statistically similar the real data. This phenomenon does not necessarily imply that these datasets are overly simple or that the generative performance is perfect. Since DS highly sensitive to the discriminator architecture, training protocol, and dataset size, zero value may also reflect limited discriminative power under the given evaluation configuration. That is why we treat DS secondary metric and interpret it alongside the other quantitative measures and qualitative visualizations.
E VISUALIZATION
Figure 4 shows the distribution plot results. The distribution plot illuminates the difference between the input and generated time series in terms of density, spread, and central tendency to show how the generated time series closely mirrors the original's ' statistics (Ang et al., 2023). The proposed I.2D- Diff demonstrates exceptional performance, consistently producing synthetic data with a distribution that closely mirrors the original, regardless of the complexity of the underlying data.
4
Densly
Original
Density
Original
Density
Original
Genersted
Generated
2
Generated
-2
-2
(a) L2D-Diff.
(b) FourierDifffssin.
(c) FourierFlow.
Original Genersted
Original Generated
Original Generated
Densky
Density
Density
2-
-2
(d) Diffusion-TS.
(e) TimeGAN.
(I) KoVAE
Figure 4: Distribution plot results on Character Trajectories.
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Published as a conference paper at ICLR 2026
Figure 5 and Figure 6 show the t-SNE results for the other datasets. As can be seen, the proposed L2D-Diff consistently delivers the best performance in producing data distributions that closely align with the true multimodal distribution.
Orgau
Drgind
-to
Orignd
-10
-32 -10
(a) Stock
Ordhrel
Orginll
-00
Orgin
-32
(b) Energy.
Ceveress
Cenelatd
Ger Garastic
Senerate
-22
(c) ETTh.
-32
-10
-40
-36
Geterted
(d) Riverflow.
Figure 5: 2D I-SNE embeddings of data (not representations) generated vs the real data. Column 1: L2D-Diff; Column 2: FourierDiffusion; Column 3: FourierFlow; Column 4: Diffusion-TS.
F IMPLEMENTATION DETAILS OF THE DENOISING NETWORK
F.1 DIFFUSION STEP'S EMBEDDINGS
For each diffusion step k, its d' dimmenssionll embedding pa is computed using two fully connected (FC) layers, as in (Rasul et al., 2021; Tashiro et al., 2021; Kong et al., 2020):
* SIL.U(CC(SI.U (Kenbe(eknnedd)))))
(16)
where SiLU is the sigmoid linear unit activation function (Elfwing et al., 2018), and
kianaadding (in 1), (10 ux4
cos(100=0))))
ox4 (), ...., cos w-1 D]
(17)
the sinusoidal position embedding (Vaswani al., 2017) with -. By default, d' is
128.
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Published as a conference paper at ICLR 2026
onghal
-20
-20
20
400
(a) TwtPamsss..
Genrred
(b) ECG5000.
Orgiu
Oopuu
Orgind
-10
-30
-00
20
-45
(c) Medical Images.
Ordinel
-30
-40
Orannll
-1
Conented
(d) Arabic Digits.
Daa
Ceneated
Orgnnl
-0.
-18
Atrial Fibrillation.
Ospel
-10
(1 Japanese Vowels.
Figure 6: 2D i-SNE embeddings of data generated vs the real data. Column 1: L2D-Diff; Column 2: FourierDiffusion; Column 3: FourierFlow: Column 4: Diffusion-TS.
19

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Published as a conference paper at ICLR 2026
F.2 DENOISING NETWORK WORKFLOW
Take the data denoising network X example. The input x² E Rx is first mapped to the embedding zº Rdd' L by input projection block consisting of several convolutional layers.
The embedding z", along with the d' d'-dimensional diffusion step embedding p (from Equation (16)), then passed to an encoder (a convolutional network) to produce the representation 2 E RRd XL Next, z" is concatenated with z (which has size of de I. after being upsampled to length by the conditioning network F, and de is the number of channels in z along the variable dimension, forming tensor of size (d. + d3)) x L. This concatenated tensor then passed decoder, also implemented as convolutional network, which outputs the denoised estimation xx(x", k, c).
In the latent-space denoising network rø, the corresponding representation z" directly fed into the decoder, which outputs the final denoised estimation r (r"r),s).
F.3 NETWORK IMPLEMENTATION
The conditioning network and denoising network's encoder/decoder are built by stacking a number of convolutional blocks. The default configuration of each convolutional block shown in Table 7.
layer
operator
default parameters
Convid
channel=256, out channe]=256, kernel size=3, stride=1, padding=]
2
BatchNormid
number of 2ca
3
LeakyReLU
negative slope=0.1
4
Dropout
dropout rate=0.1
Table 7: Configuration of the convolutional block.
G EFFECTS OF THE LATENT DIMENSION
As L2D-Diff bridges the diffusion process in the latent and data spaces, the dimension of the latent space plays a crucial role. We study its effects by varying the dimension sizes in (4, 8, 32, 64, 128). As shown in Table 8, smaller latent dimensions, such reasonable since smaller latent spaces compress data and extract the most informative representations, effectively capturing the essential structures of the time series. In contrast, higher dimensions, such as 64 or 128, tend to increase training complexity and may lead to overfitting, as they retain more redundant less informative details.
32, yield promising results. This is
Stock
Character Trajectories
C-FID
DS
PS
C-FID
DS
PS
0.334
0.052
0.048
0.308
0.169
0.363
0,310
0.048
0.041
0.284
0.171
0,333
32
0.339
0.078
0.047
0.304
0.165
0.342
64
1.121
0.095
0.049
2.614
0.175
0.372
128
0.366
0.071
0.059
2.947
0.243
0.382
Table 8: Varying the latent dimension d.
H ABLATION STUDIES ON CONDITIONING NETWORK F
In this section, present several ablation studies the conditioning network F: (i) the impact of varying the depth of F; (ii) exploring the use of MLP as an alternative for F; and (iii) examining different conditioning strategies.
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Published as a conference paper at ICLR 2026
H.1 VARYING THE DEPTH OF CONDITIONING NETWORK F
To investigate the impact of the depth of F. Table H.1 varies the depth to (1, 3, 5, 10, 20). The results indicate that the depth of F has relatively stable range; it should neither be shallow excessively deep. A single layer may result in inadequate learning of conditions, while an overly deep architecture can complicate the overall model structure and ultimately leads overfitting. Empirically, in our experiments, a depth of 5 layers shows promising performance with fewer parameters compared model with 10 layers.
Japanese Vowels
Character Trajectories
No. Layers
Stocks
Energy
0.59
0.71
1.39
0.58
3
0.34
0.62
0.58
0.42
5
0..31
0,53
0.46
0.28
10
0.30
0,53
0.51
0.24
20
0.75
0.87
0.85
0.61
Table 9: C-FID results of varying depth of F.
H.2 MLP OR CNN
Table H.2 explores the effect of replacing the CNN in F with an MLP. The results clearly show that the CNN consistently outperforms the MLP, indicating that convolutional structures are better suited for this task than MLPs this context.
Japanese Vowels
Character Trajectories
Stocks
Energy
CNN
0.31
0.53
0.46
0.28
MIP
0.48
0.61
0.57
0.43
Table 10: Types of F.
H.3 DIFFERENT CONDITIONING STRATEGIES
Finally, we analyze various conditioning strategies, including concatenation, cross-attention FiLM, and direct addition. Results are shown in Table 11. As can be seen, concatenation consistently outperforms the other three alternatives. While FiLM is also viable option, it exhibits slightly unstable performance and incurs higher inference costs. In contrast, cross-attention ineffective for bridging the latent space with the data space.
Japanese Vowels
Character Trajectories
C-FID
Stocks
Energy
Training
Inference
Concatenate
0.31
0.53
0.46
0.28
0.52
3.47
Cross-attention
0.39
0.68
0.49
0.61
0.71
4.13
FiLM
0.32
0.55
0.48
0.37
0.54
3.89
Addition
0.55
0.81
0.72
0.97
0.49
3.10
Table 11: Effects of Conditioning Modules. Training and inference times are measured milliseconds sample.
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Published as a conference paper at ICLR 2026
I TRAINIG & INFERENCE TIME
Table 12 shows the training time, inference time, and number of trainable parameters on the Energy. Stock, and Character Trajectories datasets. As be seen, L2D-Diff consistently ranks among the most efficient models, achieving the comparable results in terms of parameter count and computational cost.
Energy
Stock
Character Trajectories
training (ms/sample)
inference (ms/sample)
training (ms/sample)
inference (ms/sample)
training (ms/sample)
inference (ms/sample)
models
type
L.2D-Diff
data latent
0.32
1.28
0.44
1.95
0.52
3.47
Diffusion- TS
data
1.30
2.14
1.31
3.65
14.28
5.10
TSDE
data
0.37
6.55
0.83
8.63
2.10
5.05
TimeLDM
latent
0.28
2.11
0.35
3.12
0.51
4.85
EDDPM
latent
2.14
0.51
3.80
FourierDDiffusinn
frequency
0.19
6.10
0.28
8.11
0.36
9.66
ImagenTTime
fourier
0.67
1.11
0.84
1.64
1.22
2.82
Table 12: Training and inference times (in milliseconds per sample) of various methods.
22


Reviews:
[{"review_id": "BuMa9mZD3N", "reviewer": "Reviewer_6zx2", "summary": "This paper introduces L2D-Diff, a cascaded dual-branch diffusion framework for unconditional synthetic time series generation. The proposed model first learns high-level representation distributions by operating a diffusion process in a compressed latent space, then refines the generated time series in the data space by conditioning on these latent codes. The authors claim this approach offers a balance between preserving global semantics and capturing local fidelity, a challenge that existing methods struggle with individually. The method is evaluated on 11 diverse single- and multi-modal datasets, with ablations, efficiency analyses, and qualitative/quantitative results provided.", "questions": "While Table 2 and Figure 4/5 show improvement on complex datasets (e.g., Character Trajectories), the text primarily asserts these methods struggle on multimodal data, without actually drilling into when/why. For example, how does L2D-Diff perform on classes with small sample sizes or rare modes? Does it suffer from mode-collapse-like phenomena, or does the latent-to-data pipeline truly enable consistent coverage?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper’s strengths, I believe, lie in its dual-space cascaded architecture that combines latent- and data-space diffusion stages through a dedicated conditioning network, its strong empirical performance consistently surpassing diverse baselines across multiple datasets, and its comprehensive evaluation covering both standard and complex multimodal time-series benchmarks with supporting qualitative analyses. The work demonstrates mathematical clarity through detailed formalizations, loss functions, and architectural descriptions, while also achieving good parameter and computational efficiency. Moreover, the ablation studies carefully validate the importance of the dual-space design across varying data complexities.", "weaknesses": "- Although the cascaded dual-branch approach is well-motivated, key aspects of latent-to-data bridging have seen preliminary exploration in other domains (see Section 1 where RCG and EDDPM mentioned). The novelty for specifically time series scenario is asserted, but the adaptation from images/graphs to time series follows a relatively straightforward path (representation learning, conditional denoising). The authors could do more to clarify precisely which challenges for time series are not addressed by prior multi-stage diffusion frameworks, ideally with more rigorous empirical or theoretical separation.\n\n- Related to my first point, there is no formal theoretical analysis (e.g., convergence, identifiability, or information bottleneck trade-off) of why the cascaded approach works or how the latent encoding impacts downstream fidelity or diversity. For example, there is no proof or lemma quantifying what local/global information is lost or gained across the two denoising stages, nor any characterization of failure cases or limits of the method.\n\n- In section 3.2, the conditioning network $\\mathcal{F}$ is stated to be a CNN of \"5 layers by default\", but there is little justification or sensitivity analysis of its depth, efficacy, or alternative forms (e.g., attention, MLP, graph). Choices here can majorly affect expressivity, especially in bridging global latents to local series. Additionally, ablation on this module's architecture/hyper-parameters is also recommneded.\n\n- The latent dimension $d$ is chosen by default as 8 \"based on prior works\". However, there is minimal discussion of how this dimension influences the information bottleneck, potential tradeoffs in capacity (e.g., underfitting, overfitting), or the sensitivity of results to $d$, beyond a brief empirical study. A deeper analysis (possibly via information-theoretic measures or error bounds) would clarify how much key information is recoverable from latent diffusion before refinement, and under which conditions the cascaded architecture may fail.\n\n- I happen to know some highly relevant recent diffusion models (such as [1,2]) tailored for time series, which is missing in this work. I recommend author(s) to conduct a careful add-on survey to fill this gap. \n\n- [Minor suggestion]: some supplementary content (Appendices C, D) provides background or repeats points already made in the main text, which occasionally diffuses focus from key innovations/limitations.\n\n**Refs:**\n\n[1] Ge et al., T2S: High-resolution Time Series Generation with Text-to-Series Diffusion Models\n\n[2] Sikder et al., TransFusion: Generating long, high fidelity time series using diffusion models with transformers", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qHepy5pXgj", "reviewer": "Reviewer_GKHV", "summary": "This paper introduces L2D-Diff, a cascaded diffusion framework for unconditional multivariate time series generation. The approach combines latent-space diffusion with a data-space diffusion process. The latent representations generated by the first stage condition the data-space diffusion in the second stage via a learned conditioning network. Empirical results demonstrate SOTA or near-SOTA performance across standard quantitative metrics, with ablation and efficiency analyses supporting the value of the dual-branch architecture.", "questions": "Can the authors clarify the precise architecture and operation of the conditioning network $\\mathcal{F}$? Is it simply a fixed CNN, or does it use more advanced attention or cross-modal mechanisms, and why was this choice made?\n\nHow does the proposed framework handle very long or highly irregular time series, given that the encoder (TS2Vec) is fixed in dimension?\n\nCould the authors provide head-to-head results against the T2S model or other missing recent baselines, or explain why such comparisons are omitted?\n\nCan the authors clarify the process for baseline adaptation, especially for methods originally focusing on conditional or autoregressive tasks?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Unconditional high-fidelity time series generation, especially in multi-modal settings, is a non-trivial challenge with clear applications in data augmentation and privacy. The dual-branch latent-to-data design is visually and mathematically well documented, intuitively linking high-level semantic representation (latent) with local fidelity (data space).\n\nThe experiments demonstrate consistently superior (lower) FID on challenging benchmarks, with standout results on multi-modal datasets.", "weaknesses": "**The theoretical insight and novelty (theoretically) is limited**. The derivation and formulation of the cascaded diffusion process in Section 3 are correct but mostly extend existing latent/diffusion model frameworks without introducing substantial new theoretical principles. The dual-branch connection, while effective, does not offer a formal analysis (e.g., convergence or expressivity gains) that might help explain or generalize its empirical success.\n\nWhile the conditioning mechanism is described, the exact integration between latent embeddings and data-space denoising is vague: e.g., is $\\mathcal{F}$ simply concatenation, a form of cross-attention, or more? There is a lack of ablation or comparative analysis of alternative conditioning schemes, which could be crucial given the model’s reliance on this interface. \n\nThe paper omits explicit experimental or cited discussion of the recent and very relevant work, the T2S model. \n\nGe, Yunfeng, et al. \"T2S: High-resolution Time Series Generation with Text-to-Series Diffusion Models.\" Accepted by the 34th International Joint Conference on Artificial Intelligence (IJCAI 2025)\n\nThe paper’s main claim is a latent‑to‑data cascaded diffusion setup for unconditional time‑series generation: first sample a global latent with a latent diffusion model, then refine in data space conditioned on that sampled latent. The authors position this as the first such cascade for time series, and they contrast with prior “single‑space” models. I would call it **moderately novel**. Conceptually, it’s a domain‑appropriate consolidation of known ingredients, pretrained TS2Vec encoder, latent diffusion, and conditional data‑space denoising, with a simple conditioning interface. There isn’t claimed theoretical novelty, and the framework is close in spirit to representation‑conditioned generation in vision but re‑targeted to time series with an end‑to‑end dual‑branch training recipe.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2Vf0EKJpkC", "reviewer": "Reviewer_8LUh", "summary": "The paper proposes L2D-Diff, a cascaded diffusion framework for unconditional time-series generation. The method first learns a latent diffusion model to capture global structure and then conditions a data-space diffusion model to refine local temporal details. Experiments are performed on multiple benchmark datasets, showing improved Contextual-FID and distributional similarity metrics over existing baselines.", "questions": "Please see the weakness above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Timely and relevant topic.\n    - Time-series generation via diffusion is an emerging area, and exploring multi-stage or cascaded architectures is a meaningful direction.\n2. Conceptually intuitive design.\n   - The latent-to-data cascade nicely bridges global representation learning and fine-grained sample refinement — an idea that is simple yet effective.\n3. Comprehensive experiments.\n   - The paper evaluates on a wide range of datasets and metrics, which shows the authors’ effort to ensure empirical robustness.\n4. Clarity of motivation.\n   - The motivation for combining latent and data-space diffusion stages is clearly articulated and easy to follow.", "weaknesses": "1. Limited Theoretical Depth\n-  While the cascaded design is intuitive, the paper lacks a clear theoretical or empirical justification for why conditioning a data-space diffusion on latent representations leads to better generation quality.\n-  An analysis of information flow, representation alignment, or consistency between latent and data spaces could make the contribution more convincing.\n\n2. Insufficient Experimental Analysis\n- The experiments show promising improvements but lack in-depth ablation and diagnostic studies.\nSuggested additions include:\n  - Varying latent dimension and conditioning strength\n  - Comparing against single-stage models of similar parameter counts\n  - Evaluating temporal coherence and downstream task performance (e.g., classification or forecasting accuracy using generated samples)\n\n3. Incomplete Architectural Details\n  -  The conditioning mechanism between the latent and data diffusion stages is not well specified.\n  -  Clarify how latent vectors are integrated (e.g., concatenation, cross-attention, feature modulation) and include a schematic diagram or pseudo-code for reproducibility.\n\n4. Broader Context and Positioning\n  -  The paper could more clearly state what aspects are unique to time-series generation compared to existing cascaded diffusion frameworks in other modalities (e.g., images or audio).\n  -  Discussing challenges like temporal consistency, irregular sampling, or multi-channel correlations would better highlight the domain-specific contribution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kbMoQyiryQ", "reviewer": "Reviewer_AFRd", "summary": "This paper proposes L2D-diff, a hybrid latent and data space diffusion model that aims to capture both high-level and local fidelity. The generation process first uses latent diffusion, then passes onto data space diffusion. Through empirical experiment, L2D-Diff is on par with previous SOTA models.", "questions": "Additional questions other than weaknesses.\n\n1. In Table 7, the result for the previous SOTA methods, Diffusion-TS, seems to be very different from the original paper. \n2. Newly proposed datasets, 5 out of 8 show 0 for DS score. Does that mean those datasets are too easy?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper investigates the hybrid diffusion models for time series generation, which is a new framework.\n2. This paper conducted extensive experiments with newly added datasets.\n3. This paper is well written and easy to follow.", "weaknesses": "1. The L2D-Diff consists of multiple modules. A comprehensive ablation study is needed to justify the hybrid framework.\n2. While there are discussions of the training & inference, the discussion is limited only to a new dataset. The efficiency claim could be more compelling if the comparison can also be conducted over the sine, stock, and energy, which are more commonly compared.\n3. The t-SNE visualization is only performed over one dataset. It would be ideal to include more datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)