TimeAutoDiff: A Unified Framework for Generation, Imputation, Forecasting, and Time-Varying Metadata Conditioning of Heterogeneous Time Series Tabular Data
Authors: Namjoon Suh
Yuning Yang
Din-Yin Hsieh
Qitong Luan
Shirong Xu
Shixiang Zhu
Guang Cheng
Venue: Accepted by TMLR ()

Abstract: We present \texttt{TimeAutoDiff}, a unified latent-diffusion framework that addresses four fundamental time-series tasks—unconditional generation, missing-data imputation, forecasting, and time-varying-metadata conditional generation—within a single model that natively handles heterogeneous features (continuous, binary, and categorical). We unify these tasks through a simple masked-modeling strategy: a binary mask specifies which time feature cells are observed and which must be generated. To make this work on mixed data types, we pair a lightweight variational autoencoder (i.e., VAE)—which maps continuous, categorical, and binary variables into a continuous latent sequence—with a diffusion model that learns dynamics in that latent space, avoiding separate likelihoods for each data type while still capturing temporal and cross-feature structure.Two design choices give \texttt{TimeAutoDiff} clear speed and scalability advantages. First, the diffusion process samples a single latent trajectory for the full time horizon rather than denoising one timestep at a time; this whole-sequence sampling drastically reduces reverse-diffusion calls and yields an order-of-magnitude throughput gain. Second, the VAE compresses along the feature axis, so very wide tables are modeled in a lower-dimensional latent space, further reducing computational load. Empirical evaluation demonstrates that \texttt{TimeAutoDiff} matches or surpasses strong baselines in synthetic sequence fidelity (discriminative, temporal-correlation, and predictive metrics) and consistently lowers MAE/MSE for imputation and forecasting tasks. Time-varying metadata conditioning unlocks real-world scenario exploration: by editing metadata sequences, practitioners can generate coherent families of counterfactual trajectories that track intended directional changes, preserve cross-feature dependencies, and remain conditionally calibrated—making "what-if" analysis practical. Our ablation studies confirm that performance is impacted by key architectural choices, such as the VAE's continuous feature encoding and specific components of the DDPM denoiser. Furthermore, a distance-to-closest-record (DCR) audit demonstrates that the model achieves generalization with limited memorization given enough dataset. Code implementations of \texttt{TimeAutoDiff} are provided in https://github.com/namjoonsuh/TimeAutoDiff.

Published in Transactions on Machine Learning Research (12/2025)
TimeAutoDiff: A Unified Framework for Generation, Imputa- tion, Forecasting, and Time-Varying Metadata Conditioning of Heterogeneous Time Series Tabular Data
Namjoon ;uh'; Yuning Yang?, Din- Yin Hsieh?, Qitong Luan?, Shirong Xu Shixiang Zhu Guang Cheng²
Microsoft', UCLA", Xiamen Univ.' , CMU
Reviewed on OpenReview: https: openrevveew...w formm? id= bkUd1Dg46c
Abstract
We present TimeAutoDiff, a unified latent-diffusion framework that addresses four funda- mental time-series tasks locks-unconiitonaal generation, missing-dala imputation, forecasting, and sane-vaarinn-eetadaeee conditional generation titratiionwwithin within a single model that natively handles heterogencous features (continuous, binary, and categorical). We unify these tasks through a simple masked-modeling strategy: a binary mask specifies which time feature cells are observed and which must. be generated. To make this work on mixed data types, we pair a lightweight variational autoencoder (i.e., VAB) which maps continuous, categorical, and binary variables into continuous latent sequence-wit withh diffusion model that learns dynamics in that latent space, avoiding separate likelihoods for each data type while still capturing temporal and cross-fcature structure. Two design choices give TimeAutoDiff clear speed and scalability advantages. First, the diffusion process samples single latent trajectory for the full time horizon rather than denoising one timestep at a time; this whole-sequence sampling drastically reduces reverso-diffusion calls and yields an order-of- magnitude throughput gain. Second, the VAE compresses along the feature axis, so very wide tables are modeled in a lower-dimensional latent space, further reducing computa- tional load. Empirical evaluation demonstrates that TimeAutoDiff matches or surpasses strong basclines in synthetic sequence fidelity (discriminative, temporal-corrclation, and predictive metrics) and consistently lowers MAE/MSE for imputation and lorecasting tasks. Time-varying metadata conditioning unlocks real-world scenario exploration: by editing metadata sequences, practitioners can generate coherent families of counterfactual trajectories that track intended directional changes, preserve cross- cross-feature dependencies, and remain conditionally
"what-if" analysis practical. Our ablation studies confirm that performance impacted by key architectural choices, such the VAE's continuous feature encoding and specific components of the DDPM denoiser. Furthermore, distance- recllssss--eccord (DCR) audit demonstrates that the model achieves generalization with limited memorization given enough dataset. Code implementations of TimeAutoDiff are /iiiiuu..co/nanaoooosiill naanaoasmhn/h/////aaa TimeuutoDiiiieeiiiii...i provided in https:// https://github.com
1 Introduction
Time series with heterogeneous features ubiquitous in both engineering and scientific domains, such financial markets combining continuous stock prices with categorical market sentiment indicators (Zou
et al. 2022). and healthcare systems (Theodorou et al. 2023) integrating continuous vital signs with discrete treatment variables.
Work done while at UCLA.

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Published in Transactions on Machine Learning Research (12/2025)
Xyan
Xaie
Latent Distribution
Prior Distributtion
Forward Pracess
X Pre
Encoder
Zy
Zig
Reuu
Raisc
Rackward Process
Demoising Architecture
8 Rec
2 Pust
Decoder
Zlat
2
Time Stamps
Dota Space (VAE)
Latent Space (DDPM)
The overview of TimeAutoDiff (Unconditional Generation): the model has three components: (1) pre- and training encoder and decoder, and for projecting the pre-processed data to the latent space; (3) Diffusion model for learning the distribution of projected data latent space and generating new latent data. Notably, the feature dimension can be compressed in the latent space such that LS F.
Figure 1:
steps for the original (i.e., X and synthesized data (i.e., X XPost); (2) VAE for
The recent emergence of diffusion models (DMs) (Ho et al. 2020 Song et al. 2020b has garnered significant attention across scientific disciplines, achieving remarkable success in diverse domains, including image synthosis (Rombach et al.] 2022). tabular data generation (Zhang et al.) 2023bb, molecular modeling (Xul ct al.) 2023), and audio synthesis (Zhang al. 2023a). The diffusion model consists of two processes. In the forward process, data are progressively perturbed until they resemble pure Gaussian noise, while neural network is trained to estimate the noise added at each diffusion step. In the reverse process, the trained denoising network generates new samples by itcratively removing noise, starting from random Gaussian noises. Time series community has recently adopted DMs for downst ream tasks (Yang et al. 2024b) for their abilities ies of generating high quality, compler sequences. Nonetheless, existing models primarily focus on continuous time series data, which is suboptimal for tasks involving heterogeneous features. (The modeling efforts for time series with heterogeneous features are surprisingly sparse.) Additionally, most models lack versatility, typically being capable of performing only single task.
In this paper, propose a novel approach designed to overcome the aforementioned constraints by utilizing smooth continuous latent space, named TimeAutoDiff. Our framework is structured as a Variational Autoencoder (VAE) (Kingma & Welling 2013), with DMs functioning on the latent space. The model is designed for 'multi-task" functionalitics, capable of performing four distinct tasks on time series with heterogeneous feat ures: (1) unconditional time series generation (Yoon et al. 2019 Naiman al.] 2023) (2) missing data imputation (Tashiro al. 2021 Wang et al., (023b). (3) forecasting (Nie al. 2022 Naiman et al.) (2024), and (4) time- varying metadata conditional generation (hercafter referred to TV-MCG) (Narasimhan et al.) 2024). Technically, these four tasks can be framed within the context of conditional distribution modeling, where the goal is to generate target data given observed data, inspired by the idea used in masked language modeling @Dvvlnn 2018). To the best of our knowledge, we are the first work to incorporate heterogencous features into latent space modeling for multi-task time series generations.
A unique advantage of TimeAutoDiff that unlike previous DM methods operating in the feature do- main (Tan et al. 2023), we explicitly incorporate latent space capture the complex structures of time series with heterogencous features. This framework exhibits several strengths. First, projecting the raw data to continuous latent space avoids direct modeling of complicated likelihood of heterogeneous features in the original data space (VAE's rolc), and leverages the power of DMs for modeling distributions in the continuous space, and is therefore more expressive. Second, working on the latent space allows the model to capture complicated dependent structures along feature and temporal dimensions of data naturally without sophisticated modeling of heterogeneous issues. Third, the latent space allows for training and sampling in a lower dimensionality, reducing generative modeling complexity by compressing data along feature dimensions and facilitating efficient modeling of high-dimensional features. Lastly, the latent space
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Models
Hetero. Single-Seq.
Multi- Seq.
Cond. Gen.
Domain Agnostic
Code
Sampling Time
TimeAutoDiff
3
TimeDiff Tian et ml.
2023
X
X
Diffusion-Le Yuan Qian 2023
X
V
TSGM [Lim
X
/
TimeGAN Yoon et 2019
X
/
2
DoppelGANger Lin
2020
X
/
1
EHR-M-CAN [i
2023
X
CPAR (Zhang et al. 2022
X
J
TabGPT [Padhi al. 2021b
X
X
Table 1: A comparison table that summarizes TimeAutoDiff against bascline methods, evaluating metrics like heterogeneity, single- and multi-sequence data generation, conditional generation, domain-agnostic (i.e., whether the model is not designed for specific domains), code availability, and sampling time. Baseline models without domain specificity and with available code are used for numerical comparisons. The sampling time column ranks models by their speed, with lower numbers indicating faster sampling. (Only models with generative capabilities for time series data included in the comparison.)
approach simplifies modeling by transforming P(heterogeneous helerogeneous) into the more tractable P(contimuous heterogeneous). This enables efficient handling of diverse heterogeneous features condi- tions, crucial for tasks like imputation, forecasting, and TV-MGG. Our framework, inspired by successes in text-guided image generation (Rombach et al. 2022), offers a unified solution for complex, heterogeneous data that would be challenging to model otherwise.
2 Relevant Literatures
In this section, focus topics most pertinent study: heterogeneous tabular modeling, time series data generation, and downstream time series tasks such as forecasting, imputation, and conditional generation.
Heterogeneous Non-time series-tabular Modeling: For mixed-type tables, GAN methods such as CTGAN (Xu et al. 2019) and its extensions CTABGAN (Zhao et al. 2021 and CTABGAN+ (Zhao et al. 2022
popularized adversarial synthesis for categorieal-contiioor data (Goodfellow al. 2020). Diffusion- based
tabular synthesizers soon showed advantages, with Stasy (Kim al.; 2022] outperforming GANs on several tasks, although classical diffusion processes (Ho et al. 2020 Song et al. 2020b) are not inherently tailored to heterogeneity. Subsequent work addressed this via tailored stochastic processes and objectives e.g., Doob's h-transform for categorical variables (Liu et al. 2022), TabDDPM (Kotelnikvv et al. 2022) and CoDi (Lee et al., Miscrete/continoone 2023), which combine discrete continuous diffusion (Song et al. 2020b Hoogeboom et al. 2022) and employ
contrastive co-evolution (Schroff et al. 2015). Most recently, latent diffusion has emerged as a simple and effective unilying strategy: AutoDiff (Suh et al. (2023) and TabSyn (Zhang et al. 2023b) first compress heterogeneous tables with an autoencoder and then learn diffusion model in the continuous latent space, demonstrating strong fidclity and utility across diverse tabular benchmarks. However, those models are not explicitly designed to capture the temporal inductive biases inherent in time series tabular data.
Unconditional Time Series Data Generation: Early work adopts GANs: TimeGAN (Yoon et al, 2019) learns a latent sequence representation with an autoencoder adversarial scheme, followed by healtheare-oriented variants such as EHR-Safe Safe (Yoon et al. 2023) which integrates an decoderr with GAN, and EHR-M-GAN (Li et al. 2023) which employs type-specifie encoders; however, GANs remain sensitive to non-convergence and mode collapse [Goodfellow al. 2020). Diffusion models then gain traction: TimeDiff (Tan et al. 2023) couples multinomial diffusion for discrete variables with Gaussian diffusion for continuous ones to synthesize mixed-type EHR sequences; TSGM (Lim et al. 2023) uses latent conditional
scsrd-based diffusion for continuous scries; and Diffusion-Ts (Yuan & Qiao 2023) combines a transformer
autoencoder with latent diffusion to capture temporal dynamics. Beyond these, GPT--tyle and parametric approaches include TabGPT (Padhi et al.] 2021b), a GPT-2 based synthesizer for mixed-type time-series tables, and CPAR (Zhang al., 2022), which assigns different parametric models heterogencous variables in multi-sequence tabular settings. Despite progress, accurately modeling cross-feature correlations while handling complex temporal dependencies across data types remains challenging.
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Remark 2.1. For more in-depth understanding of our model's positioning, we provide detailed comparisons between TimeAutoDiff and related methods-incchee TimeDiff, Diffusion-TS, TabSyn, AutoDiff, and TabDDPM in Appendix D
Imputation & Forecasting for Time Series Data: Several recent methods addressed time series tasks like filling missing data and predicting future values. For filling in missing data, CSDI (Tashiro et al., 2021 studied the imputations of continuous time series tabular data through a diffusion- based framework. The main idea was to employ specially designed masks; masking the observed data, and letting the model predict the masked values in the observations, i.e. self-supervised learning. Then, the trained model could impute the real missing parts of the table by treating them as masked observations. Building on this diffusion paradigm, MG-TSD {Fan et al. (024)) was introduced to stabilize the stochastic nature of diffusion models by using the inherent multi- gramularty levels within the data as intermediate targets to guide the learning process. TimeDiT (Cao et al. 2024) further extends this by combining a diffusion transformer with a comprehensive masking strategy for imperfect data and a novel mechanism for injecting physics knowledge during inference. FiLM (Zhou et. al., 2 used a memory system and noise-flttring techniques to capture long-term patterns efficiently. Meanwhile, DLinear (Zeng et al., 2023) broke time series into seasonal and trend parts, using simple linear models to match or outperform complex Transformer models, offering a fast and lightweight option. TimesNet (Wu et al. 2022) organized time series into a 2D format to highlight repeating patterns, using advanced convolution techniques to capture both short-term and long-term trends, which helped with both predicting future values and filling in missing data. MICN (Wang et al. 2023a) blended local and global patterns in time series data using a mix of standard and specialized convolution methods across different scales, boosting long-term prediction accuracy while keeping computations fast. Time-LLM (Jin) ct al. 2023) adapted pre-trained language model using text-based guides and a special input format to predict time series values, achieving strong results with little or extra training. MOMENT (Goswami et al.) 2024) trained a large Transformer model on wide range of time series data, teaching it to fill in missing parts, creating a versatile model that worked for predicting future values, filling gaps, classifying patterns, and detecting anomalies with minimal extra training. Following this trend toward large-scale foundation models, Time-MoE (Xiaoming et al., 2025) introduced a sparse Mixture-of-Fxpert (MoF) architecture, scaling a decoder-only model to billion-scale parameters on the massive "Time-300B" dataset enhance forecasting precision while maintaining computational efficiency. TEFN (Zhan et al. 2024 used method to mcasure both fealure-specific and lime-based patterns in data, combining them to make accurate and reliable long-term predictions. Other Transformer-basee innovations include PatchTST (Nie et al., 2022), which segments time series into 'patches' and applies attention across these segments rather than individual points, and iTransformer (Liu et al., 2023). which inverts this by applying attention across entire variates (channels) as tokens to explicitly model inter-channel dependencies. Finally, TimeMixer (Wang et al., 2024b) blended patterns across different time scales using a simple, fast model (MLP-only architecture), combining specialized predictors to achieve top results for both short-term and long-term forecasting. This concept was extended in TimeMixer+ (Wang et al.) 2024a), general Time Series Pattern Machine (TSPM) that transforms multi-scale time scries into multi-resolution "time images' and uses dual-axis attention to decompose scasonal and trend patterns, achieving state-of-the-art results across eight distinct time series tasks.
The development of models like MOMENT within this literature highlights significant trend toward large-scale, pre-trained Time-Series Foundation Models (TSFMs), which aim to generalize across diverse tasks and domains. However, many of these TSFMs, including the aforementioned examples, are often limited to contimuuus-only data (assuming channel independence) or specialized tasks. TimeAutoDiff complements this emerging field by introducing a unified latent-diffusion framework that (1) natively handles heterogencous features (continuous, categorical, and binary), and (2) unifics four key tasks-gone generraion,,, imputation, forecasting, and conditional generation using a single conditional masking strategy. We provide more in-depth comparisons in this regard in the Appendix D
Remark 2.2.
The only work on TV-CGG we are aware of is Timeweaver (Narasimhan el al. 2024), which formulates conditional generation as diffusion conditioned on heterogeneous, time-varying metadata ' categorical and continuous metadata are tokenized, fused via self-attention into a time-aligned conditioning sequence, and injected into a CSDI-style denoiser to synthesis. Unlike Timeweaver, which restricts Xtar to
Time- Varcing Metadata Conditional Generation:
learn t( X enabling continuous variables, TimeAutoDiff if supports heterogeneous X (contimuous, binary, categorical).
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Target mask
Conditioning mask
Feature Axis
Temporal Axis
(A) M
(B) M D(Xe, missing.)
(C) MC, 1(t > w)
(D) M I( F)
Figure 2: Illustration of four binary-mask M (shaded cells = 1) on a T F timeseries grid: (A) Unconditional Generation (M"): all T x entries shaded. (B) Missing-Dala Imputation (M
1(Xr, is missing)): shaded cells mark missing entries, and unshaded cells indicate observed values. Forccasting (M,,- 1tt w)): only rows with > w shaded, the first w rows serve as conditioning. (D)
M, = 1(f F)): only columns corresponding to features in F are shaded.
3 Problem Setting
Let X = [x X xr| RxF be T-sequence time series tabular data with heterogencous features. Each observation X, [XDisc.]: XCont,j] is an F dimensional feature vector that includes both discrete (XDac.j) and continuous (XCont.j) variables. Throughout this paper, assume that there are B i.i.d observed sequences (i.e., (=) sampled from P(X). Every record is timestamped in the format. Notably, these timestamps (i.e, ts, E RTx4 xxx4,ie f1...., may differ across sequences within the same batch (i.e, ts; F tsj, Vi & j) and are treated as auxiliary variables that can be leveraged during training and inference. (This will be more detailed Subsection 4.4
Our goal is to model the conditional distribution Po (taa X M), where M M is binary mask that specifies which entries belong to target output (i.e., X In other words, the target data X == M @ X and the conditioning data X = (I M) @ X with I being matrix all entries are 1s. Here, O denotes element-wise matrix multiplication. A family of the binary masks M E {0, 1]TxF" provides a unified framework that supports four different time series tasks framed under conditional distributional modeling:
Unconditional Generation: Sotting M I (all entries are 1) makes X (all entries are 0) inducing unconditional generation.
2. Missing Data Imputation: The mask identifies missing entries in the data M
[Xx., is missing.), Vt E {l...., TT. F).
3. Forecasting: Given a lookback window of size w E {1,2.... 1}. the mask is defined as M = 1( (L> w) for all f ( (1....,FF), where 1l > w) is the indicator function with values 1 for t > w, and otherwise.
Metadata Conditional Generation: For fixed set of fcatures F= if (l.e....... F}} and all time steps E (l...., T}, the mask is set to M., 1(f F).
This formulation imposes parametric assumptions w, feature indices f, or missingness patterns. Experimental setlings for each case are detailed in Section 5
4
Method
In this section, we formally introduce our model, TimeAutoDiff, which extends recent advances in latent diffusion for image (Rombach et. al. 2022) and labular data generation (Zhang et. al. 2023b Suh et al., 2023). Unlike these domains, heterogeneous time series require tailored architectural choices: a customized encoder-decoder the VAE (Section 4.3) and domain-aware denoising module the diffusion model (Section 4.4) effectively capture inductive biases specific temporal data. We begin with the overall training objective in Section 4.1 followed by pre- and post-processing strategies in Section 4.2 Sections 4.3
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and 1.4 detail the VAE and DDPM components, respectively. The training and sampling procedures are summarized in Section 15
4.1 Objective function
Let = (ó. th 8) denote the parameters involved in TimeAutoDiff where each parameter character- izes the encoder (i.c., go), decoder (i.c., Pp), and diffusion prior (i.c., Po) distributions, respectively.
The model (iee, Pe) estimates the conditional distribution P(Xtar far X Mlask) with given samples (x (f)..((), M
P(X X Mtask) by minimizing the ELBO loss (i.e., LELBO LVAE + LDM) of the following negative log- oggliielihood::
log Pe (x
X M
S Eyyl log Pe
(xtar
x aa Mlask) }
+ DKL (1o (Zbat X x
|| N IrFXTF))
:=<VAE
Egg 90
- log Po (Z
X
+ Constant.
<CDM
The (KL
(PI) || Q) is a KI-divergence between two probability measures P and Q, and ITFXTR is an identity matrix of dimension TF x TF. A simple proof of the above ELBO loss provided in Appendix A
As reflected in the reconstruction term of LVAE, the VAE component takes both X and X as input to the encoder (i.e., 20), which outputs the latent representation Zt The decoder then reconstructs Xtar conditioned on Zl X and the task-specific mask Mtask, This design enables the latent representation to capture richer context from the full imput = X + X and empirically leads to more stable training compared using X alone as input the encoder. The diffusson prior, Pa. models the conditional distribution P(Z]at X The model parameter can be estimated through minimizing the ELBO loss of - log Po (Z]aat (xro),, denoted as LDM- The further descriptions on LVAE and Lpm will be specified in the following sections.
4.2 Pre- and post-processing steps in TimeAutoDiff
It is essential to pre-process the real tabular data in a form that the machine learning model can extract the desired information from the data properly. We divide the heterogencous features into two categories; (1) innui, and (2) discrete. Following is how we categorize the variables and process each feature type. Let x be the column of a table to be processed.
1. Continuous Features: A column is treated as continuous (numerical) feature if its entries are real-valued, or if they are integers with more than 25 unique values (e.g.. "Age"). The threshold of 25 a user-defined hyperparameter to distinguish high-cardinality integer foatures from discrete ones. All continuous features are normalized to the range [0, 1] using min-max scaling (Yoon et al. 2019). The processed output is denoted by XNun
2. Discrete / Categorical Features: A column x is categorized as discrete if its entries are of string type (e.g., "Gender"), or if it consists of integers with fewer than 25 distinct values. During preprocessing, each unique value x is mapped to a non-negative integer index. We further divide these into binary and categorical types: x for variables with exactly two categories, and X for those with more than three categories.
Once the TimeAutoDiff model generates synthetic data, we apply inverse trans- formations recover the original format. For continuous features, this involves reversing the min-max scaling. For discrete features, the integer-encoded values are mapped back their original categorical string representations.
post-processing:

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Learn context of features
M2
My
MLP Block
PC
FC
PC
Learn temporal dependence
RNN
RNN
RNN
Concatenate
f
f2
fr
log of
log
Jog of
f1
e2
F3
Cm
71g
T22
12g
FC
FC
FC
RNN
RNN
RNN
Xnise
Xcomt
f2
Figure 3: Schematic architccture of the encoder in the variational autoencoder (VAE). The encoder takes pre-processed multivariiae time series input X [ XProc]] E IRT' composed of m discrete and continuous features, where F= m Discrete features embedded via a lookup table e(.) RR while continuous features are transformed using frequency-based representations equation information. At each time step t, the embeddings are concatenated into E(t) E (mm+ojd and passed through an MLP to yield feature embeddings f, E FR The full sequence {fe) is then processed independently by two RNNs to model temporal dependencies: one RNN estimates the mean vector μ E TX'', and the other the log- log-variance log o² E R of the approximate posterior. The latent trajectory Zi E RRxx is obtained by sampling via the reparameterization trick: Ztat = fe + E C Σ, where Σ exp(0.5 log o²) and E g~a N((O,) This encoder compresses the feature dimension from F to L while preserving temporal resolution.
D to capture spectral
4.3 Variational Autoencoder in TimeAutoDiff
In this section, encoder, decoder, objective function and training of VAE will be introduced.
Encoder in VAE: We begin with the pre-processed input data
xta X con := [xDree x
Pror, Proc TXF
Proce E RRTxx contains discrete (binary and categorical) features, and Cont E R'xx contains continuous where Disc features, with F = m m C.
For discrete features X, (j E fl....,m.) we map their values into a d-dimensional embedding space e();(()) Rd using lookup table e(.) I&, where = 128. This embedding approach, motivated by TabTransformer (Huang et al. 2020), allows the model to easily differentiate discrete classes across columns. For continuous feature value Cont (t, i), we employ a frequency-basee representation (in short 'FR' for later use in Figure 5)
n: (v(v)= = Linear (LinenLLaerr((siinn c0s(22TT)), sin(2"mu)) cos(2? cos((2==)))) E R.
(1)
which captures high-frequency variations in continuous signals. This leverages the spectral bias of deep networks Lowards low-frequency functions [Rahaman el. al. 2019 ensuring better reconstruction fidelity for continuous features.
At each time step t, we concatenate the discrete and continuous embeddings:
Et) (=(((z(()))) :((.. na Px (t, 1)); nh Pr
Proc (t, c))l E R (m+eodd
Applying an MLP block at each timestep and stacking the outputs gives
Emb(XXrocy = [MLP ((()) M...EE(TT
(2)
[ fe
f E TXP,
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h
MLP
+
FC
Mhin
ttar
Bin
h22
hz
FC;
(i) cat
X xtari Cat
FC
ht
hy
FC - Sigmoid
M num
tar X
X
Emb
Num
batt
H dec
Figure 4: VAE decoder with conditioning and modality-specifie masking. A latent trajectory Zoaz is mapped by MLP to hidden sequence Hice Conditioning information Xoon is embedded (Emb) and projected with Lincar layer, then added to the hidden state to form H = Heec + Linear (Emb cx) (on))
injecting context al every time step. From Hasee three modality-specifie heads produce largets: Binary Linear(H (Haee) + Main + sar (logits with an additive mask that biases masked positions); Categorical
for each categorical variable i, (Hnear(i[(Haee) + M( eat CCat (class logits with an additive mask that routes masked entries to designated class, e.g. index 0); Numerical o (Linear(HHae)) C Maum Xum (values
tar,i
with a multiplicative elementwise gate). Type-specific masks (Main M M are derived from the task mask t by splitting and reshaping according to feature types, so that binary/ eeaayccategoicaa channels use additive logit biasing while numerical channels use clementwisc gating.
The output from the MLP block, fe. fr] RR fed into two separate RNNs, each unfolded over T steps, model the mean and covariance of the latent distribution. Specifically, these RNNs process {f, to capture temporal dependencies. The final hidden states then passed through fully connected (FC) layers (see Figure 3) to produce:
p [M1, , .a.....-pp PT] E RXL
log 0² = [log o log of TRL
Sampling from these parameters via the reparameterization trick yields:
z Lat : EERTx,
where E = exp( (oggo))go²) and E E T whose entries are drawn from W00, 1). Note that the compression into this lover-dimensional latent space occurs along the feature dimension, reducing it from F to L(S F).
Decoder in VAE: The decoder reconstructs the binary, categorical, and numerical components the target data taar from latent representations conditioned with X Given a latent feature tensor Zt E R through MLP block:
H Hate MTP((Z(a)) TXA
where h hiddennn__.. --ideensizee size.
The decoder incorporate conditioning information c E T This conditioning data embedded through the operation equation 2 (i.e., Emb(Xx"")) and projected into the hidden space. The resulting conditional embedding is added to Hiiess enabling context-dependent reconstruction:
H
Hage Linear(Emb(X
Additionally, handle masked values and produce modality-speeiffe reconstructions, the decoder uses the provided mask t 1 to create separate masks for binary (Man), categorical (1) and numerical (Mnum) features. These newly formed masks are adjusted such that masked inputs are effectively forced to for binary, categorical and numerical features, ensuring coherent. reconstructions in the presence of unobserved entries.
M E Txmmms Mhin R TX (1) Mathies
M
(1) cut E R Txaa:
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Here, Ahins neats: and Roums are the counts of binary, categorical, and numerical features, respectively, and cat; is the cardinality of i-th categorical feature. These mopality-specifie masks are constructed as follows:
Numerical Mask: Muum is identical in shape and values to the corresponding numerical portion of M. Thus, masked values are directly mapped to zeros, making no predictions are produced for masked numerical entries.
Main retains the same shape as the binary portion of M, but any zero-valued entries are replaced with a large negative constant. This effectively suppresses any positive predictions and forces the model to output zeros for masked binary features.
Binary Mask:
Categorical Mask: For each categorical feature i, M is expanded along the category dimension, and any vero-vllued entries of the original mask are assigned large positive constant at the first category index. This ensures that masked categorical inputs are consistently mapped to designated "masked" category (assigned index 0), concentrating the predicted probability mass on that category.
Equipped with the well-designed masks Mains {M Bin [xCat Meats star Numl:
(6) Cat h=r Mnums the final outputs of decoder is Xtar
tar Bin
Linear Hete) + Mbin E RR Txnine.
tar,i
Linear[i] + M cat E Txxat
Cat.
tar Num
o (Linear (Hcce)) @ Mnum RR Txmun
where o() denotes a sigmoid function.
The reconstruction error frecous( (ttar Xtar) in the VAE is defined as the sum of mean-squared error (MSE), binary cross entropy (BCE), and cross-entropy (CE) between the target tuple Xtar, := [xbin C Xum and the output tuple from decoder Xtar == [ X x
Obj. function & Training of VAE:
Out Out Out
BCE(XBI Qut) + CE Qut) + MSE(x far XNum).
We use V-V Higgins el al. 2017) where a coefficient, 3(2 0) balances between the reconstruction error and KL-divergence of (0, and Z! ~ N (vec(()), diag diiag(ve((2)). The notations vec(-) and diag(.) are vectorization of input matrix and diagonalization of input vector, respectively. Finally, we minimize the following objective function LVAE for training:
LVAR =- Lroeons (xtar, tar Star) + @Drr. (N(ve((j) diag(vec(o?))) || NNO,
(3)
Our model does not require the distribution of embeddings Zfat to strictly follow a standard normal dist.riuttion, as the diffusion model additionally handles the distributional modeling in the latent space. Therefore, following Zhang al. (2023b), we adopt the adaptive schedules of B with its maximum value set 0.1 and minimum as 10-5, decreasing the 8 by factor of 0.7 (i.c., gnnw 0.7gold from maximum to minimum whenever frecons fails to decrease for predefined number of epochs. The effects of 8-scheduling will be more detailed in Section 5
4.4 Diffusion Model in TimeAutoDiff
In this section, our customized DDPM and newly designed denoiser in TimeAutoDiff are introduced.
Let La E RTx be the initial latent matrix derived from the VAE, and let ZLat == [zhat Lat 2........ 2 represent the latent matrix corrupted by noises after (1,2,...,N) diffusion steps. Here, zlat E RR. is the j-th column of ZLat The noising process applied column-wise, independently for each j E [T., according to:
q( ( bat) = N (Vanzlog Lat (1 (1-āx) IIxT),
where a. m a and {a [0, 11ah is decreasing sequence. We employ linear noise schedule HHo
al. 2020 Further details DDPM provided in Appendix
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(g(t2)
Z
MLP
ANNIFC
LN/FC
LN/FC
LN/FC
MLP
PE
Concatenate
RNN
RNN
RNN
MLP
Concatenste
RNN
RNN
RNN
WE
FR
CE
MLP
N2
N2
Gatec
Farst
Figure The schematic architecture of the denoising model co (o(Z) (Zt n. t, ts) in the diffusion framework. The inputs to the model en include the noisy latent matrix ZLat at the wih diffusion step, the diffusion step index n, the normalized time points t, and the periodic timestamp embeddings ts, projected through multilayer perceptron (MLP). When conditional data X [Caise: Ceont] is available, it is embedded using a word embedding (WE) for discrete variables and frequency-basee representations (FR) for contimuous variables, producing 00n. == Emb(X
These embeddings are processed through a Bi-directional RNN (Bi-RNN) to capture temporal correlations, and the output is linearly projected and fused with Z to produce Z All components Z n, i, and ts are passed through positional encodings and MLPs before concatenation. Here, + denotes matrix summation. The final block of RNNs,, followed by layer normalization fully connected layers (LN/FC), produces the diffusion stepwise prediction of noise co (ZLat, n, ts).
Conceptually, each column of Zhat treated as a discretized univariate time series in the latent space, and noise is added independently to each column. However, this does not preclude modeling cross-featurs dependencies in Z [ilos et al.] 20023.. Although the forward diffusion corrupts columns independently, the reverse (denoising) process operates the entire latent matrix simultaneously, capturing inter-feature correlations. A similar approach has been adopted in TabDDPM (Kotelnikov et al. 2022 for categorical data modeling. Under this formulation, each ej ~ ~N(OTITT))
write: Zl.at Z0 + VI E", where E" := .e"...., ef] IR Tx and
The training objective is the evidence lower bound (ELBO) defined as:
CDM == En,E" [Z c n, ts) E"I
(4)
where Eg is neural network that predicts the noise E" at a uniformly sampled diffusion step Cnif(1,...,, N).
Given the noisy latent matrix ZTat E TXF conditional data o E RRFF, the diffusion step n E (1...., N), a normalized time vector t t= (ti....,ty' t where t4 = $. and additional timestamp information ts, goal of the denoiser Eo to predict the added noise E R at step n.
Design of egz First, we encode the diffusion step n and the normalized time vector t using positional encodings (PE), described in the Vaswani et al. (017)). These encodings make the model aware of the current diffusion progression and the temporal ordering of rows in ZLat.
Since t alone provides limited temporal context, we enhance it with cyclic date-time encodings for 'YEAR- MONTH-DATE-
These cyclical encodings are computed via sine and cosine transformations:
CE(ts)= : { (sinl Per)odx22) ), cos( Perlod x }}} Period E (YEAR, MONTH, DATE, HOURS}}
and yield an 8-dimensional representation per timestamp. Here, Period chosen based on the temporal granularity of the dataset (c.g., total number of years till today (for instance, 2025), for months, 365 for days, and 24 for hours). This augmented timestamp information is projected through an MLP to match the dimension of the other encoded inputs. See Figure
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Similarly, if conditional data X is provided, it is processed through the same embedding procedures described in the VAE encoder (i.e., discrete features via lookup embeddings and continuous features via denoting Z := Emb(((cn)
breqeency-based encodings in equation 2
Next, we integrate the conditional embeddings Zenn with the noisy latent matrix Z The relation
2 = Linear(ZZ, +
defines fused latent representation Zn In this construction, Z encodes the corrupted latent. factors at diffusion step n, while Z contains temporally aligned conditional embeddings that have been further processed through a Bi-directional RNN (Bi-RNN) (Schuster & Paliwal 1997) to capture sequence-level dependencies among the conditional variables. Each of these two components z and Bi-RNN(Z passed through its own linear mapping, ensuring both latent. representations share a compatible feature space.
Next, we concatenate all encodings Zn the positional encodings of n and t, and the augmented timestamp encodings from ts into single tensor. This concatenated representation passed through another MLP:
N = ML, PE(n), PE((), CE(ts)) E RT'
To capture temporal dependencies along the sequence length T, we feed N into a Bi-RNN, similar to the approach taken by Tian et al. (2023). The Bi-RNN processes the entire sequence (N(4)...,,N( forward and backward in time, producing a context-aware representation for each time step.
After applying layer normalization and a final fully-connected layer to the Bi-RNN outputs, c generates:
co(t)))] RTF,
which serves the estimate of the noise E at the given diffusion step n.
4.5 Training, Inference & Computational Efficiency of TimeAutoDiff
In this section, we give the detailed procedures on training and sampling of TimeAutoDiff.
Training: While the training objectives for the VAE (i.c., LvAe) and the DDPM (i.e., LDM) defined
in equation et al. 2022 or in a joint manner (Vahdat et al.) 2021). Recent research on latent DMs for image (Rombach
3 and equation it remains unclear whether to train these models sequentially (Rombach
ct al. 2022). molecule (Xu et al., 2023), and tabular (Suh et al., 2023 Zhang et al.) 2023b) generation indicates that a two-stage training approach usually achieves superior resulls, which we also observe in our experiments. Specifically, we first train the VAE with regularization and then train the latent. DMs on the latent representations produced by the pre-trained encoder. A formal description of this procedure is provided in Algorithm
Inference: A generative process, defined as (X tar ZLat X M = Po(Zaat X (x Zlat X M operates in two steps. First, given X X the trained diffussin prior samples a new latent matrix Zhat. Next, the decoder takes (X Zlat Mtask) as input and generates a new sample, where Miask M is a user-defined mask for specific time-series tasks. Finally, after post-processing, the final sample New obtained. The sampling procedure detailod in Algorithm
Computational Efficiency: Its dominant cost arises from the N-step diffusion sampling, ((NHH3 where Hai denotes the hidden size of the denoiser. By operating in compressed latent space (L < F) and requiring only a moderate number of sampling steps (N ~ 50-100, in contrast to the 1000+ steps used in data-space diffusion), our method atlains a 10 100x inference speed-up. In contrast, diffusion- based time-series synthesizers that rely on autoregressive sampling incur substantially higher computational cost cost-appron approxiiixiy ONN H since each timestamp must be generated scquentially with full prefix context. This efficiency advantage is obtained while preserving higher generative fidelity compared single-step GAN approaches. See Appendix G for complete breakdown.
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Algorithm Training Algorithm of TimeAutoDiff
Input: (X X Missk ts, t (1)
Initialize: encoder network Eg. decoder network Du, denoiser network Ep
First Stage: VAE Training
4: while Ø, v have not converged do
5:
p. o. + Ex(Xtar X
b Encoddng
E~ 10,11
Zlat + u+EOE
Reparameterization D Decoding
8:
X + Dul non Zome M
LVAE f Erecons (Xtar, X + + DEL (gs (0)
10: Φ, t Optimizer CAVB 0, 2)
11: frecons has not improved for 10 epochs then
12: 3 +maxx(Buins 0.73)
Adaptive 3 update
13: end if
14: end while
15: Second Stage: DDPM Training
16: Fix o, th and latent data z
17: while 0 has not converged do
18: TE Inif(l,.....
E" 10.
19:
T. v an D +
20: LDM = lea (Z Xeon n, ts) 1
21: 8 O-ttipimerrrCmme
22:
end while
23: return (Es, co)
Algorithm Inference of TimeAutoDiff
Input: decoder Dv, denoiser EA, X Mrssk ts, t = (+) Z ~N(00,1
2: for n = N.N N 1,...,I do
3: ~~ N0, I). reshappe(T,
D. Latent denoising
4: Eg + EB (Z)att Xcon, n, t, ts)
ZL.t f- van
ZLaat l-an
20) + 3,
Vi-an
end for
7: X New
D ( oX Zhat M
D Decoding
8: return X New = post-process (X New)
5
Experiments
This section provides a thorough empirical evaluation of our proposed TimeAutoDiff model. We compare its performance against several state-of-the-arl baselines across unconditional generation, imputation, forecasting, and TV-MCG tasks.
5.1 Unconditional Generation
This subsection evaluates the core capability of TimeAutoDiff to synthesize realistic time series data in unconditioned setting. We assess the fidelity of generated samples using comprehensive suite of quantitative metrics, including low-order statistics (e.g., feature and temporal correlations) and high-order statistics (e.g., statistical fidelity to the original data). Additionally, we evaluate the utility of the synthetic data for downstream predictive tasks, examine sampling efficioncy. Next, we provide an ablation study which disentangles the contributions of individual model components overall performance. Finally, we assess model generalizability using the Distance to the Closest Record (DCR).
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5.1.1 Experimental Setting
Datasets: We use eight real-world time-series tabular datasets in this subsection. Among them, ETThl contains only numerical features, while the others include both numerical and categorical variables: Traffic, Pollution, Hurricane, AirQuality, Energy (single-sequence), NASDAQI00, and Card Fraud (multi-sequencce) Appendix B provides details on how we preprocess both single- and milti-sequence tables for training, along with dat apel-seeiiiic statistics. For all experiments in this section except for qualitative and scalability analysis, where we use T = 100 ~ 900 we set the window size to T = 48 and stride to S = (see Appendix for details).
Baselines: To assess the quality of unconditionally generaled time series data, we use 5 baseline models: (1) GAN based methods: TimeGAN (Yoon et al.) 2019). DoppelGANger (Lin et al.) 2020). (2) Diffusion basod
methods: Diffusion-TS (Yuan & Qiao 2023), TSGM (Lim al. 2023 Parametric model: CPAR (Zhang
et al. 2022).
Evaluation Methods: For the comprchensive quantitative evaluation of the synthesized data, we mainly focus on four criteria: (1) Low-order statistic- pair-wise column correlations and row-wise temporal dependences in the table are evaluated via feature corvelation score (Kotelikkvv et al. 2022) and temporal discriminativee score (devised by us), respectively. (2) High-order statistic- the overall fidelities of the synthctic data in terms of joint distributional modeling are mcasured through discriménative score (Yoon) et al., 2 (3) The effectiveness of the synthetic data for downstream tasks is assessed through the predictive score (Yoon et al. 2019), where predictive model (i.e., regressor or classifier) is trained using synthesized data and tested on real data (Mogren) 2016). (4) Sampling times (in scc.) are compared with other baseline methods. Detailed explanations for each metric are deferred in the Appendix E (5) generalizability of the model is evaluated under "Distance to the Closest Record" (DCR; Park et al. (2018) metric to ensure it draws samples from the distribution rather than memorizing the training data points.
Model Parameter Configuration: In Appendix F we present the parameter configurations of VAE and DDPM in model. Unless otherwise specified, they are universally applied to the entire dataset in the experiments conducted in this paper. Additionally, we study how the sizes of network architectures in DDPM and VAE, training epochs for both models, and noise schedulers (linear vs quadratic) in DDPM affect the performances of the model.
5.1.2 Unconditional Generation
Experimental Results: Table 2 shows that our TimeAutoDiff consistently outperforms other baseline models in almost all metrics both for single- and multi-sequence generation tasks. It significantly improves the (temporal) discriminative and feature correlation scores in all datasets over the baseline models. TimeAutoDiff also dominates the predictive score metric. (We train a classifier to predict a column in the dataset to measure the predictive score. The columns predicted in each dataset are listed in Table some datasets, the performance gaps with the recond-best model are negligible: e.g., TimeAutoDiff vs TSGM for Hurricone and AirQuality datascts. It is intriguing to note that the predictive scores can be good even when the data fidelity is low. The GAN-based models are faster in terms of sampling time compared to the diffussin--ased models. These results are expected, as diffusion-based models require multiple denoising steps for sampling, whereas GAN-based models generate samples in a single step. Nonetheless, among diffusion- based models, our approach achieves the fastest sampling time- approxxixiiiaally 90 to 100 times faster than existing methods. In the Appendix [H] and D we provide additional experiments on more metrics such as volatility, moving averages, Maximum Mean Discrepancy (MMD) and entropy for diversity (Nikitin) et al. 2023
Appendix B But for
Qualitative Illustrations of Temporal & Feature Dependences: Aside from the quantitative evaluations under the aforementioned metrics, we provide the qualitative analysis in Fig. i further demonstrating the elfectiveness of TimeAutoDiff. The autocorrelation plots for both real datasets (AirQuality and ETTh1) and their synthetic counterparts reveal that TimeAutoDiff successfully captures complex and long-range temporal dependencies-oxtedding up to T = 500 time steps. Additionally, the similar patterns observed for each feature in the real and synthetic data demonstrate the model's ability to capture the correlations along the feature dimension. We provide more visualizations across more various datasets in the A Appendix L
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Single-Sequenee
Multi-Sequence
Metric
Model
Traffic
Pollution
Hurricane
AirQuality
Card Transaction
nasdeg 100
TimsAutaDiff
0.026(0.014)
0.016 0.90
0.047(0.016)
(.061(0.013)
0.215(0.058)
(0.066)
fiffusion on
(0202
0.153(
0.181 (0.018)
0.134(0.016)
N.A.
N.A.
Discriminative
Score
TSGM
0.500(0.000)
0.488(0.010)
0.082(0.020)
0.452(0.009)
N.A.
N.A.
TimeGAll
(0.03))
(.351(0.053)
(.254(0.062)
0.460(0.020)
0.482(0.037)
(.261))
DoppelGAKger
0.258 (0.215)
0. (0100)).103)
0.176(0.00 (99)
0.211 (16)
0.185(0.025)
(.071(0.032)
(The lower, the better)
CPAR
(0.00))
0.500(0.000)
0.500(0.000)
0.499(0.001)
0.500(0.000)
0.143().120)
Real Real
0.053(0.000)
0. 0.048)0.0177)
(0.1)())01))
0.040(0.011)
0.225(0.094)
0. 0.190)),051)
TimsAutaDiff
0.203(0.014)
0.008 0.008(0.000) 000000
(009((0.06))
0.005[0.0011
0.001(0.000)
10.863(0.716]
Diffusion-ta
0.231(0.0007
0. 0.013(0.000)
0 0.306(0.076)
0.017(0.002)
N.A.
N.A.
Predictive
Soore
ISGM
(.24770.002)
0.009(0.000)
0.290(0.007)
0.006(0.000)
N.A.
N.A.
TimeGAN
(.297(0.008)
0.00330.000)
0.180(0.002 027)
0.057(0.011)
0.130(0.022)
(.597))
DoppelGANger
0.300(0.005)
0. (0.82)
0.214 (0.0000
0.060(0.009)
0.001(00006)
11.556(1
(The lower, the better)
CPAR
(0263().003)
(.032(0.000)
0.420(0.055)
0.030(0.007)
0.132(0.035)
8. (..009)
Real Real
(.200))
(.000))000)
0.098 026)
0.005(0.001)
0.001 (0.00))
9. 9.281 (0.099)
TimeAutaDiff
0.014(0.013) 0.047(0.018)
(.026(0.02)
0.033(0.0144
0.290(0.040)
0.159)
Temporal
Diffusion-ta
0.199 025)
0. (0.65))
0,247 (0.033)
0. 0.183(0.061)
N.A.
N.A.
Discriminntive
TSGM
0.400(0.001)
(.490().001)
0.497(0.002)
0. 0.499(0.000)
N.A.
N.A.
Soore
TimeGAN
(0.2900.050)
(0.00))
0.165(00.22) 025)
0.457(0.014)
0.497(0.007)
0.119(0.140)
DoppelGANger
0.100(0.039)
0.111(0.050)
0,461(0.022 028)
0.335(0.091)
0.362(0.097)
(90000))
(The lower, the better)
CPAR
(0436(0
(.497(0.000) (0401))00021)
0.493((0.010)
0.470(0.044)
(.404(0.009)
Real Real
(0.01()
0. 0.044(0.000)
0. 0.039(0.012) 012)
0.050(0.017)
0.360(0.051)
0.1 (..000))))
TimekutoDiff
1.244: 0.002(0.014)
844)
0.07:(0.013)
O. 0.463
1377)
0.213(0.012)
Feature
Diffusion-ta
2.14
1.716(1.096) 096))
1.881 208)
0.716(0. (41)
N.A.
N.A.
Correlation
TSGM
(.082().485)
(.710(0.705)
(..2440
0.543(0.077)
N.A.
N.A.
Soore
TineGAN
(.243(
2.068(1.09) L833)
2.151(1.113)
0.865(0. 123)
2.301 (0.723)
(.88(1.069)
DoppelGAKger
(0.73)
(.37)(0.875)
2.380(0.79) 798)
L.625(0.23L)
(.550(1.31)
(.035)0.818)
(The lower, the better)
CPAR
0.538 (0.336)
(0280())981)
0.965 287)
1 1.552 220)
0.295(0.294)
(.514().445)
Real Real
0.000(0.000)
0. (0.00().000)
0.000 000)
0.000(0.000)
0.000(0.000)
o (.000))
TimekutoDiff
3,512 (0.065) >
3.917 (0.070)
3.740 (0.132)
3.915 (0.103)
3.38-((0061)
3.133(0.129)
Sampling Time (in See)
Diffusion-ta
>
N.A.
N.A.
TSCM
>+
3+
*
>>
N.A.
N.A.
TineGAN
0.127(0.056)
(0.08))
1.15(0.060)
0.131(0.060)
0.051(0.051)
0.047 0.009)
(The lower, the bester)
Deppa1GAXger
) 0.014 (0.002)
0.010(0.003)
0.017(0.003) 29.816(0.840) (0.839))3224)
0.018(0.004)
(0.010))
CPAR
17.466(0.78) (34)
(0.558)
(21.2442.4455
112.500 (12.506(2152)
Table 2: Experimental results for single- and multi-soquence time series tabular data generation are reported under Discriminative, Predictive, Temporal Discriminative, and Feature Correlation metrics. Sampling times (in Secs) over datasets are included; >> indicates times exceeding 300 seconds, and 'N.A.' denotes not applicable. Bold numbers indicate the best performance. Each metric shows the mean and standard deviation (in parentheses) over 10 synthetic datasots generated by the trained model. 'Real Data' serves a bascline with each metric computed on Real vs. Real.
Scalability of TimeAutoDiff along features axis: To validate the scalability of our approach along feature axis, particularly the benefit of the VAE's latent feature compression, we conducted experiments high-dimensional synthetic datasets (sce Appendix J for full results). These tests demonstrate that TimeAutoDiff effectively models long sequences (up to T-900) with high fidelity when the feature count is low (F=5). Conversely, when increasing the feature dimension up to F=50 (at T-200), performance noticeably degrades without compression. However, by setting the latent feature dimension to F/2, the model maintained significantly better generation quality in this high-dimensional setting. This empirically confirms that the VAE's feature compression is a key component for efficiently modeling wide, high-dinenniooall time-series tables.
Ablation:
The ablation test results are summarized in Table 3 A single model alone (i.c., only VAE or DDPM) camnot accurately capture the statistical properties of the distributions of tables, which strongly supports the motivation of our model. The components related to the diffusion model, such as timestamp encoding and Bi-RNN, impact the generative performance across most cases as models lacking these components do not exhibit optimal performance. The encodings for continuous features in the VAE notably enhance the fidelity and temporal dependences of the generated data.
Additionally, we consider the following scenarios:
1. Replacing the MLP with an RNN in the decoder of the VAE.
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Aiquulity
Sythesc
CopOrs
COURTO
PTB8521000
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6.6-
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80
50
283
116 Time Lap Owert
206
253
100
200
250
200
Time Lee thurss
Figure 6: Real (left) vs. Synthetic (right): Antocorrelation plots with a time lag of 300 (hours) for the AirQuality (top) and the ETTh1 (bottom) datasets. Sequence length is set as T = 500.
5
Schod j
god.1
B
Disc. Score (0369(0.101)
est E331
$-0.003
200.0001
5303
10
33
0,000
10 10-2
0.041(0.011)
40-
0.1
0.90
10-3
0.043(0.019)
10-
41
0.33
10-4
0.079
10
0.043
19330
20031
33
1231
10:.0
11200
2500e
0.000
520
20300
Spuca
Scheduled |
0.023 (00015)
Figure 7: KL-Divergence (left) and Reconstruction (middle) losses over 20000 training iterations of VAE on Traffic datasct. The zoomed- in panel (right) displays the scheduled-3 reaches the lowest recon- struction error stably without any spikes.
Table 4: The results of discriminative
scores with varying 8 values the Traffic dataset.
2. Replacing the two RNNs with MLP in the encoder of the VAE.
Inspired by Bilos et al. (2023). ; we explore injecting continuous noise from a stochastic process (Gaussian process) into the DDPM. Specifically, the perturbation kernel
Lat) = NN Lat (1 (,,)2) q (( 20,
is applied independently to each column of Zhat E TxF, where Σij = exp(->/t; - tzl) with = 0.2.
The experimental results indicate that none of the ablated models significantly outperformed the original configuration. In particular, the second configuration highlights the potential benefits of modeling temporal dependencies two stages within both the VAE and the DDPM. While the precise mechanisms remain to be further validated, we hypothesize the following contributing factors: (1) Hierarchical Temporal Dependency Modeling: The VAE encoder captures compact latent representations that preserve temporal structure, providing well-organized foundation for the diffusion model. This hicrarchical setup may allow the diffussion process to focus on refining fine-grained temporal patterns, rather than redundantly learning high-level structures, leading to more realistic outputs. (2) Voise-Toleran Latent Representation: Incorporating temporal dependencies carly in the VAE may yield a latent variable ZT that inherently more robust noise. This resilience could help preserve critical temporal information during the diffusion process, ultimately improving the fidelity of the generated sequences.
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Synthetic
Syrithetc
Couss: Tannig
Thaning
6.0
12
6.3
3.0
0.0
1008
2003
2008 4002 2000 6000
20
Polution TimckutoDiff
OC1
Time
Polludion Syrthetic
PollutionCCsses
Tre
Pallution Training
Thmee
Tah
1007
22
0:25
0.00
0.75
0.003
0162
1902
-003
6.00
60
30
40
30
37
20
38
40
Hurrican TimeAuteD If
Hurficane Sontheric Osta
Hulf Cario Sytteeeeeeiie Data
Murticans 0505: Taining
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0.52
20
4.5
0.45
0.35
1.0
0.92
2.5
6:00
25
0.20
0.15
6.35
OC1
Time
Tra
Vihn
0.35
0.93
0.25
12
0.25
0.22
0.19
0.13
0.10
0.35
0.08
0.6
25
50
25
18.
125
15.8
20
20
20
46
35 26
4.
22
20
20
40
20
Figure 8: The leftmost column plots the empirical distributions of distance-lo--loosst--room (DCR) for the training and test splits across four datasets (top bottom: Traffic, Pollution, Hurricone, AirQuality). For each dataset, pick two representative variables: the sccond and third columns show these variables over time for one synthetic sample (T 48), and the fourth and fifth columns show the samne variables for its nearest training record (by DCR). On Traffic and Pollution, the DCR mass lies noticeably away from zero and the synthetic trajectories visibly diverge from their nearest neighbors, indicating new patterns rather than replicas. On Hurricane and AirQuality, the DCR concentrates near zero and the synthetic trajectories closely track the nearest training records, suggesting replication.
generalization: Consistent with the phenomenon reported by ZZhang et al.) 2023c), TimeAutoDiff exhibits data-regime eett-reginee-deeedeeett behavior: when effective model capacity excoeds the available data (data-poor regime), memorization is more likely; when the data budget dwarfs model capacity (memorization regime), generalization emerges. With model size fixed in our study, Traffic and Pollution provide over 20K training sequences (generalization regime), aligning with the nonzero DCR and novel trajectories, whereas Hurricane and AirQuality provide fewer than 5K (data-poor), aligning with ncar-zero DCR and training-like generations.
Memorization
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Metric
Model
Traficc
Pollution
Hurricane
AirQuality
TimeAutoDiff
(.027(0.014)
0.035(0.010) 0.014(0.011)
0.035(0.016)
only VAE
0.476(0.010)
(.1900)
0.490(0.010)
(.494(0.000)
only DDPM
0.283(0.131)
0.313
0.252(0.034)
0.266(0.048)
Discriminative
w/o Encoding equation
0.029(0.017)
0.062(0.015)
0.063)0.018)
0.072(0.020)
Score
w/o Timestamps
0.095(0.016)
0.105(0.012)
0.171(0.085)
0.074(0.013)
w/ w/o Bi-directional RNN
0.049(0.015)
0.021(0.020)
0.019 0.300(0.036)
(The lower, the better)
RNN in decoder (VAE)
0. 0.186(0.019)
0. 0.185(0.020)
0.198(0.081)
0. 0.124(0.018)
ML.P in encoder (VAE)
0.017(0.011)
0.072(0.020)
(.067(0.025) 0.117(0.019)
Smooth Noise (DDPM)
0.015(0.009)
0.078(0.013)
0.1 0.140(0.016) 0.140(0.016)
TimeAutoDiff only VAE
(.22900
0. (0.000
3,490(0.007)
0.004
0.241(0.001)
0.008(0.000)
4.566(0.041)
0.019(0.002)
only DDPM
0.241(0.012)
0.016(0.000)
0.034(0.007)
0.009(0.002)
Predictive Score
w/o Encoding equation 1
0.219(0.011)
0.008(0.000)
3. (.611(0.216)
0.005(0.000)
w/o Timestamps
0.241(0.003)
0.008
4. 4.228(0.248))
0.004(0.000)
w/o Bi-directional RNN
(0.008)
0.008
3.549(0.047)
0.004(0.000)
(The lower, the better)
RNN in decoder (VAE)
0.232(0.008)
0.008(0.000)
3.598(0.095)
0.012(0.004)
MLP in encoder (VAE)
0.220(0.011)
0.008(0.000)
3.365(0.072)
(0060()
Smooth Noise (DDPM)
(0.01)
0.008(0.000)
0.091
0.059(0.001)
TimeAutoDiff
0.047(0.017)
0.008
0.020
0.035(0.024)
only VAE
0.368(0.107)
(0.03)
0.490(0.014)
0.493
Tempprrl
only DDPM
0.197(0.127)
0.135(0.131)
0.213(0.096)
0.242
Discriminative
w/o Encoding equation
0.036(0.016)
0.052(0.019)
0.049(0.022)
0.008 005
Score
w/o Timestamps
0.084(0.047)
0.053(0.018)
0.117
0.064(0.019)
w/o Bi-directional RNN
(.001(0.021)
0.047(0.057)
0.404(0.013)
0.023(0.015)
(The lower, the better)
RNN in decoder (VAE)
0.130(0.025)
0.139(0.019)
0.324(0.072)
(.331().130)
MLP in encoder (VAE)
(.037(0.017)
0.060(0.018)
(.099().019)
0.045(..032)
Smooth Noise (DDPM)
0.020(0.007)
0.059(0.029)
0.090(0.027)
0.091(0.027)
TimeAutoDiff only VAE
0.022(0.014)
1.100(0.900)
0.069(0.027)
0.147 230)
(.409(0.339)
(.329(0.757)
0.427(0.371)
0.702(1.001
Feature
only DDPM
2.238(1.530)
2.020(1.460)
2.380(
0.198(0.298)
Correlation
w/o Encoding equation
0.029(0.021)
1.148(0.850)
0.077(0.034)
0.266(0.405)
Score
w/o Timestamps
0.247(0.521)
1.303(0.793)
0.097(0.014)
0.231
w/o Bi-directional RNN
0.018(0.024)
1.227(0.863)
0.090(0.043)
0.155(0
(The lower, the better)
RNN in decoder (VAE)
(.413(0.544)
1.187(0.820)
0.217(0
0.913(1.302)
MLP in encoder (VAE)
(.025(0.015)
1.240(0.853)
0.122(0.058)
1.217(1.745)
Smooth Noise (DDPM)
0.059(0.037)
1.246(0.811)
(.882)1.271)
1.215(L345)
Table 3: The experimental results of ablation
test in TimeAutoDiff. The bolded number indicates the
best-performing model.
The effect of adaptive V-VAE: Motivated from (Zhang et al. 2023b), evaluate the effects of scheduling on 3 coefficients in VAE in terms of tradeoffs between reconstruction error and KL-divergence. In Fig 7 we observe that while large 3 can ensure the close distance between the embedding and standard normal distributions, its reconstruction loss is relatively larger than that of smaller ones, and vice versa. The adaptive B-scheduling ensures both the lowest reconstruction error and relatively lower KL-divergence, preserving the shape of embedding distribution. The adaptive 8-scheduling achieves the fastest and the most stable signal reconstructions among other 8-choices. Table 4 shows the effectiveness of ß-schoduling for quality of synthetic data in discriminative score.
Generalizability of TimeAutoDiff: In generative modeling, it is essential to check whether the learned model can generate the datascts not seen in the training set. If model memorizes and reproduces data points from the training dataset (Zhang el al. 2023c), this can undermine the primary motivation of data synthesizing, which is 'increasing dataset diversity" To investigate further in this regard, we design an experiment using the notion of Distance to the Closest Record (DCR) (Park et al. 2018). which computes the Euclidean distance between a data point r E T in the synthesized dataset and the closest record to r in the original table. We split the data into training (50%) / testing (50%) sets.
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Mean Absoluse Error (MAE) acress Benchmark Cases
Hoan Squared Eror (MSEI acress Benchmark Cases
13
Figure 9: Dataset-averaged benchmark performance across tasks. Bars indicate the mean error across four datasets (Bike Sharing, Traffic, Pollution, and AirQuality) for each model and benchmark case, with the total sequence length fixed at T = 48. Left: Mean Absolute Error (MAE, logarithmic scale). Right: Mean Squared Error (MSE, logarithmic scale). The six benchmark cases (from left to right on the Imputation- Block (block_len-ee [seq_len=10 and forecasting with look-back windows w E {12, 2633 For basclines lacking reported results in the original tables (c.g., see Table K in the Appendix for missing TimeLLM, Moment, MICN, and DLinear entries over w E (12, 24, 36), bars are shown with zero height); means are computed while ignoring NaN values. A logarithmic y-axis is used for MAE and MSE to reduce the dominance of large errors and to reveal relative differences among the remaining cases.
z-axis) are: Imputation-
block_widthe wid irapuuaationSequennti
DCR scores for both training and lesting dalasets can be used to evaluate the model's performance. Significant overlap between the DCR distributions of the training and testing datasets suggests that the model is drawing data from the data distribution. However, even with substantial overlap between the distributions, if the distances to the origin are small, this suggests that the patterns in the training and testing sets are alike, implying the model may have memorized specific training data points. If the DCR distribution of the training data is notably closer to zero compared to the testing data, it indicates that the model has memorized the training dataset. Last but not least, it's important to recognize that random noise can also produce similar DCR distributions. Therefore, the DCR score should be evaluated in conjunction with other measures of fidelity (i.e., discriminative score), and utility measures (i.e., predictive score), to provide comprehensive assessment of the model's generalization capabilitics. We provide the interpretations of DCR distributions of TimeAutoDiff for Traffic, Pollution, Hurricane, and AirQuality datasets in the caption of Fig. S
5.2 Imputation & Forecasting
In this subsection, we present experimental results on two conditional generation tasks: Imputation and Forecasting.
Extended Evaluation Settings: We evaluate TimeAutoDiff across a comprehensive suite of six benchmark cases, encompassing three imputation regimes and three forecasting horizons. All experiments are conducted on four real-world tabular time series datasets Bike Sharing, Traffic, Pollution, and AirQuality each of which contains both numerical and categorical variables. The sequence length is fixed to T 48 for all tasks to ensure comparability. For imputation, we simulate three types of missingness using the binary mask MI 1 (X.,f is missing): (1) MCAR (Missing Completely At Random), where entries are masked uniformly: (2) Block Missingness, where rectangular regions over time and features simulate sensor dropouts; and (3) Sequential Missingness, where temporally contiguous subsequences emulate localized corruption. For the forccasting task, we vary the look-back window size w E {12, 24, 36), defining the forecasting mask as ME = 1(t w). This setup evaluates the model's ability to extrapolate across both long- and short-range temporal horizons, while maintaining fixed total sequence length T = 48.
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Baselines: To evaluate the performance of TimeAutoDiff imputation and forccasting tasks, compare
it against. nine established baseline models: TimeLLM (Jin et al. 2023). Moment (Goswami et al. 2024).
TEFN Zhan et al. 2024), TimeMixer [Wang et al. 2024b), TimesNet (Wu et al. 2022). , MICN (Wang et al.
2023a DLinear Zeng et al. 2023); FILM (Zhou et al. 2022 1, CSDI (Tashiro et al. 2021). PatchTST (Nic jet al. 2022), ; iTransformer (Liu et al. 2023), and TimeMixer++ (Wang et al. 2024a). All selected models are designed to support both imputation and forecasting tasks. For consistency and reproducibility, we utilize the PyPOTS framework (Du, ) which provides unified implementations of all baseline models. Detailed descriptions of each baseline model are provided in the Appendix D.2 The hyperparameter settings for each model are specified in the ImputeEval. py file, which is available in our public code repository.
Evaluation Methods: For the experiments, we split the dataset into training (70%) / validation (20%) / testing (10%) sets. We independently select the best- pest-performing checkpoint for the VAE and DDPM based on validation error. For both imputation and forecasting tasks, we use two standard metrics: Mean Absolute Error (MAE) and Mcan Squared Error (MSE) for performance cvaluations.
Quantitative Summary: Based on the quantitative results, TimeAutoDiff demonstrates strong overall performance, achieving a top-2 score in 31 out of the 48 total benchmark settings (4 datasets, 6 tasks, 2 metrics). This includes 23 state-of-the-art (SOTA) scores. The model's primary strength is in forecasting, where it secures 13 out of 24 SOTA metrics, plus an additional 6 second-best. scores. In contrast, its performance in imputation is more specialized, achieving 10 out of 24 SOTA metrics (plus second-best scores), primarily concentrated in the Traffic and Pollution datasets. The model's 's performance is strongest on the Traffic dataset, where it leads in 10 out of 12 total tasks. Conversely, it shows its weakest comparative performance in the Bike Sharing imputation tasks, where it secures no SOTA second-best results, being outperformed by PatchTST, "Transformes and MICN.
Result Interpretations: Our interpretation is that TimeAutoDiff's architecture is highly effective at modeling complex, heterogeneous, and non-stationary temporal dynamics, which is critical for its strong performance in forecasting. This advantage is most pronounced on the volatile Traffic dataset. The imputation tasks, however, particularly on the highly structured Bike Sharing dataset, appear to reward different architectural specializations. Models like PatchTST, which utilize patching, excel at modeling missingness. Mcanwhile, models like
local patterns, making them robust to structured Gllock/Sequentiee "Transformet and MICN, which focus on cross-channel correlations, are better suited to impute missing values by referencing other related features (e.g., weather, season) present at the same timestamp.
5.3 Time Barying-Metadan Conditional Generation (TV-MCG)
We evaluate TimeAutoDiff on the Time- Varying Metadata Conditional Generation (TV-MCC) setting, where a metadata trajectory X is fixed and the target sequence Xtarr is generated conditionally. Note that X corresponds to subset of features from the full datasel X, selected along the feature axis, characterized by a mask M Our study is organized into two complementary parts:
Single-Sequence
Metric
Methods
Traffic
Pollution
Hurricane
AirQuality
ETTh1
Energy
Discriminative
TimeAutoDiff
0.078
0.056 0 0.017)
0.014 (00005)
0.090(0.007)
0.036 (0.08)
0.113 (0 (0.070)
Score
Real vs Real
0.091(0.021)
0..67()
0.081 (0.009)
0. 0.085 027)
0.051(0.011)
0.270
Predictive
TineAutoDiff
0.113(0.007)
0.008
0.060(0.009)
(.004(0.000)
0.048(0.003 0002)
0. (2.00)
Score
Real Real
0.107(0.001)
(0000()
0.058(0.010)
0.004(0.000)
(.051(0.001)
002303)
Temporal
TimeAutoDift
(0.23)
0.081(0.0277
0.048(0.025)
0.045(0.0155 0.116(0.018)
0 0.224 (0.013)
Discriuinative
Real Real
0..00())
0. (0.899
0.072(0.019)
0.138(0.014)
0.074(0.014)
(0.003)
Score
Feature
TimeAutoDife
0.012(0.003)
0.026(0.008)
0. 1 (75(0.032)
0.011
0.014(0.002)
0. (0.29770007)
Correlation
Real vs Real
0.000(0.000)
u. 0.000(0.000)
0. 0.000(0.000)
0.000(0.000)
0.000(0.000)
0.000(0.000)
Score
Table 5: Time varying metadata conditional generations: the experiments conducted over 6 single-sequence datasots with sequence length set as T = 96. See the caption of Figure 15 (Appendix M for output and condition pairs for each dataset used for the experiments. Overall, TimeAutoDiff performs well, achieving results comparable to the Real vs Real baseline over the test dataset.
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onwther man
temp
Condi malyy Genertted weather mang
Concil onsty Gererated terp
0.25
250
0.92
2-
0-
Figure 10: Datasets: (output variables) from top to bottom: Traffic: ('Weather main', 'temp'), The output is chosen to be heterogencous both having discrete and contimuous variables. Conditional variables are set as remaining variables from the entire features. See the list of entire features of the dataset through the link in Appendix
(1) Quantitative assessment: We evaluate TimeAutoDiff on multiple public datasets using four fidelity and dynamice-based metrics, comparing against an oracle Real-vs-Real proxy. (ii) Counterfactual scenario exploration: To verify that the model learns meaningful conditional structure, examine both synthetic settings with known rules and real-world Traffic data, generating counterfactual trajectories under alternative metadata conditions. (iii Robustness of generative model: Finally, we study robustness by introducing controlled metadata corruplion (noise for continuous variables, random flips for categorical ones) and measuring how fidelity metrics change as corruption increases. This reveals how reliably the model preserves conditional dynamics under noisy or imperfect metadata.
5.3.1 Quantitative assessment with public data
Baselines. Few prior models address conditional tabular time-series generation with fixed side information. To our knowledge, TimeWeaver (Narasimhan et al.) 2024) is conceptually the closest, but it limited generating only continuous time series data. Furthermore, public implementation of the model is available. Accordingly, report Real vs Real proxy baseline oracle reference and compare it against our conditional generator.
To evaluate the generalization capability of TimeAutoDiff to unseen conditions, we randomly split each dataset into training and test sets using an 80%/20% ratio. Synthetic samples are then generated to match the size of the lest set, with the sequence length fixed at T 96. The quality of the generated data is assessed using the evaluation metries: Discriminative Score, Predictive Score, Temporal Discriminative Score, and Feature correlation score.
Evaluation Methods:
Quantitative Results. Table 5 shows that TimeAutoDiff matches or improves upon the Real vs Real oracle across datasets. Discriminative and disoriiiidiri scores are consistently lower, indicating high conditional fidelity and well-preserved dynamics. Predictive scores are on par with or slightly better than the oracle, confirming that synthetic data retain task-relevant signal. Feature-correlation deviations remain small, evidencing faithful cross-feature structure under fixed metadata. Overall, the model reliably captures both fidelity and conditional coupling across easy (periodic) and challenging (high-variance) settings.
Dataset-level Observations. Datasets with strong exogenous structure seasonal drivers Miivers-EThi,, hirQuality-shrr show very low discriminative and predictive gaps, reflecting that TimeAutoDiff reliably repro- duces condition-aligned periodic and cross-feature patterns. Challenging, high-variance settings such as Hurricane and Energy still yield markedly low discriminative and temporal discriminative scores, underscoring the model's capacity to respect conditional dynamics even when the marginal variability is large. On Truffic and Pollution, the method remains close to (or better than) Real vs Real across all metrics, evidencing accurate conditional coupling between environmental or load conditions econ and target responses Xtar
5.3.2 Counterfactual scenario exploration
We further provide numerical validations that TimeAutoDiff indeed learn the conditional distribution ta Xeon ,MM) of both PCCCnnt Vas:" Var.) and PP-Disc Cont Var.) under synthetic data set-
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25-
Surtyy Cleashy Starry
10-
15
20
10
20
10
10 Thme
30
Raity
2.75
2.50
2.50
Ohather
2.25
2.25
3.75
3.75
1.50
3.50
2.00
Time
Figure 11: Empirical validations of the model's ability to learn the conditional probability distributions (""Tepperature"/
(Weahhrrr) (top 3 pancls) and P(Weatherr' (TentherrrTTmppaatuurrr (bottom 3 pancls).
ting. Additionally, we explore its application in counterfactual sccnario analysis with real-world Traffic data, investigating how weather sequences affect trallic volume.
Synthetic Setting: Real-world data often involves complex correlations and confounding factors, making it difficult establish strict causal relationships. To validate that TimeAutoDiff can effectively learn conditional rules, we use a synthetic dataset with variables "Temperature" and 'Weather'. The "Temperature" is generated over 10,000 time points
2xt
Temp(t) = 15 + 10 sin
+ +0(00,²)
365
where Temp(t) follows sinusoidal pattern with added Gaussian noisc. Based on the generated 'Temperature', the categorical 'Weather' variable is derived as follows: 'Sunny' if Temp > 20, 'Cloudy' if 10 < Temp < 20, and 'Rainy' if 0 < Temp < 10. We set the time window as T 48 (hours) and train the model to learn two conditional distributions: P(Temp/WWed Weather), which predicts temperature given weather, and P(Weather") (Tmm)), which predicts weather given temperature.
Fig II (top 3) demonstrates the model's ability generate 'Temperature' sequences corresponding specific weather conditions under three scenarios: (1) sequences with constant weather conditions across three consecutive 48-step windows- ''Snny', 'Cloudy', and 'Rainy', labeled 1, 2, and 3 respectively; (2) repeating pattern of weather labels (e.g., 16 "Rainy", 16 'Cloudy', 16 'Sunny'), and (3) random alternating patterns of 'Couuyy' and "Rainy. The results show distinct separations in the temperature sequences generated for each weather condition, validating the model's ability to learn P(Cont Var. Var.). Similarly, Fig 11 (bottom 3) demonstrates the reverse case. When conditioned on 'Temperature" values generated in the previous scenarios, the model correctly prodicts the corresponding 'Weather' labels at cach time step (which is surprising), further validating its ability to learn iP(Disc Var.|Cont Var.).
To evaluate TimeAutoDiff real-world data, use the Traffic dataset with TTraffic Volume' (continuous) as the output and 'Weather-main' (categorical) as the conditional variable. 'Weather-main includes labels such as ('Clear', 'Rain', 'Squall', 'Cloudy"), among others. Intuilively, we expect lower traflic volumes during adverse weather conditions (e.g., 'Squall', 'Rain') and higher traffic volumes during good weather (c.g., 'Clear', 'Cloudy'). We test the model under three weather scenarios: 'Cloudy', 'Squall', and 'Clear', using six different timestamp sequences observe patterns. As shown in the results, 'Traffic Volume' is consistently lower during 'Squall' compared to 'Cloudy' and 'Clear', while significant differences are
Traffic Data:
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Stamp
Dataset:iing Stamp
Clousy
deude
6.6
5.8
0.6
6.4
0.2
0.2
Time
Time
Time
Traffic Dataset--Tme 4
Trffic Dataiel-Time Stamp
Traffic Dataset-Time Stamp
Sauell Clear
Soual
Squall
0.5
5.6
0.4
0.1
0.0
1.0
20
40
20
32
10
20
20
Time
Time
lime
Figure 12: We choose arbitary timestamp sequences in dataset, and give the models labels of ['Cloudy', 'Squall', 'Clear"] weather-conditions. The traffic-vlumee axis is normalized.
observed between 'Clear' and 'Cloudy.. These findings confirm the model's ability to reflect expected traffic patterns under different weather conditions.
5.3.3 Impact of Metadata Corruption on Generative Robustness.
In real-world time-series generation tasks, metadata -uchh as contextual covariates, environmental indicators, categorical conditions often contain noise or missing values due to sensor drift, transmission errors, imperfect data logging.
To this end, we introduce a metadata corruption setting that perturbs both continuous and categorical 0.4,
metadata according to a corruption level a E additive Gaussian noise scaled to each feature's empirical variability:
0.4,0.6), 0.6). Contimuous metadata are corrupted by
2 = z 0026,
6 6~ N~,
where ag denotes the per-feature standard deviation. Categorical metadata are corrupted by randomly flipping cach category with probability a, replacing it with a uniformly sampled altornative:
c,
with probability 1 1-a,
Unit(C \ (c)), with probability
This corruption mechanism provides controlled degradation of contextual information, enabling evaluation of the model's robustness under increasingly unreliable metadata.
Under this noisy setting, evaluate two fidelity-based metrics--iscceee Discriminative Score and Temporal Dis- criminative Score-which quantify how closcly the generated sequences match the real data in terms of distributional similarity and temporal coherence, respectively. Lower values indicale higher fidelity. As shown in Fig. 13 all datasets (Traffic, Pollution Data, Hurricone, AirQuality, ETTh1, Energy) exhibit increasing scores as the corruption level grows, demonstrating that metadata noise degrades both distributional and temporal realism. (Recall that values approaching 0.5 correspond real and synthetic data becoming fully distinguishable. We adopt the same experimental protocol as in subsection 5.3.1
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Computton
6.5
2.4
Tempora Ospriniative Store
0.3
0.3
02
01
ArQual
AiQusiitty
ETTIZ
0.
Energy
6.3
6.8
Netadeta (nlphte
(a) Discr. Score vs. corruption level a.
(b) Temporal Disc. Score vs. corruption level a.
Figure 13: Fidelity under metadata corruption. Both metrics (lower = higher fidelity) degrade as metadata corruption a increases. For each a, scores are computed over five independent runs and then averaged.
This overall trend matches our intuition. Interestingly, the Discriminative Score reveals two characteristic behaviors across datasets: some datasets ( (Traffic, Pollution Data, Hurricane, ETTh1) display a near-lincar degradation with corruption, while others (AirQuality, Energy) show a sharp increase at small corruption levels (e.g., a = 0.1) followed by a plateau. This suggests that the degree which metadata shapes marginal feature distributions varies by dataset. In contrast, the Temporal Score exhibits less structured trends no clear lincar increase-yct the overall upward shift remains consistent across datasets. This indicates that temporal coherence is also negalively impacted, but in manner that depends more intricately each dataset's underlying dynamics than on a simple corruption fidelity proportionality.
6 Discussions future topics with relevant literature
In this subsection, we further discuss about the four possible extensions of TimeAutoDiff in sequel: (1) Privacy guarantees; (2) Interpretability of generated time series data; (3) Extension to foundational model; (4) Bias from conditional metadata gencration.
(1) Privacy Guarantees is one of the main motivations of synthetic data. Specifically, in the time series domain, data from the healthcare and financial sectors is ubiquitous, but it often comes with significant privacy concerns. We hope the synthetic data docs not leak any private information of the original data, while preserving good fidelities. TimeAutoDiff lays the foundation for guaranteeing such privacy concerns with the generated synthetic data. In the vision domain, differential privacy guarantees (Dwork 2006) of synthetic images from diffusion-based models have been investigated by several researchers (Dockhorn al. 2022 Ghalebikesabi et al. 2023 Lyu al. 2023) Specifically, Lyu et al. (2023) studied DP- OP-guarantees of latent dilfuusion model by fine-tuning the attention module of noise predictor in their diffusion model, and claim their synthetic images both have good fidelities and DP uP-guarantees.
still not clear how the same idea can be applied to time series synthetic data (or regular tabular data), as differentially private time series data frequently challenging to interpret (Yoon) et al. 2020). In this regard, another privacy criterion, s-identifiability (Yoon al. 2020) (with e [0, 1) can be considered another alternative. The distance between synthetic and original data is measured through Euclidean distance, and want at least (1 p)-proportion of the synthetic data to be distinguishable (or different enough) from the original data. Under this criterion, we conjecture TimeAutoDiff can be exlended to the hesthe with a ((heoretically-prooall around several recent results on diffusion model (Zhang et al.) 2023c Bodin et al.) 2024). Zhang et al. (2023c) showed that there exist closed-form solutions of noise prodictors for every diffusion step of noisy training data points. This means that we can trace back the latent vectors (or matrix) where the original training data points are generated from. Recent findings (Bodin al. 2024 suggest that proper linear
Nonetheless,
privacy guarantees. The idea can be underpinned
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combination of data in the latent space can produce new semantically meaningful dataset in the original space. Combining the fact. that the mapping from the latent space to the original space is Lipschitz continuous (Zhang et al. 2023c) through deterministic sampling (probabilityflow),, we might be able to have controls over the generations of time series synthetic data, whose Euclidean distances from training data points are away from the training data points. This idea is naturally related to the diversity of generated data as well.
(2) Interpretability of the generated time series data is another crucial aspect that time series synthesizer should possess. In many practical applications, for instance, in financial sector, stakeholders and domain experts may be hesitant to rely on synthesis models that are difficult to interpret, as they need to understand and trust the model's behavior, especially when dealing with critical or high-risk scenarios. The current version of TimeAutoDiff does not have the luxury of generating interpretable results, but this can be easily adopted by following the previous works. Specifically, we want to point out readers TimeVAE (Desai ct al.] (021)) and Diffusion-TS (Yuan & Qiao] (023), which both focus on building a synthesizer with interprelability. Specifically, TimeVAE adopted a sophisticatedly designed decoder in VAB, which has trend, seasonality, and residual blocks for signal decompositions. Similarly, Diffusion-TS also design sophisticated decoder for the decomposition of signals into trend, seasonality, and residual, where they employ the latent diffusion framework. Both of these ideas can be directly employed in TimeAutoDiff, where the current decoder is set as an MLP block for simplicity.
(3) Extension to foundational model is another promising route the TimeAutoDiff can take. Recently, we have been seeing a wave of foundational models research on time series domain (Cao et al. 2024 Liu et al. 2024 Das et al. 2023 Yang et al.] 2024ah. These models can accommodate multiple tables from cross domains, enabling multiple time series tasks in one model; for instance, forecasting, anomaly detection, imputation, and synthetic data generation (See Cao et al. (2024).) Among them, Cao et al. (2024) devised cleverly designed masks, which provide the unifying framework to do the four abovementioned tasks under diffusion based framework. Nonctheloss, their mothods are confined to the continuous data modality, and not clear how the model can be exlended to heterogeneous features, leaving the great future opportunities for TimeAutoDiff be extended. We also conjecture the synthetic data from TimeAutoDiff can be beneficial to improving quality of forecasting foundation model i..,, see Section 5 in (Das al. 2023).
(4) Bias from conditional metadata generation: Generated data can indeed be biased with respect to conditional metadata, arising from various factors. Bias in the training data, such as inherent associations between metadata and outputs, may lead the model to replicate these biascs, for instance, generating disproportionately high traffic volumes for 'Clear" weather even when the true relationship is less deterministic. Imbalanced metadata distributions further exacerbate this issuc, as underrepresented conditions in the training set often result in less reliable outputs for those conditions, such as biased outcomes for minority demographic groups in healthcare datasets. Simplified assumptions in the model, such as assuming linear relationships between metadata and outputs, can overlook complex dependencies, producing data that fails to reflect the true conditional distribution. Noise injection, fcature of models like diffusion models and VAEs, can introduce additional bias if the noise interacts with metadata in unexpected ways, particularly for rare metadata values. Furthermore, limitations in conditional architectures, such as inadequate metadata encoding, can prevent the model from capturing nuanced dependencics, leading misaligned outputs. To mitigate such biases, ensuring balanced training data, employing robust metadata encoding techniques, applying regularization or fairness constraints, performing post-generation bias audits, and designing disentangled latent spaces are crucial steps. While conditional generative models aim to align generated data with metadata, addressing these biases is essential to ensure fairness and reliability.
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A Derivation of ELBO loss for log Pe tar Xcon Mtask
We assume the conditional density Pe (x Zaat xoon).
X Musek) == Pe (Xtar X Zfat, Mlask) Pe (ag
Then, negative loglikelihood (i.c., log Po (
X Missk) ) can be bounded by:
log Pe (x
X Mtask)
log
P X
X Zoat Mlask)
Po (Z)at
X dZz
4ol Laat
ttar X con
log
Pel X tar
X con
Laat M Po (Zhat ooou dZ
1o
X Xtar
[con)
<
(Z)mr Lat
X tar x con
log Pe (taa tar
X con
Lat M
dZzaat Lat
log 2 (Zfat
(
Xtar _
+
U6 0 Lat
X tar
c
dz Lat
Po Lat [xoon)
Egg.
log Pe.
X tar
X con
LLat
M task) }
DKL (gol Lat
x tar con
|| Po X
Enys
log Pa..
X tar
X
zbat,
M )
+DKI. (gol (Zba
x tar c
|| (0, ITFXTF)
LvAR
+ Eyg log Po (Z]at con
Eg
log. (0, IrFXTF)]
Coust.
B Datasets and Data Processing Steps
We used six single-sequence and two multi-sequence time-scries datasets for our experiments. The statistical information of datasets used in our experiments is in Table 6
Single-sequence: For single-sequence datasets, we segment the time series with N total timestamps into overlapping windows of length T and stride S, resulting in a tensor of shape (N T + S) x T F, where F denotes the number of features in the table. The specific values of T and vary across different experimental settings.
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Dataset
# of Rows
#-Cont.
# +-Disc.
Seq. Type
Pred Score Col.
Traffic
48205
3
5
Single
traffic volume
Pollution
43825
5
3
Single
1r
Hurricane
9937
4
4
Single
seasonal AH OT rv2
AirQuality
9358
1
12
Single
ETTh1
17431
7
0
Single
Energy
19736
27
Single
Bike Sharing
731
6
6
Single
N.a.N Is Fraud?
Card Transaction 20000
2
6
Multi
nasdaq100
18231
3
4
Multi
Industry
Table 6: Datasets used for experiments. The date time column is considered neither continuous nor calegorical. The 'Seq. Type' denoles the time series data lype: single- or multi-sequence data. The 'Pred Score Col denotes columns in each dataset used for measuring predictive scores.
Traffic (UCI) is a single-sequence, mixed-type time-serics datasct describing the hourly Minncapolis- St. Paul, MN traffic volume for Westbound I-94. The dataset includes weather features and holidays for evaluating their impacts traffic volume. (URL: https://archive.. www.:uiiiuuuuuueu edu/dataset/
Pollution (UCI) is single-sequence, mixed-type time-series dataset containing the PM2.5 data in Beijing between Jan 1st, 2010 to Dec 31st, 2014. (URL: https://arcciie.. icw.shhiieeeii....oo.m.m www.uuiii.uueee..c edu/dataset////////aaa 381 3
Hurricane (NHC) is a single sequence, mixed-type time-series dataset of the monthly sales revenue (2003-2020) for the tourism industry for all 67 counties of Florida which prone to annual hurricanes. This dataset is used as spatio-temporal benchmark dataset for forecasting extreme events and anomalics (Farhangi et al. 2023). (URL: httpt:::///// //www. nhc. noaa. gov/data/)
AirQuality (UCI) is single sequence, mixed-type time-series dataset containing the hourly averaged responses from gas multisensor device deployed on the field in an Italian city. (URL: https: //archive. iccs uci seah:/aataastt36//irrrgggg
ETTh1 (Github: Zhou al. (022)) is single sequence, continuous only time-series dataset, recording hourly level ETT (ie., Electricity Transformer Temperature), which is crucial indicator in the electric power long-term deployment. Specifically, the datasct combines short-term and long-term periodical patterns, long-term trends, and many irregular patterns. (URL: https://githi: com/ shoothoyy/EEDDaaasttttreee////
Energy (Kaggle) is single sequence lime-series datasel. The dataset, spanning 4.5 months, includes 10-minute interval data on house temperature and humidity via a ZigBee sensor network, energy data from m-bus meters, and weather data from Chievres Airport, Belgium, with two random variables added for regression model testing. (URL: https://wwz www..kaggle.cm com/ code/ gaganmaahi http://ppppaeeeeeeee-rttttte-eee
Bike Sharing (UCT) is a single sequence time-series dataset collected from a bike rental service with automated stations. contains hourly and daily rental records over two years, including information on rental counts, weather conditions, seasonality, and timestamps. Each entry includes normalized temperature, humidity, wind speed, and metadata like holiday soldday/wekendd /weekend indicators, making it suitable for mobility modeling and demand forecasting tasks. (URL: http: httt:://archive.. wus.uii..i...................... edu/m1/ In our experiment, we drop "instant', 'yr', and 'mnth" columns.
The sequences the multi-soquence data vary in length from one entity to another, so we selected entities with sequences longer than T = 200 and 177 and truncated them to a uniform length of T for the 'card transaction" and 'nasdaq100' datasets.
Multi-scquence:
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Card Transaction is multi-sequence, synthetic mixed-type time-series dataset created by Padhi et al. (2021a) using rule-based generator to simulate real-world credit card transac- tions. We selected 100 users (i.e., entities) for our experiment. In the dataset, we choose ('Cand', 'Amount', 'Use Chip', 'Merchant', 'MCC', 'Errors?', 'Is Fraud?? as features for the ex- periment. (URL: httts:://gtthnni httpggithhh....o///aaaaaaaeeee com http///////aaaoorr///eeeee/eeee
nasdaq100 is a multi-sqquence,, mixod-type time-series dataset consisting of stock prices of 103 corporations (i.e., entities) under nasdaq 100 and the index value of nasdaq 100. This data covers the period from July 26, 2016 April 28, 2017, in total 191 days. (URL: /https://csew/// uesd eduu~~yag00//MMDDQQ100 _ttpaio000sssstoeeeoee data. html)
C Denoising Diffusion Probabilistic Model
Ho et al. (2020) proposes the denoising diffiusion probabilistic model (DDPM) which gradually adds fixed Gaussian noise to the observed data point Xy via known variance scales &n E (0,1), n E [I....,N] at the diffusion step n. This process is referred as forward process the diffusion model, perturbing the data point and defining a sequence of noisy data X1, X2,..... XN:
N
((xn 11= = (xn5 V 1, ((1N: xo) = II g(xn Xn-1)-
Since the transition kernel is Gaussian, the conditional probability of the x,, given its original observation Xy can be succinctly written as:
(x,, xo) = -xx,, Vān (1 a. )I),
where a, = 1 - 3n and an = 1 ag. Setting 3n to be an increasing sequence, for large enough N, leads xN to the isotropic Gaussian.
Training objective of DDPM is to maximize the evidence lower bound (in short ELBO) of the log-likelihood Ex [log Po [xo)] as follows;
N
Eg log (x x1) (xL( xo) p(xN)) - Σ DKL(g(xn-11 n=1
Xa, xo) || po xx.1)]
The first two terms in the expectation are constants, and the third KL-divergence term needs to be controlled. Interestingly, the conditional probability q(xn-1 Xn, xn) can be driven the closed-form solution:
a(an11 Xn, xo) = -V( Xn-1 1 On
Van-18p
+ Van(1 - -On-1) 1 an-1 BnI
Xn:
On
On
Noticing the covariance is constant matrix and KL-divergence between two Gaussians has closed-form solution; DDPM models px(x.-11 x,) == (xxx-1 (x,,,,, n), 1-an-1 parameterized by a neural network.
I-ā, B,I). The mean vector (o n) is
The trick used (Ho et al. 2020) is to reparameterize po(xn, n) terms of e cx(x,, n) where it predicts the noise addod to X,, from Xg. (Note that X,, Vo + vI ane with e e NO ().)
Given this, the final loss function DDPM wants minimize is:
Lait := En,e [Veexva, + vI ane,n) nn) -E|13
where the expectation is taken over ~ N(O, I) and no Uni
The generative model learns the reverse process. To generate new data from the Iearned distribution, the first step is sample a point from the casy-to-sample distribution x ~ (O, and then iteratively denoise (xN XN-1 xo) it using the above model.
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D Comparison Table of TimeAutoDiff with current literature
D.1 Baseline models for Unconditional Generation
Table compares TimeAutoDiff with other time series synthesizers in the literature under seven different aspects. Additionally, we provide further detailed comparisons between our model and Diffusion-Ti --T Yuan
& Qiao 2023 / TimeDiff (Tian et al. 2023), and TabSyn (Zhang et al. 2023b). AutoDiff (Suh et al. 2023
TabDDPM Kotehnikov et al. 2022).
Diffusion-TS (Yuan & Qiao, 2023) employs an Autoencoder + DDPM framework for interpretable time-series generation. The primary difference from our work lies in the problem setting. Diffusion-IIS assumnes the signal consists only of contimuous time scries, based on the premise that the signal is decomposable into 'trend, seasonality, and noise'. While this decomppsitiio--bssed approach is effective for continuous data, it is not well-defined for the heterogeneous features (especially discrete variables) that TimeAutoDiff addresses, making it difficult to apply directly to our problem.
Dife (Tian et al. 2023 also handles heterogeneous features in EHR datasets, but does so by integrating two distinct diffusion processes: applying DDPM for continuous variables and multinomial diffusion for discrete variables, respectively. This stands in sharp contrast to our architectural philosophy. TimeAutoDiff leverages VAE to first project the entire heterogeneous time series into a single, continuous latent space. We then simplify the modeling by using only single DDPM framework within this unified latent representation.
Our work is also partially inspired by recent. successes in labular data generation models like TabSyn Zhang it al. 2023b), AutoDiff Suh et al. (2023), and TabDDPM Kotelnikov et al. (002)). However, these models are fundamentally designed for static, ID tabular data, treating each row as an independent and identically distributed (i.i.d.) data point. This i..d. assumption is invalid for time-series data, which inherently 2D (time and feature axes) and defined by temporal dependencies. That is, these tabular methodologies do not. model the sequential relationships between rows (time steps), which is the key distinction TimeAutoDiff aims to address.
D.2 Baseline models for Imputation & Forecasting
Time-LLM (Jin et al. 2023): Time-LLM reframes numerical forecasting as a language-modele problem. Numeric sequences are first segmented into short patches and mapped to bank of text prototypes, creating token stream that frozen LLM can ingest. A natural-language Prompt-as-Prefix (PaP) explains how these synthelic tokens relate to time-series patterns and what forecast is required. The LLM reasons over this hybrid input without any weight updates; its token outputs are then linearly projected back to real values to obtain the prediction. With just the lightweight prototype mnooer//ecoode trained, Time-LLM matchos or surpasses specialist forecasting architectures and remains elfective in zero-shot and few-shot settings.
MOMENT (Goswami et al., 2024): : MOMENT is a family of large-scale foundation models for time-series analysis, designed to serve as general-purposs models across diverse tasks. Each MOMENT model is a high-capacity Transformer architecture trained the Time-series-Pile
massive repository of time-series data collected from many domains (e.g. healthcare, finance, climate). The training objective is self-supervised masked time-series prediction task, wherein the model learns to reconstruct missing portions of time-series inputs, analogous to masked language modeling but for continuous sequences. Through this pre-training, MOMENT learns rich representations of temporal patterns that can be adapted to various tasks. A single pro-trained MOMENT model can be used out-of-tho-box for forecasting, classification, anomaly detection, imputation and more, often in zero-shot or few-shot manner, while also allowing fine-tuning for further gains. Experiments show that these foundation models perform effectively with minimal task-specific data, and the authors provide a benchmark to evaluate such models' performance across multiple time-sories tasks.
TEFN (Zhan et al., 2024): The Time Evidence Fusion Network explicitly fuses two complementary evidential views of a multivariate series. First, a Basic Probability Assignment (BPA) module grounded in Rempster-Shafor theory thcory-assesses assesses the contribution and uncertainty of patterns within cach channel (spatial view) and across time (temporal view). A dedicated fusion layer then combines these sources of evidence into unified latent representation, enabling the model to exploit cross-variable interactions and temporal
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dynamics coherently. This design delivers state-of-the-art long-horizon accuracy while remaining lightweight and remarkably robust to hyper-parameter choices.
TimeMixer (Wang et al., 2024b): TimeMixer views forecasting through a multi-scale lens. Its (PDM) block decomposes the history into frequency bands (e.g. seasonal vs. trend) and mixes them bidirectionally from fine-tcoarse and coarse-fiine scales-so so that microscopic and macroscopic information reinforce each other. The Maltuur-Muliipredctto---iiii several scale-specialized heads, blending their predictions into the final forecast. Built entirely from MLP layers, TimeMixer attains SOTA accuracy on both short- and long-term tasks while keeping computational complexity linear in sequence length.
Past- Sastt-Dcoomosabbe-Miiiee
(FMM) block then ensembles
TimesNet (Wu et al., 2022): TimesNet converts a 1-D sequence inlo a 2-D lensor to isolate periodic structure. Dominant periods are detected, and the series reshaped so that rows index successive periods and columns index positions within each period-turning intra-period variations into column patterns and inter-period variations into row patterns. A parameter-effiient TimesBlock (inception-style 2-D convolution) then captures these variations jointly. This single backbone achieves competitive superior results in forecasting, imputation, classification, and anomaly detection, illustrating the power of modeling temporal data in the 2-D variation domain.
MICN (Wang et al., 2023a):: The Multi-scale Isometric Convolution Network (i.e., MICN) combines the locality of CNNs with global context modeling at linear cost. Parallel branches process progressively down-sampled versions of the input, and each branch applies (i) standard convolutions for short-range patterns and (ii) isometric dilated convolutions whose receptive field spans the entire (down-sampled) sequence. Outputs from all branches are concatenated to form the forecast, yiclding substantial improvements-1177 (multivariate) and 22% (univariale) error reductions over previous best models.
DLinear (Zeng et al., 2023): Decomposition- Linear replaces deep networks with a two-step linear pipeline. First, a simple moving-average filter separates each scries into trend and scasonal components. Sccond, two independent one-layer linear projections extrapolate these parts and their outputs are summed to oblain the forecast. Despite its minimal complexity, this seasonal trend decomposition plus linear extrapolation matches surpassos statc-of-tho-art Transformer models long-horizon benchmarks, showing that carcfully chosen inductive bias can rival far more elaborate architectures.
FiLM (Zhou et al., 2022): : The Frequency- Improveed Legendre Memory module augments Legendre polynomial projections with proces-poocessigg tricks to preserve long-range history. A Legendre basis encodes the entire past into a compact state; a learnable Fourier projector filters out high-frequence noise; and a low-rank approximation accelerates computation. FiLM can operate as a standalone predictor or as plug-in memory block, boosting long-horizon accuracy by 20% when inserted into existing architectures.
CSDI TTshiro et al., 2021): : Conditional Score-based Diffusion Imputation frames gap-filling conditional
generation. A diffusion process iteratively perturbs the full sequence, while neural score conditccndii on the observed gutries-guides reverse denoising to sample plausible completions. This yields a distribution over imputations, enabling uncertainty quantification, deterministic estimates are obtained by averaging or mode-seeking. Across healthcare and environmental datasets, CSDI reduces by 40-65% relative to previous methods, and > >10% even deterministic settings.
errors
PStchTST (Nie et al., 2022): : PST adapis Transformers for forecasting by treating non-overlapping time-series segments (patches) analogous to NLP tokens. Input is segmented, flattened, and linearly embedded, with learnable positional embeddings added. This patching significantly reduces the input sequence length for a standard Transformer encoder, allowing attention to efficiently model dependencies between patches rather than individual points. This simple mechanism allows a "vanilla" Transformer to achieve state-of-the-art long-term forecasting performance by capturing local semantic information within each patch.
+imeMixer++ (Wagg et al., 2024a): TimeMixer++ lightweight, attention- free MLP architecture based 'Decomposable Multiscale Mixing: It explicitly factorizes the time-series into components, using specialized MLPs to model 'Past' (recent variations), 'Seasonal' (periodicity), and 'Trend' (long-term tendencies) patterns separately. These representations are efficiently processed and mixed using shared MLPs,
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achieving linear complexity. This interpretable decomposition matches exceeds Transformer performance while being significantly more computationally efficient.
iTransformer (Liu et al., 2023): i'Transfrmerr inverts the standard Transformer's application for mul- tivariate forecasting. Instead of applying attention across time, it treats each entire variate (channel) as a single token by embedding its whole time series. standard Transformer encoder then applies self-attention across these variate-tokens, explicitly modeling the complex interdependencies between channels (e.g., how "temperature" relates to "humidity"). The resulting variate representations are then projected by simple lincar layers to generate the forecast, setting new, simple state-of-the-ars baseline.
D.3 Comparison with SereeS Foundation Models (TSFMs)
Our work, TimeAutoDiff, is generative framework for heterogeneous time series, and its design prin- ciples differ significantly from the emerging paradigm of Time-Serics Foundation Models (TSFMs) such MOMENT (Goswari et al. 2024), Chronos (Ansari et. al. 2024), and TimesFM (Das el al. 2023). These large-scale models focus on pre-training, often using tokenization strategies on massive, mostly continuous datasets to learn powerful, general-purpose representations for zero-shot or few-shot forccasting.
The primary distinction lies in data and lask specialization. TimeAutoDiff is fundamentally designed for the native handling of heterogeneous features. We employ a VAE to project a mix of continuous, discrete, and static data into a single continuous latent space. Our latent-diffusion process then models this unified representation directly. In contrast, many TSFMs are primarily demonstrated on (and architected for) large, homogenous datasets of continuous values, with the handling of mixed data types not being their central focus. Furthermore, our mask-based task unification represents a different objective. While TSFMs aim to create a single, powerful representation for downstream adaptation (primarily forccasting), TimeAutoDiff acts as a unified generative model for a diverse set of tasks including imputation, forecasting, and synthetic data generation- controlled via flexible masking mechanism. These approaches may be viewed as complementary rather than competing. TSFMs excel at learning broad temporal patterns from large-scale data, whereas TimeAutoDiff provides high-fidelity, specialized generative framework for complex, heterogeneous datasets where tasks beyond forecasting are critical.
E
Evaluation Metric
For the quantitative evaluation of synthesized data, mainly focus on three criteria (1) the distributional similarities of the two tables; (2) the usefulness for predictive purposes; (3) the temporal and feature dependencies; We employ the following evaluation metrics:
Discriminative Score (Yoon et al. 2019) mcasures the fidelity of synthetic time series data to original data, by training a classification model (optimizing a 2-layer LSTM) to distinguish between sequences from the original and generated datasets.
Predictive Score (Yoon et al. 2019) measures the utility of generated sequences by training a posthoc sequence prediction model (optimizing a 2-layer LSTM) to predict next-step temporal vectors under a Truin-on-Syntheeiic- Cenetbhetii--et--rree Rn-all (TSTR) framework.
Temporal Discriminative Score measures the similarity of distributions of inter-row differences between generated and original sequential data. This metric is designed to see if the generated data preserves the temporal dependencies of the original data. For any fixed integer E f1...., 1}, the difference of the 1-th row and (n + ()-th row in the table over (I...., 1} is computed for both generated and original data and discriminative score (Yoon et al. 2019 is computed over the differenced matrices from original and synthetic data. We average discriminative scores over 10 randomly selected t E (1....,T -
Feature Correlation Score measures the averaged [."ddisaace of correlation matrices computed on real and synthetic data. Following (Kotelnikov et al. 2022), to compute the correlation matrices, we use the Pearson correlation coefficient for numerical-numerics. fcature relationships, Theil's U statistics between categorical-eattegriia features, and the correlation ratio for categorical numerical features. We use the following metrics to calculate the feature correlation score:
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Pearson Correlation Coefficient:
Used for Numerical to Numerical feature relationship. Pearson's 's Correlation Coeflicient is given by
E - Zy - 11 EE
where
I' and y are samples in features X and Y, respectively
H and y are the sample means in features X and Y, respectively
Theil's Coefficient: Used for Categorical to Categorical feature relationship. Theil's Coefficient is given by
U = 2(X) - ((X
H(X)
where
entropy of feature X is defined as
H(X) =- Σ Px(x)) (x
entropy of feature conditioned on feature Y defined as
H(XIY) = = Px,r v) log Px,((2,)) Pr(w))
2,8
Px and Py are empirical PMF of X and Y, respectively Px, is the joint distribution of X and Y
Correlation Ratio: Used for Categorical to Numerical feature relationship. The correlation ratio 7) is given by
nh (Nx= 77) 11 En,( (Vars 87)
where
n2 is the number of observations of label in the categorical feature
Ysi is the i-th observation of the numerical feature with label x
Du is the mean of observed samples E Y with label
y is the sample mean of
F
Model parameter settings, Training & Hyper-parameter choices
Our model consists of two components: VAE and DDPM. We present the sizes of networks in both components that are applied entirely across the experiments in the paper.
VAE-Encoder Dimension of first FC-layer in MLP-block for encoded features:
(Num of disc var. x x128 x128+112mm Num of cont var.x16) 128,
Dimension of sccond FC-layer in MLPP-blokk for encoded features: F,
Dimension of hidden layer for the 2-RNNs for μ and 200,
Number of layers for the 2-RNNs for μ and a: 2,
Dimension of fully-connected layer topped 2-RNNs: 200 F
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Method
Disc. Score
Pred. Score
Temp. Disc Score
Feat. Correl.
TutoAut
0.015(0.012)
(.229(0.011)
0.03/(0.020)
0.003(0.000)
Latent Feature Dimension F/2
0. 0.009
0.227(0.009)
0.096)0.061)
0.055(0.000
Latent Feature Dimension = F/4
0.038(0.021)
0.233(0.007)
0.099
0.018(0.000)
Diffusion Steps 75
0.016(0.009)
0.224(0.015)
(.014(0.009)
0.039(0.000)
Diffusion Steps 50
(0.019)
(.203)0.000)
(.096)
(0.00)
Diffusion Steps 25
0.150
0. 0.248 (06)
0.111 065)
0.100(0.000)
VAE Training 15000
0.075(0.009)
(.243(0.0005
0.037(0.007)
0.091 (0.000)
VAE Training 10000
0.008))
0.242(0.007
0.038(0.038)
0.050 000)
VAE Training = 5000
(.195(0.025)
0.245
0.039(0.019)
(0.07().000)
DDPM Training 15000
0.098(0.014)
0.237(0.015)
0.062(0.038)
0.086(0.000) (000
DDPM Training = 10000
0.220(..025)
0.246(0.001)
(0.055)
0.195(0.000)
DDPM Training = 5000
0. (0.02))
O. (0205)
0.216(0.031)
0.190 (0.000
Midden Dimension of RNNS (VAE) 150
0.013(0.008)
0.240(0.000)
0.031(0.009)
0.015
Hidden Dimension of RNNs (VAE) = 100
0.030
O. 0.236(0.017)
0.017(0.011)
0.039(0.000 (0.)00)
Iidden Dimension of RNNs (VAE) 50
0.082(0.023)
(0208)
0.051
0.0641
Hidden Dimension of Bi- RNNs (DDPM)
(0.00
0.243(0.011)
(0.033)
(0.000)
Iiiden Dimension of Bi-RNNs (DDPM) 100
0.167(0.012)
0.248(0.000)
0.094(0.054)
0.119(0.000)
Hidden Dimension of Bi-R.NNs (DDPM) = 50
0.174(0.014)
(.251(0.005)
0.157(0.072)
0.132 (0.000)
Number of layers in RNNs (VAE) =
0.024(0.013)
0.245(0.009)
0.042(0.018)
0.028
Number of layers in Bi-RNNs (DDPM) = =
(.097(0.009)
0.250(0.002)
(0.09)
0.086(0.000)
Quadratic Noise Scheduler
0.109(0.017)
0.234(0.001)
0.072(
0.106(0.000)
Table 7: Performances measured with various choices of hyppr-ppaaaneee in TimeAutoDiff. The experiments are conducted on Traffic dataset with T = 24.
Diiennsoo of first FC-layer in MLP-block for latent matrix Zo: F x 128,
Dimension of second FC-layer in MLP-block for latent matrix Zo: 128 x 128)
DDPM
(Output dimensions of encodings of (ZLat, n,, t, ts): 200,
Dimension of hidden layer for the Bi-RNNs: 200,
Number of layers for the Bi-RNNs: 2,
Dimension of FC-layer of the output of Bi-RNNs: 400 x F,
Diffusion Steps: 100}
Training for both the VAE and DDPM models is set to 25,000 epochs. The batch size for VAE training is 100, while the batch size for DDPM training matches the number of diffiusion steps. We use the Adam optimizer, with a learning rate of x 10-4 docaying to 10-6 for the VAE, and lcarning rate of 10-3 3 for the DDPM. For stabilization of diffusion training, we employ Exponential Moving Average (EMA) with decay rate 0.995. We employ linear noise scheduling for Pn ;= 1 an, n E (1,2,...,. N) with 31 10-4 and BN 0.2:
Bn = 1
8
n2 N
n B
NN.
N
In the following, we investigate the robustness of our models to the various hyper-parameete choices in VAE and DDPM. Specifically, we studied the effects of (1) feature dimension of Zt (F/2, F/4), (2) number of diffusion steps (75, 50, 25), (3) training epochs of VAE and DDPM (20000, 15000, 10000), (4) dimension of hidden layers of two RNNs (for μ and o) in VAE (150, 100, 50), (5) dimension of hidden layers of Bi-RNNs in
DDPM (150, 100, 50), (6) the number of layers of two RNNs (for f and σ) in VAE (1), (7) the number of
layers of Bi-RNNs in DDPM (1). (8) the quadratic noise scheduler used in Song et al. 2020a Tashiro et al. 2021
Bn ((1-
2
V B1
3N
N
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with the minimum noise level $1 0.0001, and the maximum noise level By 0.5.
The experiments are conducted over the varying parameters (in the paranthesis), while the remaining parameters in the model are being fixed as in the above settings. The first 2000 rows of Traffic data are used for the experiments with sequence length 24. (i.e., the dimensions of tensors used in the experiments are [B, T, FJ = (1,77,24,))
Results Interpretations: Table presents the performance of the models across four metrics, with variations in hyperparameter settings. Overall, larger models yield better results. Reducing the diffusion steps, dimensions, and the number of hidden layers in RNNs within the VAE and Bi-RNN components of DDPM significantly degrades model performance. Longer training of both VAE and DDPM consistently enhances results. The linear noise scheduler outperforms the quadratic noise scheduler. While reducing the feature dimension to F/2 slightly improves discriminative and temporal discriminative scores, further compression F/4 leads to information loss during signal reconstruction, resulting in poorer performance.
G Computational Complexity of TimeAutoDiff
Let T' denote the sequence length, F the number of original features, L the latent feature dimension after VAE encoding, H the hidden size of the encoder RNN, h the hidden size of the decoder MLP, Hait the hidden size of the diffusion denoiser, and N the number of diffusion steps. All Big-O expressions computed per sequence (X1,...,xp XT) and scale lincarly with batch sizc.
1. Encoder. The encoder consists an embedding layer, an MLP, two RNNs (for j and log and a projection layer. A single RNN forward pass costs (I1 + HF)), while the projection adds H(HHL), which is typically negligible. Thus,
Costenc O(T + HF)).
The overall encoder cost grows linearly with T and F, dominated by the recurrent term H
2. Decoder. The decoder reconstructs heterogeneous outputs from the latent sequence Zf R through two-layer MLP followed by separate output heads (binary, categorical, and numerical):
Costaee OLh + +
This expression accounts for latent-to-hidden mapping, internal MLP propagation, and multi-hear output projections.
3. Diffusion Model. The diffusion denoiser operates in the latent space with input dimension L. Fach step involves RNN or MLP blocks with cost
O HaiaL))
During training, only one random timestep is used per iteration:
Costaiff.rain (H + Ham
During inference, N iterative denoising steps are performed:
Costaif.nter (NT + H)m
After denoising, the decoder is applied once to recover full-resolution sequences.
4. Overall Training and Inference. Combining all components, the total training cost per sequence is
Costtrain O + HF)) O(T + hF)) + O(T + Hain
The inference cost includes the N-step denoising and one decoder call:
Costinfer OHN + L)) O 122 hF)).
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5. Discussion. TimeAutoDiff achieves efficient generation by: (1) performing diffusion in compressed latent space (T F), and (2) adopting a moderate number of diffusion steps (N R 50-100). These design choices reduce inference time by approximately (0-100x compared to data-space diffussin models (Na 2 1000), while offering improved fidelity and flexibility relative to single-step GANs.
H Volatility and Moving Average: comparison between real and synthetic under stock data
We provide the performance of model in terms of volatility and moving average. We first provide the brief descriptions Simple Moving Average, Exponential Moving Average, and Volatility.
Simple Moving Average (SMA): The Simple Moving Average (SMA) is computed as the arithmetic mean of values over a sliding window of size w = 5. For a given time step t, the SMA is given by:
1 SMA 5
Σ
Value;
where Value, represents the value of the time scries at time i. This metric smooths short-term fluctuations and highlights the overall trend by averaging values in the specified window.
Exponential Moving Average (EMA) The Exponential Moving Average (EMA) is a weighted average of values where recent data points have exponentially greater weight. For a window size of w = 5, the smoothing
factor a is computed as: a w+1 5+1 3. The EMA at time t is then computed recursively as:
2
EMA, a Value, + (1 a1) EMA.-1
where Value, is the current value of the time series, and EMA,-1 is the EMA from the previous time step. This method emphasizes recent changes while retaining some information from the historical trend.
Volatility Volatility measures the degree of variation in the time series over a sliding window of size = 5. It is calculated as the rolling standard deviation of the percentage changes (returns). First, the percentage change (return) between consecutive values is computed as:
Return; Value; Value;11
Value;-1
For a given time step t, the volatility over the window w = 5 given by: Volatility, El (Return,- Return)² where Return is the mean of the returns within the window.
Results: We work on the stock data. The figure 14 provide clear side-by-side comparison between the synthotic and real data, with the left column displaying the synthetic data and the right column showcasing the corresponding real data for two selected features (Open & Close prices) over 200 timestamps (1..,TT200)). Fach row focuses on one feature, allowing for detailed examination of the behavior across key metrics: Simple Moving Average (SMA), Exponential Moving Average (EMA), and Volatility, The SMA and EMA curves, plotted alongside the raw time series data, highlight the ability of the syuthetic data to replicate the long-term trends (SMA) and short-term responsiveness (EMA) observed in the real data. Volatility, overlaid as a secondary y-axis in each plot, demonstrates the synthetic data's capacity to reproduce the temporal variability, including periods of high and low uncertainly, as reflected in the real data. The remarkable alignment across all metrics suggests that the synthetic data closely mirrors the real data's dynamics, effectively capturing both the overall patterns and nuanced fluctuations. This visual comparison underscores the robust and reliability of the synthetic data generation process.
Maximum Mean Discrepancy & Entropy
We used two metrics proposed by TSGM (Nikitin et al. 2023): : Maximum Mean Discrepancy (MMD) and Entropy: MMD measures the similarity (or fidelity) between hette and real time series data, while Entropy
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Syrth Feeture
Beal Feet
54. 047
221
230
666
0.10
1800
0.05
045
0.00
044
Syrtp
0.50
2y-t Tustunc
Fool Foukett
0.0
0.00
030
0.5
0.10
045
0.3
2.00
11
Figure 14: Comparison of synthetic (left) and real (right) data across two features, illustrating alignment in trends (SMA, EMA) and variability (Volatility: secondary y-axis).
assesses the diversity of the synthetic data. The results are summarized in Table S and are consistent with those in Table _ TimeAutoDiff achieves the lowest MMD scores across all four datasets, aligning with the discriminative scores reported in Table 2 This indicates that TimeAutoDiff effectively generates synthetic data that closely resembles real data. For diversity, higher Entropy values indicate dataset with more diverse samples. However, noted in (Nikitin et al.) 2023), Entropy should be considered alongside other metrics, as random noise can also result in high Entropy values. TimeAutoDiff produces synthetic data with higher Entropy values than the real data, though not as excessively as other baseline models. This suggests that our model generates synthetic data that preserves the statistical properties of the original data, maintaining diversity without introducing excessive deviation.
J Applicability to High-Dimensional Data
To demonstrate the effectiveness of feature compression in the latent space, we evaluate our model on high-dimensional synthetic datascts.
Sine Waves. We generate multivariate sinusoidal sequences with varying frequencies n and phases 8, providing continuous, periodic, and mutually independent features. For each dimension i E {1, F}. the signal is defined as (t) = sin(2 + 0), where Unif(0, 1] and 0 ~ Uniff- n, T). The model is trained on data with shape [Batch Size x Seq Len x Feature Dim], and samples are drawn with the same configuration. We record the training time (for both VAE and diffusion models), sampling time, GPU memory usage during sampling (in MB), discriminative score, and temporal discriminative score. Under the configurations described in Appendix F TimeAutoDiff successfully generates sequences of length 900 with features, achieving high fidelity (see Table g). However, when the number of features increases (e.g., from 30 to 50) while keeping the sequence length fixed at 200, performance noticeably degrades if the latent feature dimension remains the same as the original input. To mitigate this, we reduce the latent feature dimension to F/2, which significantly improves generation quality in high high-dimensional settings.
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Metric
Method
Traffic
Pollution
Hurricane
AirQuality
TimeAutoDiff
0.000629
0.000895
0.000891
0.001531
TimeGAN
0.001738
0.009791
0.002775
0.042986
MMD Score
DoppelGANer 0.000644
0.000960
0.0054189
0.017038
Diffusion-TS
0.005099
0.037102
0.078387
0.004144
(The lower, the better)
TSGM
0.001484
0.006322
0.031971
0.013777
real vs. real
0.000000
0.000000
0.000000
0,000000
TimeAutoDiff
6419.404
8472.642
7129.152
16570.016
Eutropy Score
TimeGAN 6714.156
11021.597
7804.343
15343.967
DoppelGANer
3941.083
8656.403
6946.678
8708.616
(Neods to be considered with other metrics)
Diffusion-TS
9763.042
7372.591
9861.151
15934.365
TSGM
11899.222 225
11854.764
6535.306
15766.673
Real
5983.576
6976.253
6613.284
14952.996
Table 8: Maximum Mean Discrepancy (MMD) and Entropy of TimeAutoDiff, TimeGAN, DoppelGANer, Diffusion-TS, TSGM and Real data.
Batch Size
Seq Len
VAE
Diff
Sampling
GPU Mem
Disc Scr
Temp Disc Scr
500
100
187.23
94.32
1.294
910.47
0.067 (0.034)
0.143 (0.114)
400
300
420.23
201.47
3.585
1991.75
0.040 (0.023)
0.064 (0.059)
300
500
665.69
315.92
5.511
2572.63
0.032 (0.016)
0.078 (0.078)
200
700
928.83
415.36
7.303
2466.91
0.048 (0.016)
0.193 (0.122)
100
900
1209.34
530.36
8.499
1670.75
0.16 (0.094)
0.13 (0.143)
Table 9: The number of feature is fixed as 5. The sequence length increases up to 900.
Batch Size
Feat Dim
VAE
Diff
Sampling
GPU Mem
Disc Scr
Temp Disc Scr
800
10
128.11
355.03
5.36
1696.21 21
0.24 (0.08)
0.26 (0.09)
800
20
132.48
359.32
4.12
5080.84
0.26 (0.05)
0.38 (0.08)
800
30
134.02
371.38
3.99
5540.96
0.31 (0.08)
0.33 (0.17)
800
40
134.72
364.85
3.97
6003.00
0.39 (0.14)
0.41 (0.14)
800
50
135.95
374.61
5.35
6464.41
0.48 (0.02)
0.49 (0.00)
Table 10: The sequence length is fixed as 200. The feature dimension increases up 50.
Batch Size
Feat Dim
VAE
Diff
Sampling
GPU Mem
Disc Scr
Temp Disc Scr
800
10
131.65
365.81
4.63
3288.57
0.20 (0.12)
0.23 (0.14)
800
20
128.34
344.86
4.53
3947.75
0.25 (0.13)
0.29 (0.09)
800
30
130.92
363.41
4.61
4358.76
0.17 (0.11)
0.34 (0.14)
800
40
132.03
359.15
4.58
4771.51
0.24 (0.19)
0.38 (0.09)
800
50
134.96
367.05
4.70
5185.07
0.32 (0.18)
0.41 (0.10)
Table 11: Same setting with Table
10
but the dimension of latent matrix is set 200 x F/2.
40

---

Published in Transactions on Machine Learning Research (12/2025)
K Experimental details of Imputation and Forecasting Tasks
Imputation (MCAR)
Forecasting
(w
Model
Metric
Bike Sharing
Traffic
Pollution
AirQuality
Bike Sharing
Traffic
Pollution
AirQuality
MAE
0.4620
0.0481
0.0665
0.0442
0.0186
0.5202
0.1457
0.0541
TimeAutoDiff
MSE
0.0996
0.0118
0.0110
0.0007
0.052
1.7045
0.1249
0.0097
MAE
0.3080
0.3034
0.1184
0.0524
0.1554
0.5196
0.1528
0.0661
TimeLLM
MSE
0.0413
0.0721
0.0125
0.0008
0.0800
2.1470
0.1765
0.0221
MAE
0.6015
4.7193
1.2633
0.7330
N.A.
N.A
N.A.
N.A.
Noment
MSE
0.1150
2.8289
0.2468
0.0609
N.A.
N.A.
N.A.
N.A.
MAE
0.3764
0.3131
0.1238
0.0225
0.1493
0.5333
0.1466
0.0735
TEFN
MSE
2.0219
10.6461
0.8955
1.1066
0.0641
1.9177
0.1 481485
0.0194
MAE
0.2052
0.2421
0.1128
0.0451
0.0809
0.5455
0.1576
0.0887
TimeMixer
MSE
0.0117
0.0598
0.0122
0,000S
0.0171
1.9208
0,1483
0.0243
MAE
0.2184
0.2786
0.2263
0.0494
0.1352
0.5473
0.1446
0.0766
TimesNet
MSE
0.0137
0.0592
0.0341
0.0007
0.0529
1.9260
0.1451
0.0189
MAE
1818
0.1813
0.0968
0.0407
0.1249
0.5593
0. 0.14
0.0700
MSE
0.0086
0.0434
0.0116
0,0006
0.0538
2.2501
0,2036
0.0251
MAE
0.2897
0.2945
0.1428
0.1096
0.1358
0.5215
0.1449
0.0695
DLinear
MSE
0.0317
0.0731
0.0121
0.0022
0.0659
1.9290
0.1549
0.0188
MAE
0.3944
0.3043
0.1233
0.0497
U.3135
0.5382
0.1451
0.0800
MSE
0.0652
0.0736
0.0126
0,000S
0.4062
1.9271
0.1495
0.0213
MAE
8.5081
0.0899
0.0907
0.0186
0.1339
0.5799
0.1603
0.0750
CSDI
MSE
59.2891
0.0262
0.0174
0.0004
0.1008
3.6400
0.2764
0.0302
MAE
0.4857
0.1175
0.0988
0.0200
0.5217
1.6593
0.2428
0.2850
PatchTST
MSE
0.0825
0.0251
0.0113
0.0003
0.5706
8.1854
0.2552
0.0501
MAE
0.1634
0.2672
0.1022
0.0437
0.8169
2.5419
0.4112
0.6202
iTransformer
MSE
0.0086
0.0609
0.0127
0.0007
0.9061
12.4538
0.3874
0,2580
MAE
0.6939
0.7246
0.2469
0.1749
0.5968
2.3055
0.3815
0.5198
TimeMixerPP
MSE
0.1680
0.2450
0.0243
0.0067
0.4632
11.2804
0.3801
0.1865
Table 12: Imputation (MCAR) and mid-horizon forccasting (w=24). TimeAutoDiff delivers strong
overall performance, achieving the best MAE/MSE on multiple datasets in both tasks, particularly on Traffic and Pollution for imputation and on Bike Sharing, Traffic, Pollution, and AirQualily for forecasting. Bold indicates the best score.
41

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Published in Transactions on Machine Learning Research (12/2025)
Forecasting
(w=12)
Forecasting
(w=36)
Model
Metric
Bike Sharing
Traflic
Pollution
AirQuality
Bike Sharing
Traflic
Pollution
AirQuality
MAE
0.0103
0.3942
0.1458
0.0689
0.0159
0.2544
0.1317
0.0156
TimeAutoDiff
MSE
0.0027
1.3371
0.1610
0.0221
0.0014
0.2223
0.0558
0.0004
MAE
NaN
NaN
NaN
NaN
0.1546
0.2785
0.1429
0.0655
TimeLLM
MSE
NaN
NaN
NaN
NaN
0.0823
0.6198
NaN
0.0190
MAE
0.1537
0.3725
0.1377
0.0795
0,1330
0,2776
0,1381
0.0652
TEFN
MSE
0.0724
0.9662
0.1186
0.0207
0.0640
0.5744
0.1390
0.0167
MAE
0.1430
0.3961
0.1372
0.0958
0.0574
0.3271
0.1394
0.0056
TimeMixer
MSE
0.0564
0.9710
0.1179
0.0268
0,0080
0.6007
0.1343
0.0001
MAE
0.1271
0.3785
0.1375
0.0744
0.0137
0,2820
0.1425
0.0234
TimesNet
MSE
0.0569
0.9659
0,1187
0.0204
0,0016
0.5722
0,1363
0.0027
MAE
NaN
NaN
NaN
N&N
0.1308
0.2687
0.1318
0.0616
DLinear
MSE
NaN
NaN
N&N
N&N
0,0681
0,5816
0,1438
0.0158
MAE
0,3238
0.3736
0.1377
0.0823
0.2954
0,2712
0,1345
0.0692
FiLM
MSE
0.4193
0.9620
0.1242
0.0219
0.3851
0.5615
0.1391
0.0179
MAE
0,1385
0.4616
0.1484
0.0718
0.1380
0,3025
0.1406
0.0476
CSDI
MSE
0.1176
2.0394
0.2048
0.0269
0.0990
1.0445
0.1593
0.0176
MAE
0.5193
1.3809
0.2749
0.2107
0.5535
1,2894
0.2327
0.1173
PatchTST
MSE
0.7533
11.7008
0.3984
0.0471
0.2821
3.0525
0.1208
0.0114
MAE
0.8852
2.1533
0.4912
0.6517
0.8038
2.1488
0.4141
0.5340
iTransformer
MSE
1.3915
18.0800 18.0809
0.7799
0. 4299
0.4269
4.7132
0.1799
0.1036
MAE 0.5719
1.6961
0.3926
0.4514
0.6389
1.6801
0.3468
0.4651
TimeMixerPP
MSE
0.8420
13.3529
0.6534
0.2208
0.2537
3.5940
0.1693
0.0730
Table 13: Long- and short-horizon forccasting, TimeAutoDiff achieves competitive stato-of-the- art accuracy across both horizons, showing particularly strong performance at w = 36 (lowest MAE on Traffic/ Pollution and lowest MSE on Bike Sharing/ Traffic/ Pollution). At w: = 12, it attains the best MAE on Bike Sharing/ AurQuality and best MSE on Bike Sharing, while other baselines lead selected datasets. Bold denotes the best score.
42

---

Published in Transactions on Machine Learning Research (12/2025)
Imputation
Imputation
Model
Metric
(block
len
block width=2) 2)
(seq
len= 12)
Bike Sharing
Traflic
Pollution
AirQuality
Bike Sharing
Traffic
Pollution
AirQuality
MAE
0,6210
0.0626
0.1018
0.0416
0.6275
0.0619
0.1047
0.0442
TimeAutoDiff
MSE
0.1777
0.0216
0.0167
0.0006
0.1938
0.0169
0.0163
0.0006
MAE
0.3377
0.7765
0.1509
0.0749
0.2986
0.3892
0.1798
0.0959
TimeLLM
MSE
0.0348
0.2630
0.0131
0.0011
0.0238
0.0821
0.0168
0.0016
MAE
1.1657
3.3265
0.1320
0.1644
1.08419
2.8521
0.1810
0.5845
XOMENT
MSE
0.3263
1.6206
0.0128
0.0042
0.2814
1.9517
0.0142
0.0477
MAE
0.3445
0.4689
0.1300
0.0260
0.3887
0.3169
0.1350
0.0259
TEFN
MSE
2,0100
7 7.1343
0. 0.8980
1. 1044
2.0158
14.8765
0.8609
1.1044
MAE
0.4839
0.5438
0.1389
0,03 0,0527
0,5316
0.3438
0,1463
0,0518
TimeMixer
MSE
0.0783
0.1969
0.0134
0,0008
0,0861
0.0877
0.0140
0.0008
MAE
0.2351
0.7233
0.3009
0.0660
0.2791
0.4186
0.2650
0.0579
TimesNet
MSE
0.0144
0.4586
0.0370
0.0010
0.0197
0,1282
0.0349
0.0008
MAE
0.4087
0.7839
0.2100
0.3505
0.4412
0.9407
0,2151
0.3621
MICN
MSE
0.0522
0.3624
0.0208
0.0168
0,0392
0.4592
0.0232
0.0177
MAE
0,2600
0.6430
0.1532
0.0841
0,2672
0.5241
0.1623
0.0779
DLinear
MSE
0.0255
0.2428
0.0141
0.0014
0,0242
0,1237
0.0179
0.0012
MAE
U.4382
0.5119
0.1312
0.0516
O. 4938
0.3197
0,1637
0.0509
FILM
MSE
0.0676
0.1988
0.0130
0.0007
0,0776
0,0783
0.0144
0.0007
MAE
2.0009
0.1184
0.1208
0.0238
0.7977
1043
0.1254
0.0229
CSDI
MSE
11.7243
0.0348
0.0198
0.0005
0.4599
0.0287
0.0203
0.0004
MAE
0.1899
0.1983
0.1238
0.0337
0,1878
0.1754
0.1118
0.0332
PatchTST
MSE
0.0095
0.03-18
0.0127
0.0004
0.0099
0.0291
0.0111
0.0004
MAE
0.2161
0.5851
0.1535
0.0501
0.2202
0.3511
0.1713
0.0508
iTransformer
MSE
0.0156
0.2033
0.0129
0.0007
0.0150
0.0832
0.0154
0.0007
MAE
0.6323
1.3245
0.3046
0.2598
0.6785
1.2450
0.3373
0.3353
TimeMixerPP
MSE
0.1517
0.6668
0.0332
0.0116
0.1654
0.7089
0.0364
0.0191
Table 14: Imputation with structured masks (block
sequential). Block (block_lee len-3,
block_wit wocckk=wdth_) and Sequential (seq_len=12). PatchTST shows strong performance, leading MAE/MSE on Bike Sharing (both masks) and Pollution (MSE). TimeAutoDiff consistently leads Traffic (MAE/MSE) and Pollution (MAE). CSDI and TEFN perform best on AirQuality. Bold = best.
43

---

Published in Transactions on Machine Learning Research (12/2025)
L
Additional Plots: Auto- Cuto-Correlation periodic, cyclic patterns
Antocometatoo Nettriloned Thftic
Autocorabicin fict Fariftoned Sythetic Trait:: [TimelutDDt))
1.2
0.3-
0.5
106
203
293
105
204
250
Autocorre Put Pattzioned Traffic
Autocomaaaiinn Not of Fatiilonee Sythet Traffic (TImoAurati1)
1ppa
0.4
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293
250
4.0
Autocomdation Pat fortiioned heal Esergy
Autocomnationn Rac Purki Pf oneoooo Sytholi Energy (Timokut:DiM
1.5
1.2
025-
322
432
-335
-150
-375
230
250
290
900
100
150
200
26
30
La2
Figure 15: The four plots from the top are auto-corrclation plots of lag 300 for real (left) and synthetic (right) of 'Traffic', Hurricane'' Pollution', and Energy".
44

---

Published in Transactions on Machine Learning Research (12/2025)
M Plots of Synthethic v.s. Real on TV-MCG from TimeAutoDiff
3.50
Lomd
Roal lns
Limd
Condiconly Genarakel Ins
0.25
2.50
2.25
3.75
0.10
1.50
125
Tims
year
Realtrent
Uiiullyy Cencratal year
Canduiraly
16.0
7.5
5.0
2.5
0.0
Time
NOwtl))
Rcal NO22911
Conditionaly Genented NOwTOT)
Consitorally Gevented NazHa
0.8
3.5
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3.0
5.0
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0.500
Real
3.70
Conally Cenersted LULL
Conditonaly Genertted or
3.55
3.550
3.925
2.50
3.55
5.775
2.50
9.725
3.45
No21 igftts
Rudi
Cordl Lonally Gencronet iynes
Conolounall Gereratedial
0.5
0.2
Time 60
Time
Figure 16: Datasets: (output variables) from top bottom: Pollution: ('ebwd', 'Iws') Hurricane: ('year', 'trend'), AirQuality: (NNOxGGT), "NO2()T))), ETTh1: ('LULL', 'OT',, Eneryy: (lightts,, 'rvl'). The output is chosen to be heterogeneous (except AirQuality & ETTh1) both having discrete and continuous variables. Conditional variables e are set remaining variables from the entire features. See the list of entire features of each dataset through the link in Appendix B
45


Reviews:
[{"review_id": "yKWgDM1zMi", "reviewer": "Reviewer_Mvk1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents TimeAutoDiff, a unified latent-diffusion framework for heterogeneous time-series modeling that jointly supports generation, imputation, forecasting, and time-varying metadata conditional generation within a single model. The key idea is to use a VAE to map mixed-type features into a continuous latent space and apply a diffusion model to learn dynamics in that latent space. A simple mask-based conditioning strategy unifies multiple tasks, while whole-sequence sampling and feature-axis compression significantly improve computational efficiency. Extensive experiments on diverse real-world datasets demonstrate competitive or superior performance across multiple metrics.\n\nStrengths:  \n\n1.\tThe paper tackles a timely and forward-looking problem — unifying multiple time-series tasks (generation, imputation, forecasting, and conditional synthesis) under a single latent diffusion framework — which is conceptually novel and highly relevant to current trends in generative time-series modeling.  \n\n2. The proposed integration of a VAE for heterogeneous feature encoding with a diffusion model operating in latent space is well-structured, technically sound, and efficiently implemented (e.g., whole-sequence sampling and feature-axis compression).  \n\n3. The manuscript is well-organized, with clear explanations, solid empirical validation, and informative visualizations, making it accessible to both theoretical and applied audiences.  \n\nWeaknesses:  \n\n1.\tThe discussion of prior studies on VAE-based and diffusion-based generative frameworks is somewhat limited; including more comprehensive comparisons would improve context.  \n\n2.\tRecent works on diffusion and flow-matching models for time-series synthesis are missing and should be reviewed to better situate the proposed method.  \n\n3.\tThe experimental section lacks comparison with time-series foundation models or general-purpose large pre-trained forecasting models (e.g., MOMENT, Chronos, TimeGPT, TimesFM), which are increasingly relevant to the paper’s stated goal of task unification", "claims_and_evidence_yn": "No", "claims_explanation": "Some recently proposed baselines are missing.", "audience_yn": "Yes", "audience_explanation": "Given its unified latent-diffusion formulation and strong empirical results across diverse time-series tasks, this work would be of clear interest to TMLR’s audience—particularly those focused on generative modeling, multimodal representation learning, and efficient time-series analysis.", "requested_changes": "1.\tExpand Related Work Section\nThe current related work review is informative but lacks sufficient coverage of recent diffusion-based time-series models. The authors should include a more detailed comparison with very recent works (2023–2025) that combine diffusion or flow-matching models with time-series tasks. This will help position TimeAutoDiff more clearly within the rapidly evolving diffusion modeling landscape.\n\n2.\tDeepen Discussion on VAE and Diffusion Foundations\nProvide a clearer theoretical or conceptual explanation of how VAE-based latent compression complements diffusion modeling in this framework. Explicitly highlight differences from prior latent-diffusion approaches such as TabDDPM, TabSyn, or AutoDiff, and from existing VAE-diffusion hybrids used in other modalities (e.g., image or speech).\n3.\tStrengthen Baseline Comparisons\nThe experimental section could include more recent baselines relevant to heterogeneous or conditional time-series diffusion, ensuring fair benchmarking. This includes both diffusion-based (e.g., TimeGrad++, TimeMixer-Diffusion, TimeFlow) and transformer-based generative models that have become standard in 2024–2025 literature.\n4.\tExpand Discussion on Time-Series Foundation Models\nThe paper would benefit from a clearer connection to the emerging literature on time-series foundation models (e.g., Timer, MOMENT, Chronos, TimesFM). These models share the goal of unifying multiple forecasting and generation tasks across domains, similar in spirit to TimeAutoDiff. Including a discussion on how TimeAutoDiff differs from or complements such foundation models — particularly in terms of heterogeneous feature handling, mask-based task unification, and latent diffusion modeling — would strengthen the positioning of the work.", "broader_impact_concerns": "There are no significant broader impact concerns beyond standard considerations for generative models; the proposed framework focuses on time-series modeling and does not introduce direct ethical or societal risks.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "E4oHoMeMZ6", "reviewer": "Reviewer_sDPT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces TimeAutoDiff, a unified latent-diffusion framework that can handle four major time-series tasks—generation, imputation, forecasting, and time-varying-metadata conditional generation (TV-MCG)—within a single model that supports heterogeneous features (continuous, binary, and categorical).\n\nKey innovations include:\n1. A VAE–diffusion hybrid that maps heterogeneous data into a continuous latent space, allowing diffusion to operate on unified continuous representations.\n2. A whole-sequence latent diffusion process that generates full trajectories at once, significantly improving sampling speed.\n3. Feature-axis compression via the VAE, enabling scalable modeling of wide tables.\n4. A mask-conditioned framework that unifies the four tasks under a single conditional formulation.\n\nStrengths:\n1. Clear technical motivation and elegant unification of multiple time-series tasks.\n2. Well-designed architectural details (frequency encoding, bidirectional RNNs, adaptive β-VAE, etc.).\n\nWeakness\n1. In the unconditional generation task, although TimeAutoDiff is faster than typical diffusion models, it remains significantly slower than GAN-based models, indicating that the overall computational cost of the framework is still relatively high.\n\n2. In the imputation and forecasting tasks, the set of compared baselines is too limited, lacking widely adopted models such as PatchTST and iTransformer.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The experimental section (Table 2 and ablation studies) provides strong empirical support that TimeAutoDiff outperforms diffusion-based and GAN-based baselines across metrics such as discriminative score, predictive score, and feature correlation.\n\nThe qualitative plots (e.g., autocorrelation functions) and Distance-to-Closest-Record audit indicate generalization rather than memorization.\nThe claims of speed and scalability are also backed by concrete runtime comparisons.\n\nHowever, the evaluations of forecasting and metadata-conditioned generation are relatively less comprehensive — the results are presented but lack the same level of depth and analysis as the generation experiments, leaving part of the claimed versatility only partially substantiated.", "audience_yn": "Yes", "audience_explanation": "Its unification of multiple downstream tasks, scalability to heterogeneous data, and substantial sampling acceleration make it relevant to researchers working on:\n1. Time-series generative modeling,\n2. Multimodal/heterogeneous data learning,\n3. Diffusion models for structured data, and practical “what-if” counterfactual simulations.\n\nEven practitioners in finance, healthcare, and IoT domains could find it valuable for realistic data synthesis and data-driven scenario modeling.", "requested_changes": "1. Expand the experimental comparison for imputation and forecasting tasks.\nThe current evaluation lacks several widely adopted baselines such as PatchTST, iTransformer, and other strong transformer-based models. Including these baselines is necessary to fairly assess the claimed advantages of TimeAutoDiff across multiple time-series tasks.\n\n2. Provide a clearer discussion of computational cost.\nAlthough TimeAutoDiff is faster than conventional diffusion models, it remains significantly slower than GAN-based methods. A detailed analysis or discussion (e.g., breakdown of runtime components, Flops) is needed to justify the model’s efficiency claim and clarify its scalability limits.\n\n3. Enhance the analysis of forecasting and metadata-conditioned generation results.\nWhile results are presented, the analysis is relatively shallow compared to generation and imputation experiments. The paper would benefit from additional quantitative and qualitative evaluation.", "broader_impact_concerns": "N/A", "additional_comments": "1. In Section 4.3, do different x_j share the same embedding function e()? However, the discrete variables x_j may have different numbers of possible categorical values.\n2. In Section 4.3, “**Dencoder in VAE**” should be corrected to “**Decoder in VAE**.”\n3. In Section 5.3.1, “**Table K**” should be corrected to “**Table 4**.”\n4. Why does the model primarily adopt an RNN-based architecture? Has the author tried using Transformers instead? In most domains, RNNs have already been surpassed by Transformers in terms of representation capacity and scalability.", "extra_scores": {}, "extra_text": {}}, {"review_id": "aqxp4VkN9B", "reviewer": "Reviewer_e6E8", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents **TimeAutoDiff**, a unified latent–diffusion framework that integrates **four key time-series tasks**—unconditional generation, missing-data imputation, forecasting, and time-varying metadata conditional generation—within a single, coherent model architecture.  \n\nThe authors introduce a **mask-based conditional modeling scheme** that allows all tasks to be represented under a unified probabilistic formulation. The model couples:\n1. A **Variational Autoencoder (VAE)** to embed heterogeneous time-series (continuous, categorical, and binary) into a continuous latent space while compressing along the feature dimension; and  \n2. A **latent-space diffusion model** that models temporal dynamics and conditional generation efficiently.\n\n**Major contributions include:**\n- A **unified conditional framework** enabling four heterogeneous time-series tasks via binary masks.  \n- A **latent-space diffusion process** that avoids modeling discrete likelihoods directly, improving scalability.  \n- A **whole-sequence latent sampling mechanism**, achieving substantial efficiency gains (up to 100× faster sampling).  \n- Comprehensive empirical validation across **eight real-world datasets**, outperforming strong baselines on fidelity, predictive, and correlation metrics.  \n- Rigorous **ablation studies** and analysis of adaptive β-VAE scheduling, timestamp encodings, and Bi-RNN structures.\n\n**Strengths**\n- Elegant and theoretically grounded unification of multiple tasks.  \n- Strong empirical performance and robust ablations.  \n- Significant computational efficiency gains.  \n- Clear, well-organized, and reproducible presentation.  \n\n**Weaknesses (minor)**\n- Limited discussion of applicability to irregular or asynchronous time-series.  \n- Forecasting benchmarks could include comparison to recent large pre-trained models (e.g., MOMENT, Time-LLM).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are well substantiated by both **mathematical derivations** and **empirical validation**.  \n- The unified ELBO objective is theoretically sound, clearly integrating VAE and DDPM components.  \n- The experiments span diverse datasets and metrics, demonstrating robustness and strong generalization.  \n- Quantitative and qualitative analyses (e.g., autocorrelation plots, ablation tables) convincingly support the model’s superior fidelity, scalability, and generalization performance.", "audience_yn": "Yes", "audience_explanation": "The work will attract readers from multiple subfields, including:\n- **Time-series modeling** and **multimodal generative modeling**,  \n- **Diffusion-based learning** for non-visual domains,  \n- **Unified task frameworks** bridging generation, imputation, and forecasting, and  \n- **Applied ML domains** such as finance, healthcare, and supply-chain analytics.  \n\nThe methodological unification and efficiency improvements make it of strong interest to both academic and industrial practitioners.", "requested_changes": "The work is already strong and ready for publication. Suggested **minor improvements** include:\n\n1. **Relate to large pre-trained time-series models** (e.g., Time-LLM, MOMENT) to contextualize differences in scalability and adaptability.  \n2. Provide a brief **computational complexity analysis** (e.g., $O(T)$ and $O(F)$) to quantify efficiency gains.  \n3. Add a concise **summary table of dataset modalities** (number of continuous vs categorical variables) for reproducibility.  \n4. Include a short **case example** of metadata-conditioned generation for interpretability (e.g., “policy regime change” scenario).\n\nAll are non-critical enhancements to improve clarity and completeness.", "broader_impact_concerns": "No major ethical or societal issues are identified.  \nWhile the framework could be applied to sensitive domains (e.g., healthcare or finance), it primarily focuses on **technical modeling contributions**.  \nTo further strengthen the discussion, the authors could briefly mention **privacy safeguards** (e.g., use of DCR metric or differential privacy) when applying generative models to real-world data.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "YRyI4vxMTT", "reviewer": "Reviewer_RQtB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduced TimeAutoDiff, a unified latent-diffusion framework for heterogeneous time-series tabular data that can handle four different tasks including unconditional generation, missing data imputation, forecasting, and time-varying metadata conditional generation (TV-MCG). The proposed method is structured as a combination of a VAE with a diffusion model in the latent space. The VAE part facilitates capturing complicated structures of heterogeneous data and simultaneously helps to avoid direct modeling of such data, making the model structure simple without any special consideration of heterogeneity. The diffusion model in latent space possesses expressive power that flexibly captures temporal and feature dependencies, underpinning the overall performance of the method. The proposed method demonstrated superior results on multiple datasets compared to the baseline methods across four different tasks, and the effectiveness of each setting was verified through meticulous ablation studies.", "claims_and_evidence_yn": "Yes", "claims_explanation": "This paper is well written, with the motivation and positioning clearly explained in the introduction. Relevant literatures are extensively documented, and the advantages of the proposed method and its relationship to baselines are well organized. The unified formalization of multiple tasks via binary masks is conceptually simple and easy to understand. Moreover, the combination of VAE and diffusion models effectively addresses the complexity of heterogeneous time-series data, and the validity of the approach has been comprehensively examined across datasets, baselines, and tasks. The ablation study not only clarifies recommended settings but also carefully examines how each design choice affects results, enhancing the persuasiveness of the research.", "audience_yn": "Yes", "audience_explanation": "At least some individuals in the TMLR audience will find this paper of strong interest. The unified framework for heterogeneous time-series tasks addresses an important and timely challenge, and the conceptual simplicity of the binary mask formulation makes the contribution accessible to a broad set of researchers. The extensive empirical validation, combined with clear motivation and well-documented related work, ensures that the findings are relevant for the machine learning community, particularly those working on time-series analysis, generative modeling, and diffusion-based methods.", "requested_changes": "Some limitations remain. While the authors emphasize that latent compression enables efficient processing of high-dimensional features, the experimental evidence is limited to moderately dimensional cases, leaving open the question of scalability to truly high-dimensional data. In addition, certain results that would strengthen the paper (e.g., multi-sequence analyses, clearer visualizations) are missing or not fully consistent with the explanations in the text.\n\n**Suggested Changes**\n- Add some results or at least discussion on the applicability to high dimensional data.\n- Results on multi-sequence data lack in Table 3 and Figure 9. Is there any reason for this?\n- In Figure 10, the MAE bar for the baseline method's forecasting is not visible, making it appear at first glance that the baseline method performs better than the proposed method in forecasting scenarios. This contradicts the explanation in the main text.\n- In the first sentence of Section 4.1, the font for TimeAutoDiff is not a typewriter font, as it is elsewhere in the document.\n- In the last sentence of Section 5, “This section is divided into two subsections (to be completed):” should be changed as “This section is divided into three subsections”\n- In the last sentence of page 15, “Table 8” should be “Table 4”. In the same page, “Figure 8” should be “Table 4”. This change affects the number of following tables.", "broader_impact_concerns": "No concerns identified.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)