Time-Information Hierarchical Denoising Mechanism

• The paper introduces THD, which adaptively guides trend and seasonality in diffusion steps to reduce distribution bias and improve interpretability.
• THD employs STL-based decomposition and timestep-conditioned transformers to separately modulate low-frequency trends and high-frequency seasonality.
• Empirical results show DS-Diffusion with THD significantly outperforms baselines by lowering predictive and discriminative scores while preserving stochastic details.

Time-Information Based Hierarchical Denoising Mechanism (THD) is a denoising mechanism introduced in DS-Diffusion for time-series generation. It is designed to reduce the distributional bias between generated data and real data by guiding reverse diffusion with explicitly extracted data styles, particularly trend and seasonality, while also making the inference process more interpretable. In the reported formulation, THD operates inside style-guided kernels at inference only, so the main diffusion model remains frozen and new styles can be introduced without retraining the entire framework (Sun et al., 23 Sep 2025).

1. Conceptual definition and problem setting

THD was proposed against three stated limitations of existing diffusion models for time-series generation: distributional bias between generated and real data, an uninterpretable inference process caused by complex latent spaces and denoising dynamics, and the need to retrain the full framework when specific conditional guidance is changed. The mechanism addresses these issues by exploiting a basic structural property of many time series: the coexistence of low-frequency trend and high-frequency seasonality (Sun et al., 23 Sep 2025).

The defining idea is hierarchical denoising conditioned on time information. Rather than treating denoising as a uniform process over all timesteps and all signal components, THD uses the current reverse-diffusion timestep $t$ to modulate how much guidance is applied to trend and seasonality. Early reverse steps emphasize low-frequency guidance because the signal-to-noise ratio is higher there, while later steps emphasize high-frequency details after the overall structure has been established. This suggests a staged reconstruction process in which outline is stabilized before detail is refined.

Within DS-Diffusion, the term “hierarchical” therefore refers to component-wise and time-adaptive guidance over the reverse denoising trajectory. It does not denote a spatial hierarchy over channels or sensors. That distinction matters because a separate framework, THD-BAR, uses a different hierarchy—Brain Topology Hierarchy—for EEG autoregressive modeling rather than frequency-guided diffusion denoising (Yang et al., 5 Nov 2025).

2. Formal decomposition and denoising equations

THD begins from style extraction on real data instances. For each real sample $x$, Seasonal and Trend decomposition using LOESS (STL) is applied to obtain trend and seasonal components:

$S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$

Here, $S_{tr}$ and $S_{seas}$ are the extracted styles that later guide denoising (Sun et al., 23 Sep 2025).

The backbone generative process follows standard diffusion steps, and the reported implementation uses an EDM backbone. Reverse denoising begins from Gaussian noise $x_T \sim N(0, I)$ and iteratively recovers $x_0$. At each reverse step $t$, if $\mathcal{D}$ denotes the trained denoising model and $x_t$ the current noisy sample, the denoised estimate is written as

$x$0

and the evolution is described by

$x$1

with $x$2 and $x$3 as drift and diffusion coefficients (Sun et al., 23 Sep 2025).

THD intervenes after the backbone denoising step. The current sample is first converted back to time-series space:

$x$4

STL is then applied to decompose the current sample into trend, seasonality, and residual/noise:

$x$5

The trend component is guided by the extracted trend style, and the seasonal component is guided by the extracted seasonal style:

$x$6

$x$7

where $x$8 and $x$9 are Transformers that map $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$0 to guided components. The guided sample is then reconstructed as

$S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$1

and transformed back into the model’s image space for the next reverse step (Sun et al., 23 Sep 2025).

3. Hierarchical time-information guidance

The central operational principle of THD is that different frequency components are most useful at different reverse-denoising stages. Early timesteps prioritize trend guidance because low-frequency structure remains more recoverable under heavy noise. Later timesteps prioritize seasonality because higher-frequency detail becomes meaningful only after large-scale structure has been restored. In the reported formulation, this temporal modulation is learned by the Transformers through explicit access to the timestep $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$2 (Sun et al., 23 Sep 2025).

This temporal staging differentiates THD from denoising schemes that mix structure and detail uniformly across all reverse steps. The mechanism therefore combines two hierarchies simultaneously: a frequency hierarchy, since trend and seasonality are separated and guided independently, and a temporal hierarchy, since the relative contribution of each component changes over denoising time.

The residual term $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$3 is not discarded. It is retained to preserve random details and to avoid overfitting or generating artifacts. This is a notable design choice because THD does not impose complete deterministic replacement of the denoised sample by guided styles; instead, it preserves a residual path for stochastic variation (Sun et al., 23 Sep 2025).

Operationally, the algorithmic sequence is concise: denoise with the backbone, decompose into $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$4, $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$5, and $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$6, apply the respective transformer-guided updates using extracted styles, aggregate the components, and retransform for the next step. The reported interpretation is that this makes it possible to observe how style guidance influences sample formation step by step.

4. Placement within DS-Diffusion and comparison with traditional denoising

THD is embedded in the style-guided kernels of DS-Diffusion and is used at inference only. This allows specific styles to be introduced or updated without retraining the entire diffusion model. The trend and seasonality transformers are trained as part of the style-guided kernel module, which is described as much lighter than full model retraining (Sun et al., 23 Sep 2025).

In the comparative framing used for DS-Diffusion, traditional diffusion models learn all patterns at all timesteps, often mixing low- and high-frequency treatment, and typically require retraining or auxiliary classifiers to introduce style or conditional constraints. THD instead uses explicit decomposition, time-varying guidance for structure versus detail, and inference-time style conditioning.

Aspect	THD in DS-Diffusion	Traditional denoising
Guidance	Trend/seasonality guided at each step	All info mixed at all steps
Time modulation	Early: trend; Late: seasonality	No stepwise frequency modulation
Decomposition	STL separates style and guides per component	No explicit decomposition
Conditional flexibility	No retraining needed	Retraining or classifier needed
Interpretability	Styles and steps visible	Latent/black box

A common misconception is to treat THD as a replacement for the diffusion backbone. In the reported architecture it is not the backbone itself; it is a guidance mechanism layered onto a standard diffusion process. Another misconception is to equate its “hierarchical” character with multiscale spatial topology. In DS-Diffusion, the hierarchy is over frequency components and reverse time, whereas in THD-BAR the hierarchy is over spatial EEG topology and autoregressive scale ordering (Yang et al., 5 Nov 2025).

5. Empirical evidence, distributional alignment, and ablation findings

DS-Diffusion reports that, relative to state-of-the-art models such as ImagenTime, the predictive score and the discriminative score decrease by $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$7 and $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$8, respectively, and that the distributional bias between generated and real data is further reduced while inference becomes more interpretable (Sun et al., 23 Sep 2025).

The detailed evaluation states that on Stocks, Energy, MuJoCo, and Sine datasets, DS-Diffusion outperforms ImagenTime and other baselines in prediction and discriminative scores. On real-world datasets, DS-Diffusion reduces the average pred score and disc score of ImagenTime by $S_{tr}, S_{seas} = \mathscr{S}(x), \quad x \in D$9 and $S_{tr}$0, and on Sine by $S_{tr}$1 and $S_{tr}$2, respectively. The reported interpretation is explicit: lower pred indicates higher sample quality, and lower disc indicates less distributional bias (Sun et al., 23 Sep 2025).

Distribution similarity metrics are also reported to decrease significantly. On Stocks, the values are listed as KL divergence $S_{tr}$3, JS divergence $S_{tr}$4, Wasserstein distance $S_{tr}$5, and KS statistic $S_{tr}$6. Across datasets, the reported aggregate changes are KL $S_{tr}$7, JS $S_{tr}$8, Wass $S_{tr}$9, and KS $S_{seas}$0 (Sun et al., 23 Sep 2025).

Ablation analysis identifies the trend transformer $S_{seas}$1 as critical. Removing it “massively increases” discriminative and predictive scores; the example given is MuJoCo, where disc changes from $S_{seas}$2 to $S_{seas}$3 and pred from $S_{seas}$4 to $S_{seas}$5. The seasonality transformer $S_{seas}$6 is described as auxiliary but important, since its removal also worsens metrics. For style extraction, STL is reported to outperform FT, CWT, and TG because of adaptability and robustness. For sequence modeling inside THD, Transformers outperform GRU and MLP for dependence modeling and frequency adaptation (Sun et al., 23 Sep 2025).

6. Interpretability, visualization, and position within related work

A defining claim of THD is that it makes inference more interpretable. Because trend and seasonality are explicitly extracted from real data and then used to guide reverse denoising, each generated sample’s trend and seasonality can be traced to real data styles. The denoising process can therefore be visualized as a progression in which trend guidance dominates early and seasonality guidance becomes more important later (Sun et al., 23 Sep 2025).

The reported qualitative evidence includes PCA, t-SNE, and probability density plots. These visualizations are said to show that DS-Diffusion-generated data overlaps real data distributions more closely than ImagenTime or prior models, and that both outline similarity, associated with trend, and detail similarity, associated with seasonality, are improved. Figures referenced in the source summary are also described as showing better clustering with real data in both low- and high-frequency features, together with near-overlap of probability density curves for generated versus real data (Sun et al., 23 Sep 2025).

Within related work, THD occupies a specific niche: it is a frequency-aware, time-adaptive guidance mechanism for diffusion-based time-series generation. A plausible implication is that its main contribution is not a new diffusion objective but a structured intervention into the reverse process that links denoising decisions to observable signal components. This distinguishes it both from monolithic denoisers and from unrelated hierarchical frameworks such as THD-BAR, whose hierarchy is built around Brain Topology Hierarchy, Topology-Hierarchical VQ-VAE, and “next-scale-time prediction” for EEG representation learning rather than style-guided diffusion denoising (Yang et al., 5 Nov 2025).