A Non-isotropic Time Series Diffusion Model with Moving Average Transitions

We are grateful for your valuable suggestions and insightful comments. We'd like to reply as follows:

1. Uniform sampling strategy

We've added the default DDIM sampling strategy to our experiments. Specifically, for a given sampling budget, we start with t=T and uniformly select the sampling steps. The results can be seen at (https://anonymous.4open.science/r/ICML2025_rebuttal-134C/sample_strategy.png). We can see that uniform sampling did offer a better result than random sampling, but the marginal gain on the performance decreases by $\tau/T$. Compared to both random and uniform strategies, our factor-only strategy also shows superior performance.

2. Ablation on $L$

We did an ablation study on the sequence length $L={48, 96, 192}$ and recorded the gradient similarity in this link (https://anonymous.4open.science/r/ICML2025_rebuttal-134C/ablation_L.png). We observe that for all L, gradient conflicts still exist when directly applying DDPM on time series data, and our method can keep alleviating. It should be noted that the severity of such conflicts is not changed with L varying. Instead, we think the gradient conflicts mainly come from the degradation design, and in our framework, MA always keeps more low-frequency information across different diffusion steps. These low-frequency components could serve as common information during diffusion training. Thus, the inputs for denoising models are more structurally similar, and consequently, the gradients across different steps are similar.

3. Backbone architecture

For the denoising backbone, we mainly rely on the DiT, with an adaptation to 1D time series. For the condition encoder in the forecasting task, we adopt an MLP with RevIN[1], which is widely used in TS forecasting. In our paper, we don't focus much on the neural architecture design, and we believe that the backbones can be replaced with other reasonable networks. Combining the SOTA time series architecture with our framework could also be one of our future works.

4. Applicable to video generation

We appreciate your insightful comments for inspiring us to explore more of our framework. Though we are not experts in the field of video generation, we still try to evaluate our framework on a simple "video-like" time series.

Specifically, we believe that the transition between frames in a video should be smooth and mild because even the most common frame rate (like 24 FPS) is higher than the sampling rate of time series data. Thus, we manually create a pure 2-D signal: $x(t)=\sqrt{0.5 t} + 0.7 \cos(t) + 0.3 \sin(4 t), y(t)=\sqrt{0.2 t} + 0.8 \cos(t) + 0.2 \sin(3 t)$ with $t\in[0, 4\pi]$ recorded with 1280 time steps (mimicking a 53s video in 24FPS), and concatenate them as a bivariate time series. Then, we slide temporal windows with $L={48, 96, 192}$ and record the gradient similarity of directly using DDPM and our framework. As shown in (https://anonymous.4open.science/r/ICML2025_rebuttal-134C/simple_syn.png), we found a similar phenomenon, and our framework also works to ease the gradient conflicts to some extent. However, we think more framework adaptation is needed to bridge the large modality gap for video generation.

In summary, our framework shows similar effects on our synthetic "video-like" time series, and we reckon that our high-level idea could also be applied to video generation with some necessary adaptation. We sincerely hope that our framework could also be enlightening and supportive for other fields beyond time series.

[1] Reversible Instance Normalization for Accurate TSF Against Distribution Shift

Thank you for the valuable feedback and the opportunity to strengthen our experiments. Below, we address each of the weaknesses:

1. Standard Time Series Generation

We agreed that the standard generation task is necessary for evaluating our model. Thus, we follow the setting of your mentioned benchmarks and experiment on three datasets, ETTh2, Exchange, and ECG (newly added, medical time series). We also include discriminative score and predictive score as additional metrics for evaluating the fidelity and usefulness, as the following table shows:

We can see that compared to the SOTA time series generation models, our method still shows salient improvements in discriminative score and predictive score, illustrating the capability of generating high-fidelity and useful synthetic time series samples.

2. Super-resolution(SR)

We included flow matching with Variance Preserving path (FM-VP) as our additional SR benchmark. The backbone used in FM-VP is kept the same as our original benchmark and model. The following table records the comparison:

Table.1 Consistency error comparison

Table.2 Context-FID comparison

Despite the slight inferiority in Context-FID on the Solar dataset when the SR scale goes up, our proposed method still outperformed the FM-VP model generally, especially in terms of consistency. It should be noted again that we perform SR naturally through our backward process instead of retraining the whole conditional model like benchmarks. Therefore, we think our method provides a trade-off of (1) training overheads, (2) SR quality, and (3) consistency to the low-resolution input.

3. Forecasting

Although our paper focuses on how to improve the time series diffusion model, we also agree that it's necessary to include SOTA time series forecasting methods as a reference. Therefore, we included Autoformer[1], Non-stationary Transformer (NSformer)[2], and PatchTST[3] to compare deterministic forecasting performance. We run all models in the same setting in our paper, i.e., L = {96, 192, 336, 720}.

Regarding overall performance, our model still ranks first among these benchmarks, though it is slightly inferior to ETTh2 and weather compared to PatchTST.

It should be noted that these SOTA architectures are particularly tailored for time series forecasting and well adapted to the benchmark datasets, while forecasting is one of the downstream applications of our proposed MA-TSD framework. Therefore, we think there exists great potential to accommodate the SOTA architectures into our MA-TSD framework to have a better forecasting performance in our future work.

[1] Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting

[2] Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting

[3] A Time Series is Worth 64 Words: Long-term Forecasting with Transformers

We sincerely appreciate your time and effort in reviewing, as well as your constructive feedback, which helps strengthen our work. We would like to address your questions as follows.

1. Lack of non-diffusion benchmarks for comparison

We agree that non-diffusion benchmarks are also necessary for general comparison, even if our work focuses on the improvement of time series diffusion models. We've added related experiments with Autoformer, Non-stationary transformer, and PatchTST for time series forecasting.

Regarding overall performance, our model still ranks first among these benchmarks, though it is slightly inferior to ETTh2 and weather compared to PatchTST.

It should be noted that these SOTA architectures are particularly tailored for time series forecasting and well adapted to the benchmark datasets, while forecasting is one of the downstream applications of our proposed MA-TSD framework. Therefore, we think there exists great potential to accommodate the SOTA architectures into our MA-TSD framework to have a better forecasting performance in our future work.

2. Computational demands

We'd like to analyze the computation burden of our method from two perspectives, i.e., spatially and temporally.

Spatially: It should be noted again that the training process of our framework is almost identical to the standard diffusion, i.e., sample a batched time series window, degrade it, and use a denoising network to learn. The main difference lies in the forward (degrading) process. In our framework, we need to store a series of kernel matrices $\boldsymbol{K}_t$ in advance for degrading at different diffusion time steps, which in practice is a tensor with shape $T \times L \times L$ (T=diffusion steps, L=sequence length). Thus, if we try to model an extremely long sequence, it could increase the storage demand in quadratic growth. Fortunately, $\boldsymbol{K}_t$ is sparse because only unconvolved time steps remain zeros, and such sparsity decreases by t. For example, when $T=100, L=576$, the sparsity ($1 - oc / L^2$, $oc$ is the number of non-zero entries) at $t=20,40,60,80$ is [0.979 0.945 0.879 0.694]. Therefore, though we need to store a series of kernel matrices, they are mostly highly sparse, thus not imposing much storage burden.

Temporally: The most time-consuming part of diffusion models is sampling, but we can utilize the accelerated backward process to speed up, especially in the time series super-resolution task. As we illustrated in the paper, we conduct SR naturally through our backward process without retraining the whole model like benchmarks that rely on conditional inputs.

In practice, we used only one NVIDIA 4090 24GB to conduct all the experiments. Even if we run a $L=720$ task from scratch on the largest dataset (ettm2), it only takes $**0.72**$ GPU hours to train and sample, thus no significant computational demands in our method.

3. Dataset diversity

We highly agree that dataset diversity is important for evaluation. For now, our used datasets have included Traffic (traffic system), Electricity/MFRED (power system), Exchange_rate (financial market), ETT (IoT sensors), Weather (meteorological system), and Solar/Wind (renewable energy). These are widely used benchmark datasets in the field of time series [1,2].

Beyond these, we also added an ECG dataset (medical system) during the rebuttal period for a more complete evaluation. We launched a standard time series generation experiment on the ECG dataset and included two more benchmarks, i.e., KoVAE (RNN-based) and ImagenTime (Diffusion-based). The results are shown below.

Disc. Score evaluates the fidelity by training a post-classifier to tell whether the data is real or not. Pred. Score evaluates the usefulness by training an RNN forecasting model with the generated data and testing it on the real [3]. Thus, we can see the superiority of our method on both fidelity and usefulness over benchmarks.

[1] Deep Time Series Models: A Comprehensive Survey and Benchmark

[2] A Survey on Diffusion Models for Time Series and Spatio-Temporal Data

[3] TSGBench: Time Series Generation Benchmark

We greatly appreciate your acknowledgement of our work. We would like to address your questions as follows.

Q1: The difference between Blurring Diffusion Model (BDM) and ours

From a high-level perspective, BDM and ours shared a similar idea, i.e. building the degradation process with low-pass filters (blurring in BDM, MA in ours). However, there exists clear distinctions.

1. Filtering space: Though we think it's minor due to convolution theoram, we still point out at first to pave the way for the following parts. Specifically, we filtered the data in the time space by convolution (matrix multiplication),$q(\mathbf{x}_t | \mathbf{x}_0) = \mathcal{N} (\mathbf{K}_t\mathbf{x}_0,\beta_t^2 \mathbf{I})$ while BDM blurs the images in the frequency domain and transform back to the pixel domain, i.e. $q(\mathbf{x}_t|\mathbf{x}_0) = \mathcal{N} ( \mathbf{V} \boldsymbol{\alpha}_t \mathbf{V}^\top \mathbf{x}_0, \sigma_t^2 \mathbf{I})$, where $\mathbf{V}^\top,\mathbf{V}$ are DCT and IDCT, and $\boldsymbol{\alpha}_t$ is the frequency response of Gaussian blurring kernel, a diagnoal matrix, whose each entry $\alpha_t^i \in (0, 1]$ is a coefficient of i th frequency component. For low pass filters, $\alpha_t^i$ decreases by i until (nearly) zero to suppress high frequency components.

2. Markovian or not (Major): Since $\boldsymbol{\alpha_t}$ is diagnoal, BDM proposed that for each frequency component, a standard Markovian DDPM can be constructed. The one-step transition is accordingly defined, i.e. $q(u_t | u_s) = \mathcal{N}(\alpha_{t|s}u_s, \sigma_{t | s}^2)$, where s=t-1, $u_t=V^\top x_0$ is the frequency representation and $\boldsymbol{\alpha}_{t | s}=\boldsymbol{\alpha}_t / \boldsymbol{\alpha}_s$. However, dividing $\boldsymbol{\alpha}_s$ could be problematic in practice, because $\alpha_t^i$ could become to be (nearly) zero for large i, so dividing $\boldsymbol{\alpha}_s$ is numerically unstable for all diffusion steps. Therefore, though BDM claimed to have Markov transition under the DDPM framework, we think it's improper to define $q(x_t | x_s)$ when the transition operation from $x_0$ to $x_t$ is non-invertible. In our framework, faced with similar invertible MA, we bypassed the definition of $q(x_t|x_s)$, assumed $q(x_1:T | x_0)$ non-Markovian (in the DDIM-style) and then delicately defined $q(x_s|x_t, x_0)$ to satisfy $q(x_t|x_0)$ for all t. Thus, whether it's Markovian or not is another distinct difference from ours and BDM.

3. Noise schedule: BDM designed $\boldsymbol{\alpha}_t = a_t \mathbf{d}_t$, where $\mathbf{d}_t$ is the frequency response of blurring kernel and $a_t \in [0,1]$ is an extra scaler decreasing by t, and the noise schedule $\sigma_t = 1-a_t^2$. In our framework, the noise schedule $\beta_t$ is dataset-based, chosen regarding the variance decrease caused by MA (Eq.13) on different datasets. To compare these two, we mimic BDM's noise schedule, i.e. multiplying an $a_t$ to $\mathbf{K}_t$ and set $\beta_t = 1-a_t^2$, and ran the experiments on forecasting task (L=96). We can see that test errors indeed increased, which might be because of BDM's $\sigma_t$ schedule ignored the extra effects of blurring ($\mathbf{d}_t$) and only consider the $a_t$.

In summary, despite the similar high-level idea, there exist clear differences between BDM and ours. BDM tried to fit in the standard DDPM framework, while we reformulated a framework with special adaptation on moving average filters and time series. Unfortunately, BDM didn't release their official codes, and we are unable to reproduce it on time series data for more quantitative comparison.

Q2: Moving average aligns gradient signals

By intuition, a mild data degradation in diffusion should benefit the learning of denoising networks, since the inputs could be mostly noise if degradation is too fast. For example T=500 in Fig.2, the gradients at t> ~190 (right down block) are similar because x_t are almost noise and the model can hardly learn anything. For t < ~190 (left up block), they are more informative and contribute to denoising learning more. Thus, we'd like to "prolong" the left up block to help the learning.

Meanwhile, to reconstruct a time series, low-frequency (LF) information is usually more helpful. Thus, we're motivated to use MA, a widely used TS low-pass filter, to design a mild degradation where LF is kept more and serves as common information for all diffusion steps. In this way, the inputs for denoising models are more structurally similar so that gradients conflicts at different steps could be less.

Others: Forecasting examples are in https://anonymous.4open.science/r/ICML2025_rebuttal-134C/forecast_plot.png