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Time Series Diffusion in the Frequency Domain
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Derives score-based diffusion SDEs for time series in the DFT domain (using mirrored Brownian motions and a score-matching equivalence) and shows frequency diffusion beats time diffusion on six real-world datasets, attributing this to spectral localization.
Problem
Whether representing time series in the frequency domain is a useful inductive bias for score-based diffusion models, given that the DFT is complex-valued and its components are constrained by mirror symmetry.
Approach
Applies the DFT to the forward/reverse SDEs of time-domain diffusion, showing the frequency-domain noise is a mirrored Brownian motion (complex with mirror symmetry), and adapts denoising score matching accordingly so the frequency score is equivalent to an auxiliary time-domain score loss.
Key results
- Frequency diffusion gives lower sliced Wasserstein distance than time diffusion on all six datasets in both domains, e.g. US-Droughts 0.633 vs 2.849 (frequency-domain SW, frequency vs time diffusion).
- On ECG, the frequency and time diffusion models cross at a kernel width sigma of about 7 when spectral localization is artificially removed, with the gap closing as sigma grows.
- Time series in all six datasets are more localized in the frequency domain than in the time domain, with energy concentrated at low frequencies.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · ECG | sliced Wasserstein distance (time domain): 0.021 \ \pm \ 0.000 -> 0.015 \ \pm \ 0.000 | Helped |
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · MIMIC-III | sliced Wasserstein distance (time domain): 0.211 \ \pm \ 0.006 -> 0.152 \ \pm \ 0.004 | Helped |
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · NASDAQ-2019 | sliced Wasserstein distance (time domain): 60.512 \ \pm \ 2.960 -> 43.602 \ \pm \ 2.044 | Helped |
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · NASA-Charge | sliced Wasserstein distance (time domain): 0.316 \ \pm \ 0.008 -> 0.229 \ \pm \ 0.008 | Helped |
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · NASA-Discharge | sliced Wasserstein distance (time domain): 2.942 \ \pm \ 0.134 -> 2.028 \ \pm \ 0.082 | Helped |
| represent time series in the frequency domain and diffuse there instead of in the time domain | Score-based time-series diffusion · US-Droughts | sliced Wasserstein distance (time domain): 2.913 \ \pm \ 0.092 -> 0.738 \ \pm \ 0.020 | Helped |
| smooth the ECG spectra with increasingly wide Gaussian kernels, reducing frequency localization (ablation) | Frequency diffusion versus time diffusion · ECG variants | sliced Wasserstein distance: around at \sigma \approx 7 | Mixed |
Surprising findings
- Artificially delocalizing the spectrum of ECG until time series are more localized in time makes the time diffusion model outperform the frequency model.
- Frequency diffusion outperforms time diffusion even though the two are theoretically equivalent formulations, showing the domain acts as a practical inductive bias.
Limitations
- Localization explains performance only partially: the crossover points for the localization metric and the performance metric differ (sigma approx 2 vs approx 7).
- Evaluation relies on sliced Wasserstein distances in input space because no good general representation for time series evaluation exists.
- Results are shown for a single small transformer architecture (about 3.2M parameters) and LSTM variant, not state-of-the-art models.
Experimental setup
Datasets: ECG MIMIC-III NASDAQ-2019 NASA-Charge NASA-Discharge US-Droughts
Benchmarks: ECG MIMIC-III NASDAQ-2019 NASA-Charge NASA-Discharge US-Droughts
Metrics: sliced Wasserstein distance delocalization metric (time and frequency)
Baselines: time diffusion model
Builds on: score-based SDE diffusion denoising score matching VP-SDE
Closest papers by meaning
- SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings (ICLR 2025 · similarity 0.90)
- HyFAD: Hybrid Time-Frequency Diffusion with Frequency-Aware Embedding for Time Series Imputation (NeurIPS 2026 · similarity 0.88)
- A Non-isotropic Time Series Diffusion Model with Moving Average Transitions (ICML 2025 · similarity 0.88)
- Dimension-free Score Matching and Time Bootstrapping for Diffusion Models (NeurIPS 2025 · similarity 0.88)
- Generalization through variance: how noise shapes inductive biases in diffusion models (ICLR 2025 · similarity 0.88)
- Retrieval-Augmented Diffusion Models for Time Series Forecasting (NeurIPS 2024 · similarity 0.87)
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