# Time Series Diffusion in the Frequency Domain

arXiv: https://arxiv.org/abs/2402.05933 · HTML: https://mlsanity.com/paper/2402.05933/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 21 of its 38 references that are in the map.
- First posted on arXiv 8 Feb 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs](https://mlsanity.com/paper/1706.02633/) (2017)
- [Adversarial Audio Synthesis](https://mlsanity.com/paper/1802.04208/) (2018)
- [Continuous diffusion for categorical data](https://mlsanity.com/paper/2211.15089/) (2022)
- [MIMIC-Extract: A Data Extraction, Preprocessing, and Representation Pipeline for MIMIC-III](https://mlsanity.com/paper/1907.08322/) (2019)
- [MatterGen: a generative model for inorganic materials design](https://mlsanity.com/paper/2312.03687/) (2023)
- [ECG Heartbeat Classification: A Deep Transferable Representation](https://mlsanity.com/paper/1805.00794/) (2018)
- [Diffusion Models for Time Series Applications: A Survey](https://mlsanity.com/paper/2305.00624/) (2023)
- [A Survey on Deep Learning based Time Series Analysis with Frequency Transformation](https://mlsanity.com/paper/2302.02173/) (2023)
- [Frequency Domain-based Dataset Distillation](https://mlsanity.com/paper/2311.08819/) (2023)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)

## Built on by (most cited first, 30 of 37)

- [Foundation Models for Time Series Analysis: A Tutorial and Survey](https://mlsanity.com/paper/2403.14735/) (2024)
- [Deep Time Series Models: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2407.13278/) (2024)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [A Fourier Space Perspective on Diffusion Models](https://mlsanity.com/paper/2505.11278/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [Shaping Inductive Bias in Diffusion Models through Frequency-Based Noise Control](https://mlsanity.com/paper/2502.10236/) (2025)
- [FreCT: Frequency-augmented Convolutional Transformer for Robust Time Series Anomaly Detection](https://mlsanity.com/paper/2505.00941/) (2025)
- [LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation](https://mlsanity.com/paper/2506.17039/) (2025)
- [Modeling Human Gaze Behavior with Diffusion Models for Unified Scanpath Prediction](https://mlsanity.com/paper/2507.23021/) (2025)
- [TimeBridge: Better Diffusion Prior Design with Bridge Models for Time Series Generation](https://mlsanity.com/paper/2408.06672/) (2024)
- [FLEXtime: Filterbank learning to explain time series](https://mlsanity.com/paper/2411.05841/) (2024)
- [NeuralRemaster: Phase-Preserving Diffusion for Structure-Aligned Generation](https://mlsanity.com/paper/2512.05106/) (2025)
- [WaveletDiff: Multilevel Wavelet Diffusion For Time Series Generation](https://mlsanity.com/paper/2510.11839/) (2025)
- [Channel-aware Contrastive Conditional Diffusion for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2410.02168/) (2024)
- [Learning What Matters: Steering Diffusion via Spectrally Anisotropic Forward Noise](https://mlsanity.com/paper/2510.09660/) (2025)
- [TSGDiff: Rethinking Synthetic Time Series Generation from a Pure Graph Perspective](https://mlsanity.com/paper/2511.12174/) (2025)
- [DiM-TS: Bridge the Gap between Selective State Space Models and Time Series for Generative Modeling](https://mlsanity.com/paper/2511.18312/) (2025)
- [Translation from Wearable PPG to 12-Lead ECG](https://mlsanity.com/paper/2509.25480/) (2025)
- [Spectral Regularization for Diffusion Models](https://mlsanity.com/paper/2603.02447/) (2026)
- [Accelerating Frequency Domain Diffusion Models with Error-Feedback Event-Driven Caching](https://mlsanity.com/paper/2604.22901/) (2026)
- [HyFAD: Hybrid Time-Frequency Diffusion with Frequency-Aware Embedding for Time Series Imputation](https://mlsanity.com/paper/2606.05239/) (2026)
- [Cyclostationary Phase Conditioning for Medical Time Series Diffusion](https://mlsanity.com/paper/2609.34965/) (2026)
- [Frequency-Constrained Learning for Long-Term Forecasting](https://mlsanity.com/paper/2508.01508/) (2025)
- [Preserving Spectral Structure and Statistics in Diffusion Models](https://mlsanity.com/paper/2512.17873/) (2025)
- [From Easy to Hard++: Promoting Differentially Private Image Synthesis Through Spatial-Frequency Curriculum](https://mlsanity.com/paper/2601.06368/) (2026)
- [Spectral Diffusion Models on the Sphere](https://mlsanity.com/paper/2601.20498/) (2026)
- [A Decomposable Forward Process in Diffusion Models for Time-Series Forecasting](https://mlsanity.com/paper/2601.21812/) (2026)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)
- [Parallel Complex Diffusion for Scalable Time Series Generation](https://mlsanity.com/paper/2602.17706/) (2026)
- [Deep Time-Series Forecasting in 10 Years: A Survey](https://mlsanity.com/paper/2603.19899/) (2026)

## Abstract (arXiv)

Fourier analysis has been an instrumental tool in the development of signal processing. This leads us to wonder whether this framework could similarly benefit generative modelling. In this paper, we explore this question through the scope of time series diffusion models. More specifically, we analyze whether representing time series in the frequency domain is a useful inductive bias for score-based diffusion models. By starting from the canonical SDE formulation of diffusion in the time domain, we show that a dual diffusion process occurs in the frequency domain with an important nuance: Brownian motions are replaced by what we call mirrored Brownian motions, characterized by mirror symmetries among their components. Building on this insight, we show how to adapt the denoising score matching approach to implement diffusion models in the frequency domain. This results in frequency diffusion models, which we compare to canonical time diffusion models. Our empirical evaluation on real-world datasets, covering various domains like healthcare and finance, shows that frequency diffusion models better capture the training distribution than time diffusion models. We explain this observation by showing that time series from these datasets tend to be more localized in the frequency domain than in the time domain, which makes them easier to model in the former case. All our observations point towards impactful synergies between Fourier analysis and diffusion models.