# Parallel Complex Diffusion for Scalable Time Series Generation

arXiv: https://arxiv.org/abs/2602.17706 · HTML: https://mlsanity.com/paper/2602.17706/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 16 of its 25 references that are in the map.
- First posted on arXiv 10 Feb 2026.

## Placed in

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

## Builds on (most cited first, 16 of 16)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [TimeVAE: A Variational Auto-Encoder for Multivariate Time Series Generation](https://mlsanity.com/paper/2111.08095/) (2021)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [PSA-GAN: Progressive Self Attention GANs for Synthetic Time Series](https://mlsanity.com/paper/2108.00981/) (2021)
- [Boosting Diffusion Models with Moving Average Sampling in Frequency Domain](https://mlsanity.com/paper/2403.17870/) (2024)
- [Time Series Diffusion in the Frequency Domain](https://mlsanity.com/paper/2402.05933/) (2024)
- [T2S: High-resolution Time Series Generation with Text-to-Series Diffusion Models](https://mlsanity.com/paper/2505.02417/) (2025)
- [Empowering Time Series Analysis with Synthetic Data: A Survey and Outlook in the Era of Foundation Models](https://mlsanity.com/paper/2503.11411/) (2025)
- [FourierFlow: Frequency-aware Flow Matching for Generative Turbulence Modeling](https://mlsanity.com/paper/2506.00862/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)

## Abstract (arXiv)

Diffusion models learn data distributions indirectly through denoising, making the difficulty of generative modeling closely tied to the dependency structure of data. For time series, strong temporal dependence forces the noise / score estimator to recover highly entangled cross-time relationships, leading to the curse of entanglement. We mitigate this burden by changing the topology of the diffusion space: the Discrete Fourier Transform (DFT) decomposes temporal dependencies into spectral modes, diagonalizing second-order dependency structure and better aligning the data manifold with isotropic Gaussian noise and homogeneous diffusion dynamics. However, existing frequency-aware diffusion methods mainly use the DFT to design estimator blocks under temporal DDPM/SDE frameworks, while frequency-native diffusion paths face a mathematical barrier from complex-valued dynamics. We propose PaCoDi (Parallel Complex Diffusion), a frequency-native diffusion framework that constructs the diffusion path in the spectral domain while replacing the complex-valued estimator with parallel real-valued estimators for real and imaginary components. Theoretically, we prove the statistical orthogonality of spectral Gaussian noise, establish quadrature forward transitions and conditional reverse factorization, and extend discrete PaCoDi to continuous-time spectral SDEs through a Spectral Wiener Process. We further introduce a Mean Field Theory approximation with an Interactive Correction Branch to handle marginal coupling, and exploit Hermitian symmetry to reduce 50% attention FLOPs without information loss. Extensive experiments on unconditional and conditional time series generation demonstrate superior generative quality and computational efficiency against 5 SOTA baselines in 5 benchmarks, respectively. Code is available at https://github.com/RongyaoCai/PaCoDi.