Parallel Complex Diffusion for Scalable Time Series Generation

2026-02-10

arXiv.org

AI Summary

This work addresses the training difficulties in time series diffusion models caused by strong temporal dependencies—referred to as the “curse of entanglement”—by proposing PaCoDi, a frequency-domain native diffusion framework. PaCoDi constructs the diffusion process in the spectral domain, leveraging the discrete Fourier transform to decouple temporal dependencies and employing parallel real-valued networks to model the real and imaginary components of complex spectra separately, thereby enabling the first efficient real-valued learning of complex diffusion processes. Theoretically, it establishes statistical orthogonality of Gaussian noise in the spectral domain and combines mean-field approximation with interaction correction to handle marginal couplings. Technically, it exploits Hermitian symmetry to reduce attention computation by 50%. Experiments demonstrate that PaCoDi consistently outperforms five state-of-the-art methods across five benchmark datasets in both unconditional and conditional generation tasks, achieving superior generation quality and computational efficiency.

Abstract

Modeling long-range dependencies in time series generation poses a fundamental trade-off between representational capacity and computational efficiency. Traditional temporal diffusion models suffer from local entanglement and the $\mathcal{O}(L^2)$ cost of attention mechanisms. We address these limitations by introducing PaCoDi (Parallel Complex Diffusion), a spectral-native architecture that decouples generative modeling in the frequency domain. PaCoDi fundamentally alters the problem topology: the Fourier Transform acts as a diagonalizing operator, converting locally coupled temporal signals into globally decorrelated spectral components. Theoretically, we prove the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, demonstrating that the complex diffusion process can be split into independent real and imaginary branches. We bridge the gap between this decoupled theory and data reality using a \textbf{Mean Field Theory (MFT) approximation} reinforced by an interactive correction mechanism. Furthermore, we generalize this discrete DDPM to continuous-time Frequency SDEs, rigorously deriving the Spectral Wiener Process describe the differential spectral Brownian motion limit. Crucially, PaCoDi exploits the Hermitian Symmetry of real-valued signals to compress the sequence length by half, achieving a 50% reduction in attention FLOPs without information loss. We further derive a rigorous Heteroscedastic Loss to handle the non-isotropic noise distribution on the compressed manifold. Extensive experiments show that PaCoDi outperforms existing baselines in both generation quality and inference speed, offering a theoretically grounded and computationally efficient solution for time series modeling.

Problem

• time series generation
• temporal dependence
• curse of entanglement
• diffusion models
• spectral domain

Innovation

• Parallel Complex Diffusion
• Spectral Domain Diffusion
• Time Series Generation
• Hermitian Symmetry
• Spectral Wiener Process