Parallel Complex Diffusion for Scalable Time Series Generation

TL;DR

PaCoDi introduces a spectral-native diffusion architecture that decouples time series modeling in the frequency domain, significantly reducing computational costs while improving generation quality and speed.

Contribution

This work presents the first spectral-native diffusion model for time series, leveraging Fourier transforms and complex diffusion theory to enhance scalability and efficiency.

Findings

Achieves 50% reduction in attention FLOPs without information loss.

Outperforms baselines in generation quality.

Provides a rigorous theoretical foundation for spectral diffusion.

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.