Title: Parallel Complex Diffusion for Scalable Time Series Generation

Abstract:

Diffusion models typically learn data distributions via an indirect denoising process, meaning the complexity of generative modeling is intrinsically linked to the data’s dependency structure. In the context of time series, intense temporal dependencies compel the noise or score estimator to reconstruct highly entangled cross-time relationships, resulting in what is known as the "curse of entanglement." To alleviate this challenge, we alter the topology of the diffusion space. By employing the Discrete Fourier Transform (DFT), we decompose temporal dependencies into spectral modes. This approach diagonalizes the second-order dependency structure, thereby better aligning the data manifold with isotropic Gaussian noise and ensuring homogeneous diffusion dynamics.

While current frequency-aware diffusion methods primarily leverage the DFT to design estimator blocks within temporal DDPM or SDE frameworks, frequency-native diffusion paths have historically been hindered by the mathematical complexities of complex-valued dynamics. To address this, we introduce PaCoDi (Parallel Complex Diffusion), a frequency-native framework that constructs the diffusion path directly in the spectral domain. PaCoDi replaces the complex-valued estimator with parallel real-valued estimators dedicated to the real and imaginary components.

Theoretically, we demonstrate the statistical orthogonality of spectral Gaussian noise and establish quadrature forward transitions alongside conditional reverse factorization. Furthermore, we extend the discrete PaCoDi model to continuous-time spectral SDEs by utilizing a Spectral Wiener Process. To manage marginal coupling, we incorporate a Mean Field Theory approximation enhanced by an Interactive Correction Branch. Additionally, by exploiting Hermitian symmetry, we achieve a 50% reduction in attention FLOPs without any loss of information. Extensive experiments conducted on both unconditional and conditional time series generation tasks reveal that PaCoDi outperforms five state-of-the-art baselines across five benchmarks in terms of both generative quality and computational efficiency. The source code is publicly available at https://github.com/RongyaoCai/PaCoDi.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC