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Parallel Complex Diffusion for Scalable Time Series Generation

Rongyao Cai, Yuxi Wan, Kexin Zhang, Ming Jin, Zhiqiang Ge, Qingsong Wen, Yong Liu

TL;DR

Theoretically, PaCoDi proves the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, and 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.

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.

Parallel Complex Diffusion for Scalable Time Series Generation

TL;DR

Theoretically, PaCoDi proves the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, and 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.

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 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.
Paper Structure (80 sections, 8 theorems, 121 equations, 3 figures, 9 tables)

This paper contains 80 sections, 8 theorems, 121 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Temporal Diffusion Modeling
  3. Forward Temporal Transition
  4. Structural Properties of Temporal Manifold
  5. Parallel Complex Diffusion Framework
  6. Forward Diffusion in Complex Manifold
  7. Conditional Factorization of Reverse Dynamics
  8. Parallel Modeling via Mean Field Theory
  9. Bypassing Holomorphic Constraints
  10. Continuous-time Perspective: Frequency SDEs
  11. Forward SDE and Spectral Wiener Process
  12. Bridging Complex Scores and Parallel Real-valued SDEs
  13. Ideally Decoupled Reverse Dynamics
  14. MFT Approximation and Interactive Score Matching
  15. Spectral-Native Optimization and Efficiency
  16. ...and 65 more sections

Key Result

Proposition 2.1

(Temporal Marginal Distribution). Let $\alpha_t = 1 - \beta_t$ and $\bar{\alpha}_t = \prod_{s=1}^t \alpha_s$ be the cumulative signal retention factors. The marginal distribution of the state at step $t$ is given by:

Figures (3)

  • Figure 1: Architecture evolution from temporal to parallel complex diffusion. DiT blocks are visualized in a simplified form (Attn. and FFN only) for brevity. (a) Standard temporal diffusion. (b) Conditional complex diffusion with independent quadrature components under fixed conditional boundary $\mathcal{X}_0$ in frequency manifold. (c) PaCoDi: Our proposed framework that utilizes Interactive Correction Branch to recover correlation constraints marginalized by the Mean Field Theory (MFT) approximation.
  • Figure 2: Data Distribution Visualization on ETTh1 ($L = 64$). Rows (1)–(3) correspond to PaCoDi DDPM, Diffusion-TS, and vanilla Diffusion. Columns (a)–(c) display Data Density Estimation, PCA, and t-SNE projections. Red and blue markers denote real and synthetic samples, respectively.
  • Figure 3: Computational Complexity Analysis

Theorems & Definitions (9)

  • Proposition 2.1
  • Theorem 3.1
  • Proposition 3.2
  • Definition 4.1
  • Theorem 4.2
  • Theorem A.1
  • Lemma C.1
  • Lemma F.1
  • Lemma F.2