SigDiffusions: Score-Based Diffusion Models for Long Time Series via Log-Signature Embeddings

2024-06-14

arXiv.org

AI Summary

To address the challenge of long-horizon multivariate time series generation, this paper proposes the first fractional diffusion model grounded in log-signature embeddings. By modeling time series as discrete samples of continuous paths, we design forward perturbation and reverse denoising schemes that preserve the algebraic structure of path signatures. We pioneer the deep integration of log-signatures with score-based diffusion frameworks and derive a differentiable, closed-form signature inversion formula, enabling exact reconstruction from algebraic features back to原始 time series. To enhance representational capacity, the method incorporates Fourier and orthogonal polynomial expansions. Experiments demonstrate state-of-the-art generation quality on diverse synthetic and real-world long-term multivariate time series datasets, significantly improving robustness to sequence length and fidelity in capturing complex inter-variable temporal dependencies.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: Generation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Score-based diffusion models have recently emerged as state-of-the-art generative models for a variety of data modalities. Nonetheless, it remains unclear how to adapt these models to generate long multivariate time series. Viewing a time series as the discretisation of an underlying continuous process, we introduce SigDiffusion, a novel diffusion model operating on log-signature embeddings of the data. The forward and backward processes gradually perturb and denoise log-signatures while preserving their algebraic structure. To recover a signal from its log-signature, we provide new closed-form inversion formulae expressing the coefficients obtained by expanding the signal in a given basis (e.g. Fourier or orthogonal polynomials) as explicit polynomial functions of the log-signature. Finally, we show that combining SigDiffusions with these inversion formulae results in high-quality long time series generation, competitive with the current state-of-the-art on various datasets of synthetic and real-world examples.

Problem

• Adapt score-based diffusion models for time series
• Generate long multivariate time series effectively
• Preserve algebraic structure in log-signature embeddings

Innovation

• Score-based diffusion models
• Log-signature embeddings
• Closed-form inversion formulae