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

TL;DR

SigDiffusion introduces a novel score-based diffusion model operating on log-signature embeddings to generate high-quality long multivariate time series, effectively capturing their continuous underlying processes.

Contribution

The paper presents SigDiffusion, a new diffusion model that uses log-signature embeddings and provides closed-form inversion formulas for reconstructing signals, enabling effective long time series generation.

Findings

Achieves high-quality long time series generation

Competitive performance on synthetic and real-world datasets

Introduces closed-form inversion formulas for log-signature coefficients

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.