SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings
Barbora Barancikova Zhuoyue Huang Cristopher Salvi
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.
diffusion models, path signatures, time series
We introduce SigDiffusion, a novel Lie algebra preserving diffusion model operating on log-signature embeddings of a time series; we also provide new closed-form signature inversion formulae.