  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 discretization 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 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 SigDiffusion with these inversion formulae results in highly
realistic time series generation, competitive with the current state-of-the-art
on various datasets of synthetic and real-world examples.
