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

arXiv: https://arxiv.org/abs/2406.10354 · HTML: https://mlsanity.com/paper/2406.10354/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 12 of its 79 references that are in the map.
- First posted on arXiv 14 Jun 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 12 of 12)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs](https://mlsanity.com/paper/1706.02633/) (2017)
- [C-RNN-GAN: Continuous recurrent neural networks with adversarial training](https://mlsanity.com/paper/1611.09904/) (2016)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [A Data-driven Market Simulator for Small Data Environments](https://mlsanity.com/paper/2006.14498/) (2020)
- [Reducing the Prior Mismatch of Stochastic Differential Equations for Diffusion-based Speech Enhancement](https://mlsanity.com/paper/2302.14748/) (2023)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Lecture notes on rough paths and applications to machine learning](https://mlsanity.com/paper/2404.06583/) (2024)
- [The insertion method to invert the signature of a path](https://mlsanity.com/paper/2304.01862/) (2023)

## Built on by (most cited first, 4 of 4)

- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)
- [Population Aware Diffusion for Time Series Generation](https://mlsanity.com/paper/2501.00910/) (2025)
- [Sparse Signature Coefficient Recovery via Kernels](https://mlsanity.com/paper/2412.08579/) (2024)
- [Tensor-to-Tensor Models with Fast Iterated Sum Features](https://mlsanity.com/paper/2506.06041/) (2025)

## Abstract (arXiv)

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.