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SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings
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Introduces SigDiffusion, which runs score-based diffusion on step-4 log-signature embeddings of time series and recovers the series with new closed-form Fourier/orthogonal-polynomial signature inversion formulae.
Problem
Adapting score-based diffusion models to generate long multivariate time series viewed as discretisations of continuous paths, and recovering a path from its signature for generation.
Approach
Run a score-based diffusion model in the Lie algebra (log-signature space), where the log-signature is a compact Euclidean embedding of a path, and invert back to the time series with new closed-form formulae expressing Fourier or orthogonal-polynomial coefficients as polynomial functions of the log-signature.
Key results
- On length-1000 series, SigDiffusion achieves the best discriminative score on HEPC (0.070) and Exchange Rates (0.278) and best predictive score on Exchange Rates (0.057) and Weather (0.168), while DDO is best on Sines discriminative score (0.040).
- SigDiffusion uses about 229K parameters and trains in 8 minutes on Sines, versus 4.12M parameters and 3.6 hours for DDO and 4.18M for Diffusion-TS.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · Sines | Discriminative Score: 0.040$\pm$.025 -> 0.100$\pm$.026 | Hurt |
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · Sines | Predictive Score: 0.187$\pm$.003 -> 0.191$\pm$.004 | Hurt |
| Use SigDiffusion instead of CSPD-GP (RNN) | Time-series generation · Predator-prey | Discriminative Score: 0.168$\pm$.024 -> 0.184$\pm$.058 | Hurt |
| Use SigDiffusion instead of DDO (γ = 10) | Time-series generation · Predator-prey | Predictive Score: 0.056$\pm$.000 -> 0.056$\pm$.000 | No clear effect |
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · HEPC | Discriminative Score: 0.081$\pm$.019 -> 0.070$\pm$.032 | Helped |
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · HEPC | Predictive Score: 0.044$\pm$.001 -> 0.050$\pm$.012 | Hurt |
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · Exchange Rates | Discriminative Score: 0.326$\pm$.102 -> 0.278$\pm$.062 | Helped |
| Use SigDiffusion instead of DDO (γ = 1) | Time-series generation · Exchange Rates | Predictive Score: 0.094$\pm$.004 -> 0.057$\pm$.001 | Helped |
| Use SigDiffusion instead of DDO (γ = 10) | Time-series generation · Weather | Discriminative Score: 0.356$\pm$.196 -> 0.350$\pm$.080 | Helped |
| Use SigDiffusion instead of DDO (γ = 10) | Time-series generation · Weather | Predictive Score: 0.307$\pm$.007 -> 0.168$\pm$.001 | Helped |
Surprising findings
- On Predator-prey, CSPD-GP (RNN) attains a better discriminative score (0.168) than SigDiffusion (0.184).
Limitations
- Inversion quality is bounded by signature truncation level, so high-frequency signals need high truncation and grow the feature dimension exponentially.
- Signature inversion formulae are for 1-D channels applied channel by channel, and cross-channel log-signature terms are discarded in experiments.
Experimental setup
Datasets: Sines Predator-prey (Lotka-Volterra) Household Electric Power Consumption (voltage) Exchange Rates Weather
Benchmarks: Sines Predator-prey HEPC Exchange Rates Weather
Metrics: discriminative score predictive score Kolmogorov-Smirnov test parameter count training time sampling time
Baselines: Diffusion-TS CSPD-GP (RNN) CSPD-GP (Transformer) Denoising Diffusion Operators (DDO)
Builds on: score-based diffusion (SDE) Diffusion-TS signature inversion (Insertion method)
Closest papers by meaning
- Time Series Diffusion in the Frequency Domain (ICML 2024 · similarity 0.90)
- Dimension-free Score Matching and Time Bootstrapping for Diffusion Models (NeurIPS 2025 · similarity 0.89)
- Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis (ICLR 2025 · similarity 0.88)
- Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models (NeurIPS 2024 · similarity 0.87)
- O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions (ICLR 2025 · similarity 0.87)
- LSCD: Lomb--Scargle Conditioned Diffusion for Time series Imputation (ICML 2025 · similarity 0.87)
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