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Learning with Expected Signatures: Theory and Applications
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Proves consistency and asymptotic normality of the discrete-time expected signature estimator for canonical geometric stochastic processes, and proposes a martingale control-variate correction that lowers MSE, tested on existing expected-signature ML models.
Problem
Expected signature estimators are computed from discretely observed paths, but their statistical properties relative to the latent continuous-time expected signature were not established for irregular partitions, dependent samples, or non-bounded-variation processes.
Approach
Establishes L^m in-fill convergence and long-span consistency/CLT for the empirical expected signature estimator under continuity and mixing conditions, and introduces a martingale control-variate (Itô-based) correction that is unbiased for martingales and reduces variance.
Key results
- On the OU synthetic dataset, GPES-MC improves accuracy from 62.20% to 88.26% over GPES (Bidim: 79.33% to 88.97%).
- On the FBM dataset, GPES-MC (95.26%) does not significantly differ from GPES (95.62%), p=0.15.
- Ideal gas and rough volatility SES experiments show MC gives mild, statistically non-significant MSE changes (all p>0.2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add martingale correction (ablation) | GPES · FBM | accuracy: 95.62 (0.18) -> 95.26 (0.70) | Hurt |
| add martingale correction (ablation) | GPES · OU | accuracy: 62.20 (0.70) -> 88.26 (0.31) | Helped |
| add martingale correction (ablation) | GPES · Bidim | accuracy: 79.33 (0.46) -> 88.97 (0.44) | Helped |
| add martingale correction (ablation) | SES · Ideal gas, r1 = 0.35 × cube root of V/N | MSE: $1.27\ (0.23)$ -> $1.31\ (0.45)$ | No clear effect |
| add martingale correction (ablation) | SES · Ideal gas, r2 = 0.65 × cube root of V/N | MSE: $0.09\ (0.03)$ -> $0.07\ (0.02)$ | No clear effect |
| add martingale correction (ablation) | SES · Rough volatility, N=20 | MSE: 1.49 (0.39) -> 1.26 (0.48) | No clear effect |
| add martingale correction (ablation) | SES · Rough volatility, N=50 | MSE: 0.33 (0.13) -> 0.31 (0.09) | No clear effect |
| add martingale correction (ablation) | SES · Rough volatility, N=100 | MSE: 0.20 (0.08) -> 0.19 (0.05) | No clear effect |
Surprising findings
- Martingale correction improves GPES substantially on OU and Bidim datasets, even though most processes are not martingales.
Limitations
- Martingale correction introduces bias when the process is not a martingale; benefit is empirical.
- Strong mixing assumption is restrictive for long-span asymptotics; fBm increments are not strongly mixing so CLT does not apply under chop sampling.
- Ideal gas and rough volatility improvements are not statistically significant.
Experimental setup
Datasets: FBM synthetic OU synthetic Bidim synthetic ideal gas simulation rough volatility simulation
Benchmarks: FBM OU Bidim ideal gas rough volatility
Metrics: predictive accuracy mean squared error estimator variance t-test p-value
Baselines: GPES SES classic expected signature estimator
Builds on: signature transform rough path theory stochastic sewing lemma GPES SES
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