Expected Signature Estimators

This repo provides code for estimating expected signatures

$$\phi(T) := \mathbb{E}\Big[S(\mathbb{X})_{[0,T]}\Big]\in T((\mathbb{R}^d)),$$

from a collection of paths $\mathbb{X}^{\pi, 1}, \ldots, \mathbb{X}^{\pi, n}$. The paths may be obtained by chopping-and-shifting a single long observation.

We implement both the naive estimator

$$\hat{\phi}^{N, \pi}(T) := \frac{1}{N} \sum_{n=1}^N S\left(\mathbb{X}^{n, \pi}\right)_{[0,T]},$$

and the martingale-corrected estimator

$$\hat{\phi}^{N, \pi, c}(T) := \frac{1}{N} \sum_{n=1}^N S\left(\mathbb{X}^{n, \pi}\right)_{[0,T]} + \hat{c} S^c\left(\mathbb{X}^{n, \pi}\right)_{[0,T]},$$

where $S^c\left(\mathbb{X}^{n, \pi}\right)_{[0,T]}$ is an Ito correction.

The code is compatible with numpy arrays and torch tensors, using iisignature and signatory for signature computations.

These functions can be used directly into more general ML pipelines/models, as illustrated in the forks:

• conditional-sig-wasserstein-gans
• distribution-regression-streams
• gp-esig-classifier

References

Paper: Learning with Expected Signatures: Theory and Applications.

Spotlight Poster and Oral Presentation at ICML 2025.