Learning with Expected Signatures: Theory and Applications
Abstract
The expected signature maps a collection of data streams to a lower dimensional representation, with a remarkable property: the resulting feature tensor can fully characterize the data generating distribution. This "model-free" embedding has been successfully leveraged to build multiple domain-agnostic machine learning (ML) algorithms for time series and sequential data. The convergence results proved in this paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value, allowing for a more complete probabilistic interpretation of expected signature-based ML methods. Moreover, when the data generating process is a martingale, we suggest a simple modification of the expected signature estimator with significantly lower mean squared error and empirically demonstrate how it can be effectively applied to improve predictive performance.
Cite
@article{arxiv.2505.20465,
title = {Learning with Expected Signatures: Theory and Applications},
author = {Lorenzo Lucchese and Mikko S. Pakkanen and Almut E. D. Veraart},
journal= {arXiv preprint arXiv:2505.20465},
year = {2025}
}