Learning with Expected Signatures: Theory and Applications

TL;DR

This paper explores the theoretical foundations and practical applications of expected signatures in machine learning, providing convergence results, a modified estimator for martingale data, and demonstrating improved predictive performance.

Contribution

It offers new convergence proofs linking empirical and theoretical expected signatures, and introduces a simplified estimator for martingale processes with enhanced accuracy.

Findings

Convergence established between empirical and theoretical expected signatures.

Modified estimator reduces mean squared error for martingale data.

Empirical results show improved predictive performance using the new estimator.

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.