The Exponentially Weighted Signature

TL;DR

The paper introduces the Exponentially Weighted Signature (EWS), a novel path representation that incorporates contextual temporal weighting and richer memory dynamics, enhancing expressivity over traditional signatures.

Contribution

It generalizes the EFM signature using linear operators, enabling cross-channel coupling, oscillatory and regime-dependent behaviour, and integrates into a learnable, algebraically structured framework.

Findings

EWS outperforms classical signature and EFM in regression tasks.

EWS captures complex path dynamics more effectively.

Framework allows efficient computation and gradient-based learning.

Abstract

The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.