The Exponentially Weighted Signature

2026-03-19

AI Summary

This work proposes the Exponentially Weighted Signature (EWS) to overcome the limitation of classical path signatures, which treat historical information uniformly without dynamic weighting. By introducing cross-channel coupling and context awareness through a general bounded linear operator, EWS preserves the algebraic advantages of signatures while enabling complex memory behaviors—such as oscillatory, growing, and state-dependent dynamics. The method extends exponentially decaying memory to non-diagonal linear operators and unifies state-space models with Laplace/Fourier transforms of paths. Built upon a tensor-algebraic framework of linear controlled differential equations and parameterized via semigroup generators, EWS supports gradient-learnable, group-like computational structures. Experiments demonstrate that EWS significantly outperforms classical signatures and the Exponential Family Model (EFM) in two stochastic differential equation regression tasks, exhibiting superior expressive capacity.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• signature
• temporal weighting
• memory dynamics
• path representation
• time series

Innovation

• Exponentially Weighted Signature
• tensor algebra
• linear controlled differential equation
• memory dynamics
• signature method