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The Exponentially Weighted Signature

Alexandre Bloch, Samuel N. Cohen, Terry Lyons, Joël Mouterde, Benjamin Walker

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.

The Exponentially Weighted Signature

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.
Paper Structure (30 sections, 25 theorems, 198 equations, 1 figure, 3 tables)

This paper contains 30 sections, 25 theorems, 198 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The Exponentially Weighted Signature
  3. Dynamics of the EWS
  4. The Flow Operator
  5. The derivation operator
  6. The EWS is the Solution of a Linear CDE
  7. Further Interpretations of the EWS
  8. Equivalence to the Signature of an Exponentially Weighted Path
  9. Relationship to Neural Controlled Differential Equations
  10. Relationship to State Space Models
  11. The EWS Generalises the Fourier & Laplace Transforms of the Path
  12. Benefits of Non-Diagonal Operators: A Duffing Oscillator Example
  13. Analytic Well-posedness
  14. Algebraic Properties of the EWS
  15. Chen's Identity
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $K \subset \mathcal{V}^p([a, b], V)$ be a compact set of paths with $p < 2$. Let $\mathcal{F} \subset C(K, \mathbb{R})$ be the set of continuous functions such that $F(X) = F(Y)$ whenever $S(X) = S(Y)$. Then the set of linear functionals of the signature is dense in $\mathcal{F}$. That is, for a

Figures (1)

  • Figure 1: Five representative test trajectories for the coupled oscillatory SDE task. Ground truth (black) is shown alongside predictions from the EWS (blue), EFM (orange), and classical signature (green).

Theorems & Definitions (67)

  • Definition 1.1: Signature lyons2007differential
  • Theorem 1.2: Universality
  • Definition 1.3: Exponentially Fading Memory Signature jaber2025exponentiallyfadingmemorysignature
  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 57 more