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Fractional signature: a generalisation of the signature inspired by fractional calculus

José Manuel Corcuera, Rubén Jiménez

TL;DR

The paper introduces a fractional generalization of the path signature, $S^{\alpha}(X)$, built on the Riemann-Liouville integral and linked to Caputo fractional differential equations, recovering the classical signature at $\alpha=1$. To enable practical ML use, it also defines a discrete fractional signature ${}_{d}DS^{\alpha}(X)$ for piecewise linear paths, which satisfies a Chen-type identity and aligns with the fractional signature on basic cases. The authors demonstrate the utility of the new signatures in a toy MNIST digit-recognition task, showing that the discrete fractional signature can outperform the classical signature when used with gradient-boosting models, particularly for certain choices of $\alpha$ and level truncation. Overall, the work provides a new path-based representation that potentially improves modeling of fractional-order dynamics and offers practical ML benefits, motivating further theoretical and applied exploration of uniqueness, invariance properties, and broader applications.

Abstract

In this paper, we propose a novel generalisation of the signature of a path, motivated by fractional calculus, which is able to describe the solutions of linear Caputo controlled FDEs. We also propose another generalisation of the signature, inspired by the previous one, but more convenient to use in machine learning. Finally, we test this last signature in a toy application to the problem of handwritten digit recognition, where significant improvements in accuracy rates are observed compared to those of the original signature.

Fractional signature: a generalisation of the signature inspired by fractional calculus

TL;DR

The paper introduces a fractional generalization of the path signature, , built on the Riemann-Liouville integral and linked to Caputo fractional differential equations, recovering the classical signature at . To enable practical ML use, it also defines a discrete fractional signature for piecewise linear paths, which satisfies a Chen-type identity and aligns with the fractional signature on basic cases. The authors demonstrate the utility of the new signatures in a toy MNIST digit-recognition task, showing that the discrete fractional signature can outperform the classical signature when used with gradient-boosting models, particularly for certain choices of and level truncation. Overall, the work provides a new path-based representation that potentially improves modeling of fractional-order dynamics and offers practical ML benefits, motivating further theoretical and applied exploration of uniqueness, invariance properties, and broader applications.

Abstract

In this paper, we propose a novel generalisation of the signature of a path, motivated by fractional calculus, which is able to describe the solutions of linear Caputo controlled FDEs. We also propose another generalisation of the signature, inspired by the previous one, but more convenient to use in machine learning. Finally, we test this last signature in a toy application to the problem of handwritten digit recognition, where significant improvements in accuracy rates are observed compared to those of the original signature.
Paper Structure (5 sections, 6 theorems, 20 equations, 1 figure, 1 table)

This paper contains 5 sections, 6 theorems, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Fractional signature
  3. Discrete fractional signature
  4. Application to machine learning
  5. Conclusions

Key Result

Proposition 1

Let $[a,b] \subseteq \mathbb{R}$ be an interval $(a < b)$, let $\alpha > 0$ be a parameter and let $X:[a,b] \longrightarrow \mathbb{R}^{d}$ be a linear path between the points $(A_{1}, \dots, A_{d})$ and $(B_{1}, \dots, B_{d}) \in \mathbb{R}^{d}$, that is, $X_{t}^{i} = \frac{1}{b-a}(A_{i}(b-t) + B_{ where $\beta$ denotes the usual beta function.

Figures (1)

  • Figure 1: Accuracy as a function of $\alpha$

Theorems & Definitions (16)

  • Definition 1: Signature of a path
  • Definition 2: Riemann-Liouville fractional integral
  • Definition 3: Fractional signature of a path
  • Proposition 1: Fractional signature of a linear path
  • proof
  • Proposition 2
  • proof
  • Definition 4: Solution of a Caputo fractional initial value problem
  • Theorem 1: Picard's theorem for linear Caputo fractional differential equations
  • Theorem 2
  • ...and 6 more