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

TL;DR

This paper introduces a new generalization of the path signature inspired by fractional calculus, enhancing its application in solving fractional differential equations and improving machine learning tasks like digit recognition.

Contribution

It presents a novel fractional signature generalization and demonstrates its effectiveness in both mathematical modeling and practical machine learning applications.

Findings

Improved accuracy in handwritten digit recognition.

Effective description of solutions to fractional differential equations.

Enhanced feature representation for machine learning.

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.