🤖 AI Summary
Classical path signatures struggle to characterize solutions of fractional-order dynamical systems. Method: This work introduces Caputo-type fractional calculus into path signature theory for the first time, proposing a generalized path signature that exactly represents solutions of linear fractional-order differential equations. A computationally efficient simplified variant is further designed, balancing mathematical rigor with practical machine learning applicability. The framework maps temporal paths to fractional-order feature embeddings, enhancing temporal modeling capacity. Results: On the MNIST handwritten digit recognition task, the proposed method achieves a statistically significant improvement in classification accuracy over classical path signatures, empirically validating its effectiveness and generalizability as a novel temporal feature representation.
📝 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.