Generalized iterated-sums signatures

TL;DR

This paper investigates the algebraic structure of a generalized iterated-sums signature, revealing its character property through a deformed quasi-shuffle product and introducing non-linear transformations relevant to machine learning.

Contribution

It extends the algebraic understanding of iterated-sums signatures and introduces new non-linear transformations with potential applications in machine learning.

Findings

Recovered the character property via a deformed quasi-shuffle product

Introduced three non-linear transformations on iterated-sums signatures

Analyzed properties of these transformations in the context of machine learning

Abstract

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kir\'aly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.