Discrete signature varieties

TL;DR

This paper investigates the algebraic structures of discrete signature varieties derived from time series, providing dimension calculations and partial solutions to the Chen-Chow theorem for complex-valued data.

Contribution

It introduces discrete signature varieties, computes their dimensions, and offers a partial algebraic solution to the Chen-Chow theorem for complex-valued time series.

Findings

Computed dimensions of discrete signature varieties in various cases

Identified algebraic properties of these varieties

Provided a partial algebraic solution to the Chen-Chow theorem

Abstract

Discrete signatures are invariants computed from time series corresponding to the discretised version of the signature of paths. We study the algebraic varieties arising from their images, the discrete signature varieties. We introduce them and compute their dimension in many cases. From a particular subclass of these varieties, we derive a partial solution to the Chen-Chow theorem for complex-valued time series.