Varieties of Signature Tensors

TL;DR

This paper explores the algebraic geometric structure of signature tensors from various deterministic and stochastic paths, including polynomial, piecewise linear, and Brownian motion, revealing new varieties and their properties.

Contribution

It introduces algebraic varieties of signature tensors for both deterministic and stochastic paths, connecting rough path theory with algebraic geometry.

Findings

Defined new varieties of signature tensors for different path classes

Analyzed algebraic properties of signature tensors from Brownian motion

Connected rough path signatures with algebraic geometric structures

Abstract

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.