Decomposing Tensor Spaces via Path Signatures

TL;DR

This paper explores the mathematical structure of path signatures, revealing how tensor spaces decompose through representation theory and identifying constraints on tensor rank and symmetry, advancing understanding in stochastic analysis.

Contribution

It introduces a novel decomposition of tensor spaces associated with path signatures using representation theory and links this to path invariants and tensor constraints.

Findings

Decomposition of tensor space via representation theory

Constraints on rank and symmetry of signature tensors

Connection between signature variety parametrization and tensor structure

Abstract

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.