Rank and symmetries of signature tensors

TL;DR

This paper systematically studies the properties of signature tensors of paths, including rank, symmetries, and conciseness, providing bounds, characterizations, and structural insights.

Contribution

It establishes a sharp upper bound on the rank of signature tensors for piecewise linear paths and characterizes symmetry and conciseness properties.

Findings

Sharp upper bound on rank of signature tensors for piecewise linear paths

No skew-symmetric signature tensors of order three or more

Paths with non-concise signature tensors have a geometric characterization

Abstract

The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We prove a sharp upper bound on the rank of signature tensors of piecewise linear paths. We show that there are no skew-symmetric signature tensors of order three or more, and we also prove that specific instances of partial symmetry can only happen for tensors of order three. Finally, we give a simple geometric characterization of paths whose signature tensors are not concise.