Signature matrices of membranes

TL;DR

This paper investigates the algebraic properties of membrane signatures, revealing that membranes share the same signature matrices and lack algebraic relations, unlike path signatures, with an efficient algorithm for their computation.

Contribution

It generalizes path signature results to membranes, showing they have identical signature matrices and no algebraic relations, with a new linear time computation algorithm.

Findings

Membranes and paths share the same signature matrices.

No algebraic relations exist among membrane signature matrices.

A linear time algorithm computes signature tensors for piecewise bilinear membranes.

Abstract

The signature of a membrane is a sequence of tensors whose entries are iterated integrals. We study algebraic properties of membrane signatures, with a focus on signature matrices of polynomial and piecewise bilinear membranes. Generalizing known results for path signatures, we show that the two families of membranes admit the same set of signature matrices and we examine the corresponding affine varieties. In particular, we prove that there are no algebraic relations on signature matrices of membranes, in contrast to the path case. We complement our results by a linear time algorithm for the computation of signature tensors for piecewise bilinear membranes.