Signature matrices of membranes

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.

💡 Research Summary

The paper investigates the second‑level iterated‑integral signature—called the signature matrix—of two‑parameter objects known as membranes. While the path signature (one‑parameter case) has been extensively studied, its two‑parameter analogue, the id‑signature introduced in recent works, is far less understood. The authors focus on two natural families of membranes: (i) polynomial membranes whose coordinate functions are bivariate polynomials of prescribed degrees ((m,n)), and (ii) piecewise bilinear membranes obtained by bilinear interpolation of data on an (m\times n) grid.

A central technical device is the “moment membrane” (\operatorname{Mom}{m,n}(s,t) = (s,s^{2},\dots,s^{m})\otimes(t,t^{2},\dots,t^{n})). Every polynomial membrane of order ((m,n)) can be written as a linear transformation of this dictionary membrane, (X = A,\operatorname{Mom}{m,n}), where the matrix (A) encodes the coefficients of the polynomial. Using the equivariance of signatures under linear maps (Lemma 2.6), the signature matrix transforms by congruence: \