An analytic and algebraic understanding of iterated integral signatures associated to continuous paths has played a central role in in a wide range of mathematical areas, such as the construction of stochastic integration for non-martingales with rough paths theory, to formal representations and expansions of solutions to (partial) differential equations. In recent years, the signature has proven to be an efficient feature map for machine learning tasks, where the learning task is related to time series data, or data streams.

In contrast to time series data, image data can naturally be seen as two-parameter fields taking values in multi-dimensional space, and in recent years there has been some research into the extension of the path signature to multi-parameter fields (see e.g. Chouk/Gubinelli 14, Lee/Oberhauser (21 and 23)). 

In this talk I will propose a new extension of the path signature to two-parameter fields motivated by expansions of solutions to certain hyperbolic PDEs with multiplicative noise. The algebraic structure of this object turns out to be rather complicated and I will discuss our current understanding of the challenges with going from 1 to 2 parameters, and provide some interesting observations related to a Chen type relation and a Shuffle type relation. At last I will briefly discuss the universality of the 2D signature, providing a universal approximation theorem, and discuss some open problems.  

This talk is based on forthcoming joint work with Joscha Diehl, Kurusch Ebrahimi-Fard,  and Samy Tindel, and is part of the Signatures for Images project for 2023/2024 at CAS.