A multiplicative surface signature through its Magnus expansion

In the last decade, the concept of path signature has achieved significant success in data science applications. It offers a powerful set of features that effectively capture and describe the characteristics of paths or sequential data. This is partly explained by the fact that the signature of a path can be computed in linear time, using a dynamic programming principle based on Chen's identity. The path signature can be viewed as a specific example of a product or time-/path-ordered integral. In other words, it represents a one-parameter object built on iterated integrals over a path. Defining a signature over surfaces requires considering iterated integrals over these surfaces, effectively introducing an additional parameter, resulting in a two-parameter signature. This extended signature is intrinsically connected to a non-commutative generalization of Stokes' theorem, which is fundamentally connected to the concept of crossed modules of groups. The latter provides a well-established framework in higher gauge theory, where crossed modules with feedback maps exhibiting non-trivial kernels, combined with multiparameter iterated integrals, play a pivotal role. Building on Kapranov's work, we explore the surface analog of the log-signature for paths by introducing a Magnus-type formula for the logarithm of the surface signature. This expression takes values in a free crossed module of Lie algebras, defined over a free Lie algebra. We furthermore prove a non-commutative sewing lemma applicable to the crossed module setting and give a definition of rough surface in the so-called Young-H\"older regularity regime along with a corresponding continuous extension theorem. This approach enables the analysis and computation of surface features that go beyond what can be expressed by computing line integrals along the boundary of a surface.