A multiplicative surface signature through its Magnus expansion

TL;DR

This work extends rough-path theory to two dimensions by developing a non-commutative, two-parameter sewing framework in the setting of crossed modules. It establishes a 2D sewing lemma with approximate Chen and Stokes identities, proves extension theorems for both paths and surfaces in homogeneous groups, and defines a robust theory of rough surfaces with a Magnus-type view of surface log-signatures. The results yield existence and uniqueness of higher-level surface data (2-cocycles) and provide continuous extension maps, enabling surface features to be computed beyond boundary integrals. Through the Magnus expansion lens and Kapranov’s framework, the paper connects surface signatures to higher gauge theory constructions and opens avenues for multi-parameter, algebraically rich surface data in stochastic analysis and geometric topology.

Abstract

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-Hölder 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.

Figures (6)

• Figure 1: Examples of the five cases of the midway partition of a dyadic rectangle. The blue rectangle is $\square_A$ and the green rectangle is $\square_B$.
• Figure 2: The first $4$ iterations of the midway partition of the dyadic rectangle $[0,\frac{3}{4}]\times [0,\frac{1}{4}] \in \mathsf{Rect}^0$. The rectangles colored red are elementary squares in $\mathsf{Rect}^1$ that partition the original rectangle. These correspond to $b_i$ in \ref{['eq:lassos']}.
• Figure 3: Examples of surface splitting. In the right figure, the mapping $\psi_B$ collapses the bottom left triangle of the left square to the green line on the boundary of the top square; in particular, the entire brown diagonal line in the left square is mapped to the intersection point of the green and blue lines in the top square.
• Figure 4: Two ways of subdividing the same square.
• Figure 5: $\Gamma^\cl_m$ etc.

Theorems & Definitions (48)

• lemma 1
• remark 1
• definition 1
• remark 2
• proposition 1
• lemma 2
• proof
• lemma 3
• proof
• proof : Proof of \ref{['prop:subcontrol']}