A Topological Approach to Mapping Space Signatures

Abstract

A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0, 1] → R n and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0, 1] d → R n for any d ∈ N, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d ≥ 2. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.

2-d signature of images and texture classification

“…(ii) Get a better grasp on the underlying algebraic structures related to 2-d signatures. This study should go beyond the investigation lead in [Giu+22] (restricted to the d 12 x differentials introduced in (8)), and will probably involve advanced structures such as Hopf algebras. One of the main objective in this direction is to construct all signatures (up to a given order) in a systematic way.…”

“…We refer to [CG14, Section 5.1] for an account on these relations. See also the aforementioned article [Giu+22].…”

“…Since the very definition of S(x) for paths indexed by (s, t) is still open, the literature on its algebra is obviously scarce. The recent preprint [Giu+22] gives an account on the algebraic structure generated by the increments…”