We revisit (higher-order) translation operators on rough paths, in both the geometric and branched setting. As in Hairer's work on the renormalization of singular SPDEs we propose a purely algebraic view on the matter. Recent advances in the theory of regularity structures, especially the Hopf algebraic interplay of positive and negative renormalization of Bruned-Hairer-Zambotti (2016), are seen to have precise counterparts in the rough path context, even with a similar formalism (short of polynomial decorations and colourings). Renormalization is then seen to correspond precisely to (higher-order) rough path translation.

We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. It allows us to describe the relation between the coefficients of the signature of a piecewise continuously differentiable path transformed by a polynomial map and the coefficients of the signature of the initial path.

We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver’s moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in $${\mathbb {R}}^d$$ R d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in $${\mathbb {R}}^d$$ R d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.

No abstract