Conjugation, loop and closure invariants of the iterated-integrals signature

TL;DR

This paper characterizes invariants of iterated-integrals signatures of paths, focusing on loop invariants, conjugation invariants, and closure invariants, with applications to shape analysis and time series.

Contribution

It provides a comprehensive characterization of loop, conjugation, and closure invariants within the iterated-integrals signature framework, connecting algebraic structures to geometric invariants.

Findings

Identifies features invariant under path starting point shifts.

Relates invariants to conjugation and closure properties.

Applies findings to piecewise linear trajectories and time series.

Abstract

Given a feature set for the shape of a closed loop, it is natural to ask which features in that set do not change when the starting point of the path is moved. For example, in two dimensions, the area enclosed by the path does not depend on the starting point. In the present article, we characterize such loop invariants among all those features known as interated integrals of a given path. Furthermore, we relate these to conjugation invariants, which are a canonical object of study when treating (tree reduced) paths as a group with multiplication given by the concatenation. Finally, closure invariants are a third class in this context which is of particular relevance when studying piecewise linear trajectories, e.g. given by linear interpolation of time series. Keywords: invariant features; concatenation of paths; combinatorial necklaces; shuffle algebra; free Lie algebra; signed area; signed volume; tree-like equivalence.