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

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.