I gave this blackboard talk on May 13, 2022 at the 15th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis.
The talk was largely based on joint work with Joscha Diehl, Micheal Ruddy, Jeremy Reizenstein and Nikolas Tapia.
Looking at the action of the orthogonal group, we apply Fels-Olver’s moving frame method paired with the log-signature transform to construct a set of integral invariants for curves in 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 R^d under rigid motions and an explicit method to compare curves up to these transformations. In this talk, we furthermore present a so far unpublished, more explicit new description of the moving frame via the QR-decomposition of a certain matrix build from the level two signature data of the curve.
Furthermore, this talk discusses the new result that such a full characterization of a path up to group action (and tree-like equivalence) via iterated-integral invariants does not exist for the special linear group. We instead hint a so far only conjectured kind of “Determinantensatz” for the iterated-integral signature which would describe the equivalence relation on paths given by only looking at special linear iterated-integral invariants.