Orthogonal and special linear invariants of paths via the iterated-integral signature

01.07.2022, 11:00  –  Golm, Haus 9, 2.22 und online
Arbeitsgruppenseminar Analysis

Rosa Preiss (University of Potsdam)

Looking at the action of the orthogonal group on a path in a finite dimensional real vector space, 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 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.

Based on joint work with Joscha Diehl, Michael Ruddy, Nikolas Tapia

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks, please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.