06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
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
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