Abstract for the talk on 29.05.2019 (11:00 h)Seminar on Nonlinear Algebra
Weijun Xu (University of Oxford)
Inverting the signature of a path
The signature of a path is a sequence of iterated coordinate integrals along the path. We aim at reconstructing a path from its signature. In the special case of lattice paths, one can obtain exact recovery based on a simple algebraic observation. For general continuously differentiable curves, we develop an explicit procedure that allows to reconstruct the path via piecewise linear approximations. The errors in the approximation can be quantified in terms of the level of signature used and modulus of continuity of the derivative of the path. The main idea is inspired by the simple procedure for lattice paths, but with serious involvement of analysis. A key ingredient is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales.