The insertion method to invert the signature of a path

The signature is a representation of a path as an infinite sequence of its iterated integrals. Under certain assumptions, the signature characterizes the path, up to translation and reparameterization. Therefore, a crucial question of interest is the development of efficient algorithms to invert the signature, i.e., to reconstruct the path from the information of its (truncated) signature. In this article, we study the insertion procedure, originally introduced by Chang and Lyons (2019), from both a theoretical and a practical point of view. After describing our version of the method, we give its rate of convergence for piecewise linear paths, accompanied by an implementation in Pytorch. The algorithm is parallelized, meaning that it is very efficient at inverting a batch of signatures simultaneously. Its performance is illustrated with both real-world and simulated examples.