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.

翻译：摘要：路径签名是一种将路径表示为其迭代积分无穷序列的方法。在某些假设下，路径签名表征了路径，而且可以从截断的路径签名信息中重构路径，这是一个非常重要的问题。本文从理论和实践角度探讨了最近被Chang和Lyons(2019)提出的插入方法。在描述我们的方法之后，给出了在分段线性路径情况下的收敛速度，并给出了使用Pytorch的实现。该算法是并行化的，意味着同时反演一批签名非常高效。我们通过真实世界和模拟示例展示了该算法的性能。