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：Signature是将路径表示为其迭代积分的无限序列的方法。在某些假设下，Signature可以唯一地刻画路径（平移和参数化除外）。因此，一个重要的问题是开发高效的算法来反演Signature，即从其截断的Signature信息中重构路径。在本文中，我们从理论和实用两个角度研究了Chang和Lyons（2019）最初提出的插入法。在描述我们的算法之后，我们给出了分段线性路径的收敛速率，并提供了它在Pytorch中的实现。该算法是并行化的，因此可以同时高效地反演一批Signature。本文通过实际和模拟示例来说明其性能。