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：签名是将一条路径表示为无穷级数形式的迭代积分序列。在特定假设下，签名（在平移和重参数化意义下）唯一确定该路径。因此，一个关键研究问题在于开发高效的反演算法，即从（截断）签名信息中重构原始路径。本文从理论与实践两个层面系统研究由Chang与Lyons（2019）提出的插入方法。在描述本文改进版本后，我们给出了该方法对分段线性路径的收敛速率，并提供了基于PyTorch的实现。该算法采用并行化设计，可高效同步反演批量签名。其性能通过实际数据与模拟示例得到验证。