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：签名是将路径表示为迭代积分构成的无穷序列。在特定假设下，签名可唯一确定路径（平移与重参数化不计）。因此核心问题之一是开发高效算法实现签名的求逆，即从（截断）签名信息重构路径。本文从理论与应用两个维度研究由Chang与Lyons (2019) 提出的插入式方法。在阐述该方法改进版本后，我们给出了针对分段线性路径的收敛速率，并提供了Pytorch实现。该算法具备并行化特性，可高效同时求解批量签名。其性能通过真实数据与模拟案例得到验证。