# The insertion method to invert the signature of a path

arXiv: https://arxiv.org/abs/2304.01862 · HTML: https://mlsanity.com/paper/2304.01862/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2023.

## Built on by (most cited first, 3 of 3)

- [SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings](https://mlsanity.com/paper/2406.10354/) (2024)
- [Path Signatures and Graph Neural Networks for Slow Earthquake Analysis: Better Together?](https://mlsanity.com/paper/2402.03558/) (2024)
- [Probabilistic Signature Inversion: Learning Conditional Distributions from Truncated Signatures](https://mlsanity.com/paper/2606.15332/) (2026)

## Abstract (arXiv)

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.