Insertion algorithm for inverting the signature of a path

Understand

In this article we introduce the insertion method for reconstructing the path from its signature, i.e.

• inverting the signature of a path.
• For this purpose, we prove that a converging upper bound exists for the difference between the inserted n-th term and the (n+1)-th term of the normalised signature of a smooth path, and we also show that there exists a constant lower bound for a subsequence of the terms in the normalised signature of a piecewise linear path.
• We demonstrate our results with numerical examples.