What is appropriate distance between signatures?

Question

This question is related to rough path theory. A link to wikipedia is: https://en.wikipedia.org/wiki/Rough_path

I can understand to an extent how to calculate signatures given a path. I am interested to know what kind of distance measure possible between two signatures arises from two different paths?

Answer 1

Here are three wooly suggestions for definitions of distance. Just definitions, not necessarily practical distances to calculate for given data. I assume your two paths X and Y have the same number of dimensions.

1. If you have the signatures truncated to a fixed level m, you can concatenate the step-m signature of X with the step-m signature of (Y backwards) to get a step-m signature Z. Then define the distance as the length of the unique shortest Lipschitz path whose step-m signature is Z. This is called the Carnot-Caratheodory distance on page 10 of the book "Multidimensional Stochastic Processes as Rough Paths: Theory and Applications" by Friz and Victoir.

2. If the signatures are truncated to level m, they just live in a finite dimensional vector space. You can pick any norm you like on that space and its value on the difference of the signatures will be a distance. In particular, you could take the homogenous norm, the maximum over i of (the $\ell_i$ norm of level $i$), which gives you an equivalent topology to the previous idea, as explained on page 12 of the same book.

3. Suppose you had the whole of each path, and not just its signature. Then you could take the signature of any sub-interval of the path. If, further, the paths were defined on the same interval, then you could pick a p and take the p-variation metric in the rough path sense as explained on wikipedia. This distance would change if you reparametrised one of the paths. Now, you can find a canonical path which agrees with each of your signatures (the tree reduced one if you have the full signature, or the shortest one if they are truncated at level m). If you can find a canonical parametrisation (easy in the truncated case - use length, but I don't know how to do this if you specify the whole signature), and sort out scaling the parametrisation to make both be over the same interval, then you can use this metric.