How to calculate signatures numerically in rough path theory.

Question

The question is related to the link:https://en.wikipedia.org/wiki/Rough_path#Signature I search the internet looking for tutorial on how to calculate the signature sequence, but could not find a clear document. The first term is one the second term seems to be $X_t−X_s$ but after the 2nd term how can I calculate the other integral? Consider a simplest curve, I have a set of numbers which form a line and sorted properly. This is a function also a very simple curve. For such curve can we have a signature? How to calculate the terms? Any hint or reference would be helpful.

Answer 1

A rough path postulates the iterated integrals. So given a regular path, you must postulate or define (part of) the signature to make it into a rough path.

These are not unique! This is the point of rough path theory. Given a second order rough path, $(X_{st},\Bbb X_{st})$ with $X\in C^\alpha$ for $\alpha\in (1/3,1/2]$ you can define another rough path $(X_{st},\Bbb X_{st}+F_t-F_s)$ where $F$ is a $2\alpha$ Hoelder function.

These both have the same underlying path. However in the first case we have

$$\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}=\Bbb X_{st}$$

And in the second case:

$$\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}=\Bbb X_{st}+F_t-F_s$$

Both of these have the same underlying path but you have different iterated integrals.

The entire point of rough path theory is that you CANNOT CALCULATE ITERATED INTEGRALS FROM $X_t$ ALONE.

In rough paths, you postulate those values, and there are infinitely many ways to do this.

However from Lyons Victoir extension theorem, for $C^\alpha$ paths with $\alpha\in(1/3,1/2)$ is is always possible to construct a rough path lift (i.e. $\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}$) and there are other cases where we can define that iterated integral.

Answer 2

I would like to add to the other good answer that there are cases in which we can "compute" the integrals.

In ML applications, we usually look at paths as streams where we know the value of the path at certain times $t_1 < t_2, \ldots $ and we interpolate linearly in between. Thanks to Chen's identity, the signature of a concatenated path is the tensor product of the signatures (if you know what it means, this identity is a homomorphism between the two spaces!). This leads to easily computable signatures.

More details can be found in Appendix A of https://arxiv.org/abs/1905.08494 or directly in the text book from Terry Lyons: Differential Equations Driven by Rough Paths (Theorem 2.9 for Chen's identity). In the paper, they corroborate the other answer's statement:

However one of the insights of rough path theory [10] is that a path needs more than just its pointwise values to be fully determined.