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My background is in rough paths theory.

In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are needed to construct these integrals are the iterated integrals $\int_s^t\int_s^{r_2}df(r_1)\otimes df(r_2)$, … where the amount of integrals needed increase the more irregular $f$ becomes. In particular if $f$ is $\alpha$-Hölder then you must construct $\lfloor 1/\alpha \rfloor$ many iterated integrals. The point is that these iterated integrals cannot be defined in a canonical way therefore they must be constructed in other ways and there has been a great deal of effort into constructing these iterated integrals. Once you construct the first $\lfloor 1/\alpha \rfloor$ many iterated integrals, the remaining integrals don't matter from the perspective of rough paths theory.

The collection of all iterated integrals of $f$ is known as the signature of $f$.

However recently in machine learning there has been work done on the signature method. All data is discrete, maybe smoothly interpolated in different ways, but ultimately everything is smooth. Therefore all iterated integrals are defined canonically.

This seems at odds with my background in rough paths theory. In rough paths theory, it is only the noncanonically defined iterated integrals that provide new information.

My question isn't about the signature method or machine learning per se but more broad:

What does the signature of a smooth path tell you about the path? What can you do with the signature of a smooth path?

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The signature of a bounded variation path characterizes this path up to tree like equivalence

B. Hambly, T. Lyons: Uniqueness for the signature of a path of bounded variation and the reduced path group

and one can reconstruct the path from its signature, see also this link.

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  • $\begingroup$ Thanks for the post. I've seen these papers before but I'm not completely convinced. These papers are about the signature on a fixed time interval. I am convinced that the signature on a fixed time interval tells you more than just the value of a function on a fixed time interval. But what if you know the entire path? The signature is then defined canonically from the path, so in some sense it can't contain more information right? $\endgroup$
    – user479223
    Commented Aug 19 at 19:25
  • $\begingroup$ To clarify, to me the signature of a path is the map $(s,t)\mapsto S(s,t):=(1,\int_s^tdf, ...)$. It is trivial to reconstruct $f$ from its signature. What makes the papers linked nontrivial is that you can reconstruct the path from just the element $S(0,1)$. Under what circumstance would you know $S(0,1)$ but not $f$? $\endgroup$
    – user479223
    Commented Aug 21 at 13:33
  • $\begingroup$ @user479223 Maybe you can think of it as a dimensional reduction result? If you have the signature, which is just a list of numbers, you can reconstruct the whole path canonically, which a priori is a lot more complicated. I think in numerical applications this tractability is crucial. Your map is very difficult I think to actually compute, and it would be a lot of wasted resources. $\endgroup$
    – Nate River
    Commented Aug 22 at 13:31
  • $\begingroup$ It is definitely interesting. I think the “full path” signature you have has a lot of redundant info, precisely because of this reconstruction result… $\endgroup$
    – Nate River
    Commented Aug 22 at 13:35
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    $\begingroup$ @NateRiver So your point is not that the signature gives any extra information, but that the signature on a fixed time interval is a way of representing the information contained already in the path. That is helpful. $\endgroup$
    – user479223
    Commented Aug 22 at 14:02

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