My apologies if the question is a tad vague—I did try to search the literature for more, but didn't find anything (the similarity between the keywords "Takens" and "taken" on Google may be partly to blame).
If I understood correctly the Wikipedia page on Takens' theorem, and some discussions I had with people in applied (physics-related) fields that mentioned it to me, Taken's theorem essentially states that
if one has a system parameterized by a state (time-dependent) $x(t)\in\mathbb{R}^d$, then under some natural assumptions on $x$ (in particular, smooth enough) there exists $k$—related to the underlying dimension of the manifold on which $x$ "lives— and some "lag" $\tau$ such that $$ y(t) \stackrel{\rm def}{=} (x_1(t), x_1(t+\tau),\dots,x_1(t+(k-1) \tau)\in\mathbb{R}^k $$ is information-theoretically equivalent to the original parameterization $x$; and moreover, this new parameterization, now living in a "shadow manifold," preserves the topology of the original manifold.
(All imprecisions above definitely mine, I cannot claim I fully understand the theorem.)
This does strike me as a very useful tool, similar in spirit to some type of dimension reduction:
- bringing in domain-specific knowledge (to figure out $\tau$ and $k$, although apparently there have been recently methods designed to find these in a more automated way)
- allowing easy measuring (no need to perform a computationally heavy mapping for the change of parameters: instead, focusing in a single lagged coordinate of the data is enough)
My question is: has this theorem, or some version of it, been applied or explored in the context of computational learning? If so, where can I find references of it? If not, is there a reason why—e.g., "it does not apply in the usual settings we consider because [...]"?