# Applications of Takens' theorem to TCS?

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 [...]"?

• It's not clear how to incorporate the temporal aspect -- the theorem, after all, deals with dynamical systems, while in classic dimensionality reduction applications (PCA, Johnson-Lindenstrauss), one is given a static point set. – Aryeh Jul 27 '17 at 13:32
• @Aryeh Good point... however, there must be some relevant setting in CLT or theoretical ML where one has a time series, a Markov chain, or some process that can be cast under that time-varying formulation. – Clement C. Jul 28 '17 at 14:19
• Certainly CLTs for Markov chains are known. – Aryeh Jul 28 '17 at 14:47
• @Aryeh My bad, that was an acronym overload. By "CLT," I meant Computational Learning Theory (not Central Limit Theorem). – Clement C. Jul 28 '17 at 14:53
• @ClementC. I would suggest the following query on google search, "takens -taken", without the quotes. – Tayfun Pay Jul 29 '17 at 13:58