Disclaimer: I know next to nothing about this.
But I think it’s something like: Looking at the data from a, possibly stochastic, dynamical system. and hoping to infer cool things about the kinds of hidden states it has, in some general sense, such as some measure of statistical of computational complexity, or how complicated or “large” the underlying state space, in some convenient representation, is.
TBH I don’t understand this framing, but possibly because I don’t come from a dynamical systems group; I just dabble in special cases thereof. Surely you either do physics, and work out the dynamics of your system from experiment, or you do statistics and select an appropriate model to minimise some estimated predictive loss trading off data set, model complexity and algorithmic complexity. I need to read more to understand the rationale here, clearly.
Anyway, tools seem to include inventing large spaces of hidden states (Takens embedding); does this get us some nice algebraic properties? Also, how does delay embedding relate? Is that the same? Sample complexity results seem to be scanty, possibly because they usually want their chaos to be deterministic and admitting noise would be fiddly.
OTOH, from a statistical perspective there are lots of useful techniques to infer special classes of dynamical systems state-space. It is especially interesting inmgrammatical inference of formal syntax where there are many lovely and faintly depressing computational complexity results.
Stuff that I might actually use
Hirata’s reconstruction looks like good clean decorative fun — you can represent graphs by an equivalent dynamical system.