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 here 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 here 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, even with nonlinear dynamics. e.g. in plain old model-based count time series such as branching processes, and grammatical inference of formal syntax, and nonlinear system identification.
I would be interested to see a compelling new insight from the dynamical system perspective on these problems. New estimators; models outside the ken of Kalman filters?
Stuff that I might actually use
Hirata’s reconstruction looks like good clean decorative fun — you can represent graphs by an equivalent dynamical system.