  Reservoir computing is a machine learning framework that has been shown to be
able to replicate the chaotic attractor, including the fractal dimension and
the entire Lyapunov spectrum, of the dynamical system on which it is trained.
We quantitatively relate the generalized synchronization dynamics of a driven
reservoir during the training stage to the performance of the trained reservoir
computer at the attractor reconstruction task. We show that, in order to obtain
successful attractor reconstruction and Lyapunov spectrum estimation, the
largest conditional Lyapunov exponent of the driven reservoir must be
significantly more negative than the most negative Lyapunov exponent of the
target system. We also find that the maximal conditional Lyapunov exponent of
the reservoir depends strongly on the spectral radius of the reservoir
adjacency matrix, and therefore, for attractor reconstruction and Lyapunov
spectrum estimation, small spectral radius reservoir computers perform better
in general. Our arguments are supported by numerical examples on well-known
chaotic systems.
