  Reservoir Computing (RC) has become popular in recent years due to its fast
and efficient computational capabilities. Standard RC has been shown to be
equivalent in the asymptotic limit to Recurrent Kernels, which helps in
analyzing its expressive power. However, many well-established RC paradigms,
such as Leaky RC, Sparse RC, and Deep RC, are yet to be analyzed in such a way.
This study aims to fill this gap by providing an empirical analysis of the
equivalence of specific RC architectures with their corresponding Recurrent
Kernel formulation. We conduct a convergence study by varying the activation
function implemented in each architecture. Our study also sheds light on the
role of sparse connections in RC architectures and propose an optimal sparsity
level that depends on the reservoir size. Furthermore, our systematic analysis
shows that in Deep RC models, convergence is better achieved with successive
reservoirs of decreasing sizes.
