  The Echo State Network (ESN) is a specific recurrent network, which has
gained popularity during the last years. The model has a recurrent network
named reservoir, that is fixed during the learning process. The reservoir is
used for transforming the input space in a larger space. A fundamental property
that provokes an impact on the model accuracy is the Echo State Property (ESP).
There are two main theoretical results related to the ESP. First, a sufficient
condition for the ESP existence that involves the singular values of the
reservoir matrix. Second, a necessary condition for the ESP. The ESP can be
violated according to the spectral radius value of the reservoir matrix. There
is a theoretical gap between these necessary and sufficient conditions. This
article presents an empirical analysis of the accuracy and the projections of
reservoirs that satisfy this theoretical gap. It gives some insights about the
generation of the reservoir matrix. From previous works, it is already known
that the optimal accuracy is obtained near to the border of stability control
of the dynamics. Then, according to our empirical results, we can see that this
border seems to be closer to the sufficient conditions than to the necessary
conditions of the ESP.
