  Numerical evaluations of the memory capacity (MC) of recurrent neural
networks reported in the literature often contradict well-established
theoretical bounds. In this paper, we study the case of linear echo state
networks, for which the total memory capacity has been proven to be equal to
the rank of the corresponding Kalman controllability matrix. We shed light on
various reasons for the inaccurate numerical estimations of the memory, and we
show that these issues, often overlooked in the recent literature, are of an
exclusively numerical nature. More explicitly, we prove that when the Krylov
structure of the linear MC is ignored, a gap between the theoretical MC and its
empirical counterpart is introduced. As a solution, we develop robust numerical
approaches by exploiting a result of MC neutrality with respect to the input
mask matrix. Simulations show that the memory curves that are recovered using
the proposed methods fully agree with the theory.
