We argue that the success of reservoir computing lies within the separation
capacity of the reservoirs and show that the expected separation capacity of
random linear reservoirs is fully characterised by the spectral decomposition
of an associated generalised matrix of moments. Of particular interest are
reservoirs with Gaussian matrices that are either symmetric or whose entries
are all independent. In the symmetric case, we prove that the separation
capacity always deteriorates with time; while for short inputs, separation with
large reservoirs is best achieved when the entries of the matrix are scaled
with a factor $\rho_T/\sqrt{N}$, where $N$ is the dimension of the reservoir
and $\rho_T$ depends on the maximum length of the input time series. In the
i.i.d. case, we establish that optimal separation with large reservoirs is
consistently achieved when the entries of the reservoir matrix are scaled with
the exact factor $1/\sqrt{N}$. We further give upper bounds on the quality of
separation in function of the length of the time series. We complement this
analysis with an investigation of the likelihood of this separation and the
impact of the chosen architecture on separation consistency.

通过研究谱分解和矩阵的矩的关系，我们论证了储水池计算的成功取决于储水池的分离能力。对于具有高斯矩阵且对称或独立的储水池，我们分别证明了分离能力随时间的恶化，以及当矩阵条目与输入时序的最大长度相关时，大储水池在短输入情况下的最佳分离效果。通过研究时间序列长度的影响，我们进一步给出了分离质量的上限，并探究了选择架构对分离一致性的影响。