  Echo State Networks (ESNs) are a special type of the temporally deep network
model, the Recurrent Neural Network (RNN), where the recurrent matrix is
carefully designed and both the recurrent and input matrices are fixed. An ESN
uses the linearity of the activation function of the output units to simplify
the learning of the output matrix. In this paper, we devise a special technique
that take advantage of this linearity in the output units of an ESN, to learn
the input and recurrent matrices. This has not been done in earlier ESNs due to
their well known difficulty in learning those matrices. Compared to the
technique of BackPropagation Through Time (BPTT) in learning general RNNs, our
proposed method exploits linearity of activation function in the output units
to formulate the relationships amongst the various matrices in an RNN. These
relationships results in the gradient of the cost function having an analytical
form and being more accurate. This would enable us to compute the gradients
instead of obtaining them by recursion as in BPTT. Experimental results on
phone state classification show that learning one or both the input and
recurrent matrices in an ESN yields superior results compared to traditional
ESNs that do not learn these matrices, especially when longer time steps are
used.
