This article extends previous analysis of the gradient decay to a class of discrete-time fully recurrent networks, called dynamical recurrent neural networks, obtained by modeling synapses as finite impulse response (FIR) filters instead of multiplicative scalars. Using elementary matrix manipulations, we provide an upper bound on the norm of the weight matrix, ensuring that the gradient vector, when propagated in a reverse manner in time through the error-propagation network, decays exponentially to zero. This bound applies to all recurrent FIR architecture proposals, as well as fixed-point recurrent networks, regardless of delay and connectivity. In addition, we show that the computational overhead of the learning algorithm can be reduced drastically by taking advantage of the exponential decay of the gradient.

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