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This paper proposes several definitions of measures of complexity of a recurrent neural network. They measure 1) recurrent depth (degree of multilayeredness as a function of time of recursive connections) 2) feedforward depth (degree of multilayeredness as a function of input > output connections) 3) recurrent skip coefficient (degree of directness, like the inverse of multilayeredness, of connections) In addition to the actual definitions, there are two main contributions:  The authors show that the measures (which are limits as the number of time steps > infinity) are well defined.  The authors correlate the measures with empirical performance in various ways, showing that all measure of depth can lead to improved performance. This paper provides 3 measures of complexity for RNNs. They then show experimentally that these complexity measures are meaningful, in the sense that increasingly complexity seems to correlated with better performance. The authors first present a rigorous graphtheoretic framework that describes the connecting architectures of RNNs in general, with which the authors easily explain how we can unfold an RNN. The authors then go on and propose tree architecture complexity measures of RNNs, namely the recurrent depth, the feedforward depth and the recurrent skip coefficient. Experiments on various tasks show the importance of certain measures on certain tasks, which indicates that those three complexity measures might be good guidelines when designing a recurrent neural network for certain tasks.
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