Readouts for Echo-state Networks Built using Locally Regularized Orthogonal Forward Regression

Echo state network (ESN) is viewed as a temporal non-orthogonal expansion with pseudo-random parameters. Such expansions naturally give rise to regressors of various relevance to a teacher output. We illustrate that often only a certain amount of the generated echo-regressors effectively explain the variance of the teacher output and also that sole local regularization is not able to provide in-depth information concerning the importance of the generated regressors. The importance is therefore determined by a joint calculation of the individual variance contributions and Bayesian relevance using locally regularized orthogonal forward regression (LROFR) algorithm. This information can be advantageously used in a variety of ways for an in-depth analysis of an ESN structure and its state-space parameters in relation to the unknown dynamics of the underlying problem. We present locally regularized linear readout built using LROFR. The readout may have a different dimensionality than an ESN model itself, and besides improving robustness and accuracy of an ESN it relates the echo-regressors to different features of the training data and may determine what type of an additional readout is suitable for a task at hand. Moreover, as flexibility of the linear readout has limitations and might sometimes be insufficient for certain tasks, we also present a radial basis function (RBF) readout built using LROFR. It is a flexible and parsimonious readout with excellent generalization abilities and is a viable alternative to readouts based on a feed-forward neural network (FFNN) or an RBF net built using relevance vector machine (R VM). Introduction ESNs are a novel class of recurrent neural networks (RNN) [1]. Their easy construction and simple training procedure are appealing and have attracted the attention of many researchers. Vector function f is applied element-wise to its arguments. The most common choice forf is either a vector of sigmoid or identity functions. The expansion is carried out so that diverse echoes of an input and teacher signal are generated (hence the name echo-state). This diversity, which should appropriately "explain" a variance of a teacher signal, is the key to the successful training of an ESN.