An automatic estimation of the ridge parameter for extreme learning machine

Extreme learning machine (ELM) is an emerging learning method with a single-hidden layer feed-forward neural network that involves obtaining a solution to the system of linear equations. Unlike traditional gradient-based back-propagating neural networks, ELM is computationally efficient with fast training speed and good generalization capability. However, most of the time when applied to real-time problems, the linear system becomes ill-posed in the structure and needs the inclusion of a ridge parameter to obtain a reliable solution, and hence, the selection of the ridge parameter ( C ) is a crucial task. The ridge parameter is chosen heuristically from a predefined set. The generalized cross-validation is a widely used technique for the automatic estimation of the same, which is computationally expensive as it involves inversion of large matrices. The focus of the proposed work is on pragmatic aspects of the time-efficient automatic estimation of ridge parameter that result in a better generalization performance. In this work, methods are proposed that use the L-curve and U-curve techniques to automatically estimate the ridge parameter, and these methods are effective in the estimation of the ridge parameter even for systems with larger data. Through extensive numerical results, it is shown that the proposed methods outperform the existing ones in terms of accuracy, precision, sensitivity, specificity, F 1-score, and computational time on various benchmark binary as well as multiclass classification data sets. Finally, the proposed methods are statistically analyzed using the nonparametric Friedman ranking test, which is also proving the effectiveness of the proposed method as it is providing a better rank for the same over existing methods.