Pruning Multilayered ELM Using Cholesky Factorization and Givens Rotation Transformation

Abstract

Extreme learning machine is originally proposed for the learning of the single hidden layer feedforward neural network to overcome the challenges faced by the backpropagation (BP) learning algorithm and its variants. Recent studies show that ELM can be extended to the multilayered feedforward neural network in which the hidden node could be a subnetwork of nodes or a combination of other hidden nodes. Although the ELM algorithm with multiple hidden layers shows stronger nonlinear expression ability and stability in both theoretical and experimental results than the ELM algorithm with the single hidden layer, with the deepening of the network structure, the problem of parameter optimization is also highlighted, which usually requires more time for model selection and increases the computational complexity. This paper uses Cholesky factorization strategy and Givens rotation transformation to choose the hidden nodes of MELM and obtains the number of nodes more suitable for the network. First, the initial network has a large number of hidden nodes and then uses the idea of ridge regression to prune the nodes. Finally, a complete neural network can be obtained. Therefore, the ELM algorithm eliminates the need to manually set nodes and achieves complete automation. By using information from the previous generation’s connection weight matrix, it can be evitable to re-calculate the weight matrix in the network simplification process. As in the matrix factorization methods, the Cholesky factorization factor is calculated by Givens rotation transform to achieve the fast decreasing update of the current connection weight matrix, thus ensuring the numerical stability and high efficiency of the pruning process. Empirical studies on several commonly used classification benchmark problems and the real datasets collected from coal industry show that compared with the traditional ELM algorithm, the pruning multilayered ELM algorithm proposed in this paper can find the optimal number of hidden nodes automatically and has better generalization performance.