As a fast algorithm for training single-hidden layer feedforward neural networks, extreme learning machine (ELM) has been successfully applied to various classification and regression problems. In recent years, regularization techniques have been widely used in ELM to improve its stability, sparsity and generalization capability. In order to determine the appropriate number of hidden layer nodes, the ELM regularized by l1/2\documentclass[12pt]{minimal}
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\begin{document}$$l _{\,\!1/2 }$$\end{document} quasi-norm (L1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{1/2 }$$\end{document}-ELM) was developed to prune the redundant hidden nodes. However, in multi-dimensional output tasks, L1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{1/2 }$$\end{document}-ELM only removes redundant weights of hidden nodes but cannot guarantee the sparsity at the node level. In this paper, we present the L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM, which is regularized by L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$L _{2,1/2 }$$\end{document} quasi-norm to achieve sparsity in multi-dimensional output problems. With the generalization of L1/2\documentclass[12pt]{minimal}
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\begin{document}$$L _{1/2 }$$\end{document} regularization to L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$L _{2,1/2 }$$\end{document} regularization, L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM can prune the corresponding hidden nodes by setting some rows of the output weight matrix to zero. Since the proximal operator corresponding to L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$L _{2,1/2 }$$\end{document} regularization has a closed-form solution, the powerful alternating direction method of multipliers (ADMM) is employed to achieve a fast solution of L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM. Furthermore, to face the challenge of distributed computing, we extend L2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM to its distributed version, namely DL2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM. DL2,1/2\documentclass[12pt]{minimal}
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\begin{document}$$_{2,1/2 }$$\end{document}-ELM is solved by the consensus ADMM algorithm. Experiments on multi-classification and multi-target regression datasets demonstrate that our proposed algorithms can achieve competitive sparsity without compromising accuracy.
