  Extreme learning machine (ELM) is a network model that arbitrarily
initializes the first hidden layer and can be computed speedily. In order to
improve the classification performance of ELM, a $\ell_2$ and $\ell_{0.5}$
regularization ELM model ($\ell_{2}$-$\ell_{0.5}$-ELM) is proposed in this
paper. An iterative optimization algorithm of the fixed point contraction
mapping is applied to solve the $\ell_{2}$-$\ell_{0.5}$-ELM model. The
convergence and sparsity of the proposed method are discussed and analyzed
under reasonable assumptions. The performance of the proposed
$\ell_{2}$-$\ell_{0.5}$-ELM method is compared with BP, SVM, ELM,
$\ell_{0.5}$-ELM, $\ell_{1}$-ELM, $\ell_{2}$-ELM and $\ell_{2}$-$\ell_{1}$ELM,
the results show that the prediction accuracy, sparsity, and stability of the
$\ell_{2}$-$\ell_{0.5}$-ELM are better than the other $7$ models.
