  Single-hidden layer feed forward neural networks (SLFNs) are widely used in
pattern classification problems, but a huge bottleneck encountered is the slow
speed and poor performance of the traditional iterative gradient-based learning
algorithms. Although the famous extreme learning machine (ELM) has successfully
addressed the problems of slow convergence, it still has computational
robustness problems brought by input weights and biases randomly assigned.
Thus, in order to overcome the aforementioned problems, in this paper, a novel
type neural network based on Gegenbauer orthogonal polynomials, termed as GNN,
is constructed and investigated. This model could overcome the computational
robustness problems of ELM, while still has comparable structural simplicity
and approximation capability. Based on this, we propose a regularized weights
direct determination (R-WDD) based on equality-constrained optimization to
determine the optimal output weights. The R-WDD tends to minimize the empirical
risks and structural risks of the network, thus to lower the risk of over
fitting and improve the generalization ability. This leads us to a the final
GNN with R-WDD, which is a unified learning mechanism for binary and
multi-class classification problems. Finally, as is verified in the various
comparison experiments, GNN with R-WDD tends to have comparable (or even
better) generalization performances, computational scalability and efficiency,
and classification robustness, compared to least square support vector machine
(LS-SVM), ELM with Gaussian kernel.
