Extreme learning machine (ELM) is an emerging machine learning technique for training single hidden layer feedforward networks (SLFNs). During the training phase, ELM model can be created by simultaneously minimizing the modeling errors and norm of the output weights. Usually, squared loss is widely utilized in the objective function of ELMs, which is theoretically optimal for the Gaussian error distribution. However, in practice, data collected from uncertain and heterogeneous environments trivially result in unknown noise, which may be very complex and cannot be described well using any single distribution. In order to tackle this issue, in this paper, a robust ELM (R-ELM) is proposed for improving the modeling capability and robustness with Gaussian and non-Gaussian noise. In R-ELM, a modified objective function is constructed to fit the noise using mixture of Gaussian (MoG) to approximate any continuous distribution. In addition, the corresponding solution for the new objective function is developed based on expectation maximization (EM) algorithm. Comprehensive experiments, both on selected benchmark datasets and real world applications, demonstrate that the proposed R-ELM has better robustness and generalization performance than state-of-the-art machine learning approaches.

极限学习机（ELM）是一种新兴的机器学习技术，用于训练单隐藏层前馈网络（SLFN）。在训练阶段，可以通过同时最小化建模误差和输出权重范数来创建ELM模型。通常，平方损耗被广泛用于ELM的目标函数中，这在理论上对于高斯误差分布是最佳的。但是，实际上，从不确定和异构环境中收集的数据会微不足道地导致未知噪声，这可能非常复杂，无法使用任何单一分布很好地描述。为了解决这个问题，在本文中，提出了一种鲁棒的ELM（R-ELM），以提高建模能力和鲁棒的高斯和非高斯噪声。在R-ELM中，使用高斯混合（MoG）构造近似的目标函数来拟合噪声，以近似任何连续分布。此外，基于期望最大化（EM）算法，为新的目标函数开发了相应的解决方案。在选定的基准数据集和实际应用中进行的综合实验表明，与最新的机器学习方法相比，所提出的R-ELM具有更好的鲁棒性和泛化性能。