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Extreme learning machine (ELM) is a popular learning algorithm for single hidden layer feedforward networks (SLFNs). It was originally proposed with the inspiration from biological learning and has attracted massive attentions due to its adaptability to various tasks with a fast learning ability and efficient computation cost. As an effective sparse representation method, orthogonal matching pursuit (OMP) method can be embedded into ELM to overcome the singularity problem and improve the stability. Usually OMP recovers a sparse vector by minimizing a least squares (LS) loss, which is efficient for Gaussian distributed data, but may suffer performance deterioration in presence of non-Gaussian data. To address this problem, a robust matching pursuit method based on a novel kernel risk-sensitive loss (in short KRSLMP) is first proposed in this paper. The KRSLMP is then applied to ELM to solve the sparse output weight vector, and the new method named the KRSLMP-ELM is developed for SLFN learning. Experimental results on synthetic and real-world data sets confirm the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]