Extreme learning machine (ELM) has attracted widespread attention due to its simple, quick and good performance. In this work, in order to deal with cross data quickly and efficiently, we first propose generalized eigenvalue proximal extreme learning machine (GEPELM). It takes the form of ratio into consideration to seek for two non-parallel separating hyperplanes in ELM feature space, each of which is close to the samples of its own class and far away from the others simultaneously. Then generalized eigenvalue algorithm is adopted to solve, which incurs GEPELM to enjoy faster calculation speed than TELM. Further, improved generalized eigenvalue proximal extreme learning machine (IGEPELM) is proposed, which uses minus instead of ratio to avoid singular value phenomenon and further mitigate the computational burden by solving two standard eigenvalue problems. To further improve classification performance, generalized eigenvalue proximal extreme learning machine based on inter-class graph (GGEPELM) is proposed, which incorporates the geometric structure information of dissimilar samples into the guideline of GEPELM. In addition, the proposed classifiers are all extended to kernel ELM framework to handle non-linear data more precisely. Moreover, Sherman-Morrison-Woodbury formula is utilized to reduce time complexity of matrix inversion. Simultaneously, a quick solution strategy is incorporated into GEPELMs and GGEPELMs to mitigate the burden of solving large-scale problems. The numerical simulations are carried out on three databases including a benchmark database, an artificial database and a practical application database, which demonstrates the proposed classifiers enjoy high computational speed, good generalization performance and insensitivity to parameters.
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