Abstract-Due to the simplicity of their implementations, least square support vector machine (LS-SVM) and proximal support vector machine (PSVM) have been widely used in binary classification applications. The conventional LS-SVM and PSVM cannot be used in regression and multiclass classification applications directly, although variants of LS-SVM and PSVM have been proposed to handle such cases. This paper shows that both LS-SVM and PSVM can be simplified further and a unified learning framework of LS-SVM, PSVM, and other regularization algorithms referred to extreme learning machine (ELM) can be built. ELM works for the "generalized" single-hidden-layer feedforward networks (SLFNs), but the hidden layer (or called feature mapping) in ELM need not be tuned. Such SLFNs include but are not limited to SVM, polynomial network, and the conventional feedforward neural networks. This paper shows the following: 1) ELM provides a unified learning platform with a widespread type of feature mappings and can be applied in regression and multiclass classification applications directly; 2) from the optimization method point of view, ELM has milder optimization constraints compared to LS-SVM and PSVM; 3) in theory, compared to ELM, LS-SVM and PSVM achieve suboptimal solutions and require higher computational complexity; and 4) in theory, ELM can approximate any target continuous function and classify any disjoint regions. As verified by the simulation results, ELM tends to have better scalability and achieve similar (for regression and binary class cases) or much better (for multiclass cases) generalization performance at much faster learning speed (up to thousands times) than traditional SVM and LS-SVM.Index Terms-Extreme learning machine (ELM), feature mapping, kernel, least square support vector machine (LS-SVM), proximal support vector machine (PSVM), regularization network.

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This special issue includes eight original works that detail the further developments of ELMs in theories, applications, and hardware implementation. In "Representational Learning with ELMs for Big Data," Liyanaarachchi Lekamalage Chamara Kasun, Hongming Zhou, Guang-Bin Huang, and Chi Man Vong propose using the ELM as an auto-encoder for learning feature representations using singular values. In "A Secure and Practical Mechanism for Outsourcing ELMs in Cloud Computing," Jiarun Lin, Jianping Yin, Zhiping Cai, Qiang Liu, Kuan Li, and Victor C.M. Leung propose a method for handling large data applications by outsourcing to the cloud that would dramatically reduce ELM training time. In "ELM-Guided Memetic Computation for Vehicle Routing," Liang Feng, Yew-Soon Ong, and Meng-Hiot Lim consider the ELM as an engine for automating the encapsulation of knowledge memes from past problem-solving experiences. In "ELMVIS: A Nonlinear Visualization Technique Using Random Permutations and ELMs," Anton Akusok, Amaury Lendasse, Rui Nian, and Yoan Miche propose an ELM method for data visualization based on random permutations to map original data and their corresponding visualization points. In "Combining ELMs with Random Projections," Paolo Gastaldo, Rodolfo Zunino, Erik Cambria, and Sergio Decherchi analyze the relationships between ELM feature-mapping schemas and the paradigm of random projections. In "Reduced ELMs for Causal Relation Extraction from Unstructured Text," Xuefeng Yang and Kezhi Mao propose combining ELMs with neuron selection to optimize the neural network architecture and improve the ELM ensemble's computational efficiency. In "A System for Signature Verification Based on Horizontal and Vertical Components in Hand Gestures," Beom-Seok Oh, Jehyoung Jeon, Kar-Ann Toh, Andrew Beng Jin Teoh, and Jaihie Kim propose a novel paradigm for hand signature biometry- for touchless applications without the need for handheld devices. Finally, in "An Adaptive and Iterative Online Sequential ELM-Based Multi-Degree-of-Freedom Gesture Recognition System," Hanchao Yu, Yiqiang Chen, Junfa Liu, and Guang-Bin Huang propose an online sequential ELM-based efficient gesture recognition algorithm for touchless human-machine interaction

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The increasing demands of location-based services have spurred the rapid development of indoor positioning system and indoor localization system interchangeably (IPSs). However, the performance of IPSs suffers from noisy measurements. In this paper, two kinds of robust extreme learning machines (RELMs), corresponding to the close-to-mean constraint, and the small-residual constraint, have been proposed to address the issue of noisy measurements in IPSs. Based on whether the feature mapping in extreme learning machine is explicit, we respectively provide random-hidden-nodes and kernelized formulations of RELMs by second order cone programming. Furthermore, the computation of the covariance in feature space is discussed. Simulations and real-world indoor localization experiments are extensively carried out and the results demonstrate that the proposed algorithms can not only improve the accuracy and repeatability, but also reduce the deviation and worst case error of IPSs compared with other baseline algorithms.