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Abstract: Extreme learning machine (ELM) is one of the most popular and important learning algorithms. It comes from single-hidden-layer feedforward neural networks. It has been proved that ELM can achieve better performance than support vector machine (SVM) in regression and classification. In this paper, mathematically, with regression problem, the step 3 of ELM is studied. First of all, the equation are reformulated as an optimal model. With the optimality, the necessary conditions of optimal solution are presented. The equation is replaced by . We can prove that the latter must have one solution at least. Second, optimal approximation solution is discussed in cases of is column full rank, row full rank, neither column nor row full rank. In the last case, the rank-1 and rank-2 methods are used to get optimal approximation solution. In theory, this paper present a better algorithm for ELM. [Copyright &y& Elsevier]

Abstract: Recently, graph-based spectral clustering algorithms have been developing rapidly, which are proposed as discrete combinatorial optimization problems and approximately solved by relaxing them into tractable eigenvalue decomposition problems. In this paper, we first review the current existing spectral clustering algorithms in a unified-framework way and give a straightforward explanation about spectral clustering. We also present a novel model for generalizing the unsupervised spectral clustering to semi-supervised spectral clustering. Under this model, prior information given by some instance-level constraints can be generalized to space-level constraints. We find that (undirected) graph built on the enlarged prior information is more meaningful, hence the boundaries of the clusters are more correct. Experimental results based on toy data, real-world data and image segmentation demonstrate the advantages of the proposed model. [Copyright &y& Elsevier]

Abstract: Geometric methods provide an intuitive and theoretically solid viewpoint for the solution of many optimization problems in the fields of pattern recognition and machine learning. The support vector machine (SVM) classification is a typical optimization task that has achieved excellent generalization performance in a wide variety of applications. In this paper, the notion of “scaled convex hull” (SCH) is presented, through which the nonseparable SVM classifications can be approximately transformed to separable ones: by a suitable selection of the reduction factor, the initially overlapping SCHs (each is generated by the training patterns of each class) can be reduced to become separable, then the maximal margin classifier between them can be trained, which is an approximation of the standard nonseparable SVM. As a practical application of the SCH framework, the popular Gilbert''s algorithm has been generalized to approximately solve general (linear and nonlinear, separable and nonseparable) SVM classification problems both accurately and efficiently. The experiments show that the proposed method may achieve better performance than the state-of-the-art methods, an improved sequential minimal optimization and Gilbert''s algorithm based on the reduced convex hull (RCH), in terms of the number of kernel evaluations and the execution time. [Copyright &y& Elsevier]