Abstract
Since labeling all the samples by the user is time-consuming and fastidious, we often obtain a large amount of unlabeled examples and only a small number of labeled examples in classification. In this context, the classification is called semi-supervised learning. In this paper, we propose a novel semi-supervised learning methodology, named Laplacian mixed-norm proximal support vector machine Lap-MNPSVM for short. In the optimization problem of Lap-MNPSVM, the information from the unlabeled examples is used in a form of Laplace regularization, and \(l_p\) norm (\(0\,<\,p\,<\,1\)) regularizer is introduced to standard proximal support vector machine to control sparsity and the feature selection. To solve the nonconvex optimization problem in Lap-MNPSVM, an efficient algorithm is proposed by solving a series systems of linear equations, and the lower bounds of the solution are established, which are extremely helpful for feature selection. Experiments carried out on synthetic datasets and the real-world datasets show the feasibility and effectiveness of the proposed method.
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\(\Vert x\Vert _p (0 < p < 1)\) is a quasi-norm, which satisfies the norm axioms except the triangle inequality.
Sparsity is here defined as the number of nonzero components in the normal vector \(w\). This means that more zero components in \(w\), more sparse the hyperplane.
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This paper was supported by National Natural Science Foundation of China (No. 11301535, No. 11371365).
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Zhang, Z., Zhen, L., Deng, N. et al. Manifold proximal support vector machine with mixed-norm for semi-supervised classification. Neural Comput & Applic 26, 399–407 (2015). https://doi.org/10.1007/s00521-014-1728-4
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DOI: https://doi.org/10.1007/s00521-014-1728-4