Multimodal sparse support tensor machine for multiple classification learning

This paper develops new multiple classification approaches in a direct manner for high-dimensional multimodal data. Firstly, we construct multiple hyperplanes via the tensor multimodal product and design a novel piecewise quadratic loss function ‘ℓcC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{cC}$$\end{document}’ in the soft-margin scheme to propose the multimodal support tensor machine model (MSTM). Furthermore, to alleviate the overfitting phenomenon in small-size sampling instances, we construct a multimodal sparsity constrained support tensor machine model (MSSTM) by subtly imposing the sparsity constraint on the decision variables of the dual problem. In addition, the Newton method and subspace Newton method are employed to solve the MSTM and MSSTM models from the dual perspective, taking advantage of the differentiation properties of ‘ℓcC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{cC}$$\end{document}’ and the hard-thresholding operator, respectively. Numerical experiments on four image datasets demonstrate the efficiency of the proposed methods in terms of classification accuracy and training time for multiple classification.