Learning a tensor subspace for semi-supervised dimensionality reduction

The high-dimensional data is frequently encountered and processed in real-world applications and unlabeled samples are readily available, but labeled or pairwise constrained ones are fairly expensive to capture. Traditionally, when a pattern itself is an n1 × n2 image, the image first has to be vectorized to the vector pattern in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Re^{{n_{1} \times n_{2} }} $$\end{document} by concatenating its pixels. However, such a vector representation fails to take into account the spatial locality of pixels in the images, which are intrinsically matrices. In this paper, we propose a tensor subspace learning-based semi-supervised dimensionality reduction algorithm (TS2DR), in which an image is naturally represented as a second-order tensor in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Re^{{n_{1} }} \otimes \Re^{{n_{2} }} $$\end{document} and domain knowledge in the forms of pairwise similarity and dissimilarity constraints is used to specify whether pairs of instances belong to the same class or different classes. TS2DR has an analytic form of the global structure preserving embedding transformation, which can be easily computed based on eigen-decomposition. We also verify the efficiency of TS2DR by conducting unbalanced data classification experiments based on the benchmark real-word databases. Numerical results show that TS2DR tends to capture the intrinsic structure characteristics of the given data and achieves better classification accuracy, while being much more efficient.