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 n ₁ × n ₂ image, the image first has to be vectorized to the vector pattern in \( \Re^{{n_{1} \times n_{2} }} \) 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 \( \Re^{{n_{1} }} \otimes \Re^{{n_{2} }} \) 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.