Learning with Tensor Representation

Most of the existing learning algorithms take vectors as their input data. A function is then learned in such a vector space for classification, clustering, or dimensionality reduction. However, in some situations, there is reason to consider data as tensors. For example, an image can be considered as a second order tensor and a video can be considered as a third order tensor. In this paper, we propose two novel algorithms called Support Tensor Machines (STM) and Tensor Least Square (TLS). These two algorithms operate in the tesnor space. Specifically, we represent data as the second order tensors (or, matrices) in Rn1 ⊗Rn2 , where Rn1 and Rn2 are two vector spaces. STM aims at finding a maximum margin classifier in the tensor space, while TLS aims at finding a minimum residual sum-of-squares classifier. With tensor representation, the number of parameters estimated by STM (TLS) can be greatly reduced. Therefore, our algorithms are especially suitable for small sample cases. We compare our proposed algorithms with SVM and the ordinary Least Square method on six databases. Experimental results show the effectiveness of our algorithms.