Piecewise Multi-linear Support Tensor Machine

Support vector machine is a new learning method developed in recent years based on the foundations of Statistical Learning Theory(SLT). It is gaining popularity due to many attractive features and promising empirical performance in the fields of small-sample statistics, nonlinear and high dimensional pattern recognition. Because of their excellent learning performance, they have successful applications in many fields, such as: face detection, handwriting digit recognition, text auto-categorization, etc.But SVM also have many shortcomings that need to be researched. The model of linear support vector machine is sample and has a high training efficiency. But it cannot deal with the problem that the training data are impossible to be separated by a linear classifier. The kernel support vector machine can handle the non-separable data set effectively. But the use of kernel function also makes the model complexity and reduces the efficiency of training. Besides, stretching the tensor to vector will destroy the structure of tensor and leads to the increase of vector dimension.In order to solve these problems, piecewise multi-linear support tensor machine(PML-STM) has been proposed to deal with tensor data in this paper, which is an extension of piecewise multi-linear support vector machine. In PML-STM, the positive(negative) training samples is divided into many subsets that each subset is linear separable to the negative training samples. For each subset of the positive training samples, a linear support tensor machine is trained. By doing so, the model complexity is reduced and the efficiency of training is improved. The piecewise multi-linear support tensor machine constructed by a number of linear support tensor machines is a nonlinear classifier and can handle the non-separable data set effectively. Besides, rank-one decomposition is introduced to efficiently solve PML-STM, which greatly reduces the complexity of the space and time.Experiments are conducted on three datasets, such as Yale-B、ORL and CMU PIE. PML-STM is compared with STM and SMA. Experimental results show that PML-STM can effectively improve training efficiency. Besides, the classification accuracy of the algorithm is stable when compared with SMA.